Use the Rectangular Coordinate System

Plot Points on a Rectangular Coordinate System

Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. The rectangular coordinate system is also called the xy-plane or the ‘coordinate plane’.

The horizontal number line is called the x-axis. The vertical number line is called the y-axis. The x-axis and the y-axis together form the rectangular coordinate system. These axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise.

In the rectangular coordinate system, every point is represented by an ordered pair. The first number in the ordered pair is the x-coordinate of the point, and the second number is the y-coordinate of the point.

The phrase ‘ordered pair’ means the order is important. What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is \((0,0)\). The point \((0,0)\) has a special name. It is called the origin.

We use the coordinates to locate a point on the xy-plane. Let’s plot the point \((1,3)\) as an example. First, locate 1 on the x-axis and lightly sketch a vertical line through \(x=1\). Then, locate 3 on the y-axis and sketch a horizontal line through \(y=3\). Now, find the point where these two lines meet, that is the point with coordinates \((1,3)\).

Notice that the vertical line through \(x=1\) and the horizontal line through \(y=3\) are not part of the graph. We just used them to help us locate the point \((1,3)\).

How do the signs affect the location of the points? You may have noticed some patterns as you graphed the points in the previous example.

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Verify Solutions to an Equation in Two Variables

Up to now, all the equations you have solved were equations with just one variable. In almost every case, when you solved the equation you got exactly one solution. The process of solving an equation ended with a statement like \(x=4\). (Then, you checked the solution by substituting back into the equation.)

Here’s an example of an equation in one variable, and its one solution.

\[\begin{array}{lll}3x+5 & = & 17 \\ 3x & = & 12 \\ x & = & 4\end{array}\]

But equations can have more than one variable. Equations with two variables may be of the form \(Ax+By=C\). Equations of this form are called linear equations in two variables.

Notice the word line in linear. Here is an example of a linear equation in two variables, \(x\) and \(y\).

The equation \(y=-3x+5\) is also a linear equation. But it does not appear to be in the form \(Ax+By=C\). We can use the Addition Property of Equality and rewrite it in \(Ax+By=C\) form.

	\(\ y\ =\ -3x+5\)
Add to both sides.	\(y+3x\ =\ -3x+5+3x\)
Simplify.	\(y+3x\ =\ 5\)
Use the Commutative Property to put it in \(Ax+By=C\) form.	\(3x+y\ =\ 5\)

By rewriting \(y=-3x+5\) as \(3x+y=5\), we can easily see that it is a linear equation in two variables because it is of the form \(Ax+By=C\). When an equation is in the form \(Ax+By=C\), we say it is in standard form.

Most people prefer to have \(A\), \(B\), and \(C\) be integers and \(A\ge 0\) when writing a linear equation in standard form, although it is not strictly necessary.

Example

Try it.

Determine which ordered pairs are solutions to the equation \(x+4y=8\).

ⓐ \((0,2)\) ⓑ \((2,-4)\) ⓒ \((-4,3)\)

Solution

Substitute the x- and y-values from each ordered pair into the equation and determine if the result is a true statement.

Example

Try it.

Which of the following ordered pairs are solutions to the equation \(y=5x-1\)?

ⓐ \((0,-1)\) ⓑ \((1,4)\) ⓒ \((-2,-7)\)

Solution

Substitute the x- and y-values from each ordered pair into the equation and determine if it results in a true statement.

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Complete a Table of Solutions to a Linear Equation in Two Variables

In the examples above, we substituted the x- and y-values of a given ordered pair to determine whether or not it was a solution to a linear equation. But how do you find the ordered pairs if they are not given? It’s easier than you might think, you can just pick a value for \(x\) and then solve the equation for \(y\). Or, pick a value for \(y\) and then solve for \(x\).

We’ll start by looking at the solutions to the equation \(y=5x-1\) that we found in . We can summarize this information in a table of solutions,

\(y=5x-1\)
\(x\)	\(y\)	\((x,y)\)
0	\(-1\)	\((0,-1)\)
1	4	\((1,4)\)

To find a third solution, we’ll let \(x=2\) and solve for \(y\).

The ordered pair \((2,9)\) is a solution to \(y=5x-1\). We will add it to .

\(y=5x-1\)
\(x\)	\(y\)	\((x,y)\)
0	\(-1\)	\((0,-1)\)
1	4	\((1,4)\)
2	9	\((2,9)\)

We can find more solutions to the equation by substituting in any value of \(x\) or any value of \(y\) and solving the resulting equation to get another ordered pair that is a solution. There are infinitely many solutions of this equation.

Example

Try it.

Complete to find three solutions to the equation \(y=4x-2\).

\(y=4x-2\)
\(x\)	\(y\)	\((x,y)\)
0
\(-1\)
2

Solution

Substitute \(x=0\), \(x=-1\), and \(x=2\) into \(y=4x-2\).

The results are summarized in .

\(y=4x-2\)
\(x\)	\(y\)	\((x,y)\)
0	\(-2\)	\((0,-2)\)
\(-1\)	\(-6\)	\((-1,-6)\)
2	6	\((2,6)\)

Example

Try it.

Complete to find three solutions to the equation \(5x-4y=20\).

\(5x-4y=20\)
\(x\)	\(y\)	\((x,y)\)
0
	0
	5

Solution

Substitute the given value into the equation \(5x-4y=20\) and solve for the other variable. Then, fill in the values in the table.

The results are summarized in .

\(5x-4y=20\)
\(x\)	\(y\)	\((x,y)\)
0	\(-5\)	\((0,-5)\)
4	0	\((4,0)\)
8	5	\((8,5)\)

Find Solutions to a Linear Equation

To find a solution to a linear equation, you really can pick any number you want to substitute into the equation for \(x\) or \(y.\) But since you’ll need to use that number to solve for the other variable it’s a good idea to choose a number that’s easy to work with.

When the equation is in y-form, with the y by itself on one side of the equation, it is usually easier to choose values of \(x\) and then solve for \(y\).

Example

Try it.

Find three solutions to the equation \(y=-3x+2\).

Solution

We can substitute any value we want for \(x\) or any value for \(y\). Since the equation is in y-form, it will be easier to substitute in values of \(x\). Let’s pick \(x=0\), \(x=1\), and \(x=-1\).

Substitute the value into the equation.
Simplify.
Simplify.
Write the ordered pair.	(0, 2)	(1, −1)	(−1, 5)
Check.
\(\ y=-3x+2\)	\(\ y=-3x+2\)	\(\ y=-3x+2\)
\(2≟-3⋅0+2\)	\(-1≟-3⋅1+2\)	\(5≟-3(-1)+2\)
\(2≟0+2\)	\(-1≟-3+2\)	\(5≟3+2\)
\(2=2✓\)	\(-1=-1✓\)	\(5=5✓\)

So, \((0,2)\), \((1,-1)\) and \((-1,5)\) are all solutions to \(y=-3x+2\). We show them in .

\(y=-3x+2\)
\(x\)	\(y\)	\((x,y)\)
0	2	\((0,2)\)
1	\(-1\)	\((1,-1)\)
\(-1\)	5	\((-1,5)\)

We have seen how using zero as one value of \(x\) makes finding the value of \(y\) easy. When an equation is in standard form, with both the \(x\) and \(y\) on the same side of the equation, it is usually easier to first find one solution when \(x=0\) find a second solution when \(y=0\), and then find a third solution.

Example

Try it.

Find three solutions to the equation \(3x+2y=6\).

Solution

We can substitute any value we want for \(x\) or any value for \(y\). Since the equation is in standard form, let’s pick first \(x=0\), then \(y=0\), and then find a third point.

Substitute the value into the equation.
Simplify.
Solve.
Write the ordered pair.	(0, 3)	(2, 0)	\((1,\frac{3}{2})\)

Check.
\(3x+2y=6\)	\(3x+2y=6\)	\(3x+2y=6\)
\(3⋅0+2⋅3≟6\)	\(3⋅2+2⋅0≟6\)	\(3⋅1+2⋅\frac{3}{2}≟6\)
\(0+6≟6\)	\(6+0≟6\)	\(3+3≟6\)
\(6=6✓\)	\(6=6✓\)	\(6=6✓\)

So \((0,3)\), \((2,0)\), and \((1,\frac{3}{2})\) are all solutions to the equation \(3x+2y=6\). We can list these three solutions in .

\(3x+2y=6\)
\(x\)	\(y\)	\((x,y)\)
0	3	\((0,3)\)
2	0	\((2,0)\)
1	\(\frac{3}{2}\)	\((1,\frac{3}{2})\)

Key Concepts

• Sign Patterns of the Quadrants
  \(\begin{array}{llllllllll}\text{Quadrant I} & & & \text{Quadrant II} & & & \text{Quadrant III} & & & \text{Quadrant IV} \\ (x,y) & & & (x,y) & & & (x,y) & & & (x,y) \\ (+,+) & & & (\text{-},+) & & & (\text{-},\text{-}) & & & (+,\text{-})\end{array}\)
• Points on the Axes
  
  • On the x-axis, \(y=0\). Points with a y-coordinate equal to 0 are on the x-axis, and have coordinates \((a,0)\).
  • On the y-axis, \(x=0\). Points with an x-coordinate equal to 0 are on the y-axis, and have coordinates \((0,b).\)
• Solution of a Linear Equation
  
  • An ordered pair \((x,y)\) is a solution of the linear equation \(Ax+By=C\), if the equation is a true statement when the x- and y- values of the ordered pair are substituted into the equation.

Use the Rectangular Coordinate System

Plot Points in a Rectangular Coordinate System

In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.

Try it.

1. ⓐ \((-4,2)\)
2. ⓑ \((-1,-2)\)
3. ⓒ \((3,-5)\)
4. ⓓ \((-3,5)\)
   ⓔ \((\frac{5}{3},2)\)

Solution

Try it.

1. ⓐ \((-2,-3)\)
2. ⓑ \((3,-3)\)
3. ⓒ \((-4,1)\)
4. ⓓ \((4,-1)\)
5. ⓔ \((\frac{3}{2},1)\)

Try it.

1. ⓐ \((3,-1)\)
2. ⓑ \((-3,1)\)
3. ⓒ \((-2,2)\)
4. ⓓ \((-4,-3)\)
5. ⓔ \((1,\frac{14}{5})\)

Solution

Try it.

1. ⓐ \((-1,1)\)
2. ⓑ \((-2,-1)\)
3. ⓒ \((2,1)\)
4. ⓓ \((1,-4)\)
5. ⓔ \((3,\frac{7}{2})\)

In the following exercises, plot each point in a rectangular coordinate system.

Try it.

1. ⓐ \((-2,0)\)
2. ⓑ \((-3,0)\)
3. ⓒ \((0,0)\)
4. ⓓ \((0,4)\)
5. ⓔ \((0,2)\)

Solution

Try it.

1. ⓐ \((0,1)\)
2. ⓑ \((0,-4)\)
3. ⓒ \((-1,0)\)
4. ⓓ \((0,0)\)
5. ⓔ \((5,0)\)

Try it.

1. ⓐ \((0,0)\)
2. ⓑ \((0,-3)\)
3. ⓒ \((-4,0)\)
4. ⓓ \((1,0)\)
5. ⓔ \((0,-2)\)

Solution

Try it.

1. ⓐ \((-3,0)\)
2. ⓑ \((0,5)\)
3. ⓒ \((0,-2)\)
4. ⓓ \((2,0)\)
5. ⓔ \((0,0)\)

In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system.

Verify Solutions to an Equation in Two Variables

In the following exercises, which ordered pairs are solutions to the given equations?

Try it.

\(2x+y=6\)

1. ⓐ \((1,4)\)
2. ⓑ \((3,0)\)
3. ⓒ \((2,3)\)

Solution

a, b

Try it.

\(x+3y=9\)

1. ⓐ \((0,3)\)
2. ⓑ \((6,1)\)
3. ⓒ \((-3,-3)\)

Try it.

\(4x-2y=8\)

1. ⓐ \((3,2)\)
2. ⓑ \((1,4)\)
3. ⓒ \((0,-4)\)

Solution

a, c

Try it.

\(3x-2y=12\)

1. ⓐ \((4,0)\)
2. ⓑ \((2,-3)\)
3. ⓒ \((1,6)\)

Try it.

\(y=4x+3\)

1. ⓐ \((4,3)\)
2. ⓑ \((-1,-1)\)
3. ⓒ \((\frac{1}{2},5)\)

Solution

b, c

Try it.

\(y=2x-5\)

1. ⓐ \((0,-5)\)
2. ⓑ \((2,1)\)
3. ⓒ \((\frac{1}{2},-4)\)

Try it.

\(y=\frac{1}{2}x-1\)

1. ⓐ \((2,0)\)
2. ⓑ \((-6,-4)\)
3. ⓒ \((-4,-1)\)

Solution

a, b

Try it.

\(y=\frac{1}{3}x+1\)

1. ⓐ \((-3,0)\)
2. ⓑ \((9,4)\)
3. ⓒ \((-6,-1)\)

Complete a Table of Solutions to a Linear Equation

In the following exercises, complete the table to find solutions to each linear equation.

Try it.

\(y=2x-4\)

\(x\)	\(y\)	\((x,y)\)
0
2
\(-1\)

Solution

\(x\)	\(y\)	\((x,y)\)
0	\(-4\)	\((0,-4)\)
2	0	\((2,0)\)
\(-1\)	\(-6\)	\((-1,-6)\)

Try it.

\(y=3x-1\)

\(x\)	\(y\)	\((x,y)\)
0
2
\(-1\)

Try it.

\(y=\text{-}x+5\)

\(x\)	\(y\)	\((x,y)\)
0
3
\(-2\)

Solution

\(x\)	\(y\)	\((x,y)\)
0	5	\((0,5)\)
3	2	\((3,2)\)
\(-2\)	7	\((-2,7)\)

Try it.

\(y=\text{-}x+2\)

\(x\)	\(y\)	\((x,y)\)
0
3
\(-2\)

Try it.

\(y=\frac{1}{3}x+1\)

\(x\)	\(y\)	\((x,y)\)
0
3
6

Solution

\(x\)	\(y\)	\((x,y)\)
0	1	\((0,1)\)
3	2	\((3,2)\)
6	3	\((6,3)\)

Try it.

\(y=\frac{1}{2}x+4\)

\(x\)	\(y\)	\((x,y)\)
0
2
4

Try it.

\(y=-\frac{3}{2}x-2\)

\(x\)	\(y\)	\((x,y)\)
0
2
\(-2\)

Solution

\(x\)	\(y\)	\((x,y)\)
0	\(-2\)	\((0,-2)\)
2	\(-5\)	\((2,-5)\)
\(-2\)	1	\((-2,1)\)

Try it.

\(y=-\frac{2}{3}x-1\)

\(x\)	\(y\)	\((x,y)\)
0
3
\(-3\)

Try it.

\(x+3y=6\)

\(x\)	\(y\)	\((x,y)\)
0
3
	0

Solution

\(x\)	\(y\)	\((x,y)\)
0	2	\((0,2)\)
3	4	\((3,1)\)
6	0	\((6,0)\)

Try it.

\(x+2y=8\)

\(x\)	\(y\)	\((x,y)\)
0
4
	0

Try it.

\(2x-5y=10\)

\(x\)	\(y\)	\((x,y)\)
0
10
	0

Solution

\(x\)	\(y\)	\((x,y)\)
0	\(-2\)	\((0,-2)\)
10	2	\((10,2)\)
5	0	\((5,0)\)

Try it.

\(3x-4y=12\)

\(x\)	\(y\)	\((x,y)\)
0
8
	0

Find Solutions to a Linear Equation

In the following exercises, find three solutions to each linear equation.

Try it.

\(5x-2y=10\)

Solution

Answers will vary.

Try it.

\(4x-3y=12\)

Try it.

Explain in words how you plot the point \((4,-2)\) in a rectangular coordinate system.

Solution

Answers will vary.

Try it.

How do you determine if an ordered pair is a solution to a given equation?

Try it.

Is the point \((-3,0)\) on the x-axis or y-axis? How do you know?

Solution

Answers will vary.

Try it.

Is the point \((0,8)\) on the x-axis or y-axis? How do you know?

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Plot Points on a Rectangular Coordinate System

Many maps, such as the Campus Map shown in , use a grid system to identify locations. Do you see the numbers \(1,2,3,\) and \(4\) across the top and bottom of the map and the letters A, B, C, and D along the sides? Every location on the map can be identified by a number and a letter.

For example, the Student Center is in section 2B. It is located in the grid section above the number \(2\) and next to the letter B. In which grid section is the Stadium? The Stadium is in section 4D.

Example

Try it.

Use the map in .

1. ⓐ Find the grid section of the Residence Halls.
2. ⓑ What is located in grid section 4C?

Solution

1. ⓐ Read the number below the Residence Halls, \(4,\) and the letter to the side, A. So the Residence Halls are in grid section 4A.
2. ⓑ Find \(4\) across the bottom of the map and C along the side. Look below the \(4\) and next to the C. Tiger Field is in grid section 4C.

Just as maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. To create a rectangular coordinate system, start with a horizontal number line. Show both positive and negative numbers as you did before, using a convenient scale unit. This horizontal number line is called the x-axis.

Now, make a vertical number line passing through the \(x\text{-axis}\) at \(0.\) Put the positive numbers above \(0\) and the negative numbers below \(0.\) This vertical line is called the y-axis.

Vertical grid lines pass through the integers marked on the \(x\text{-axis}.\) Horizontal grid lines pass through the integers marked on the \(y\text{-axis}.\) The resulting grid is the rectangular coordinate system.

The rectangular coordinate system is also called the \(x\text{-}y\) plane, the coordinate plane, or the Cartesian coordinate system (since it was developed by a mathematician named René Descartes.)

The \(x\text{-axis}\) and the \(y\text{-axis}\) form the rectangular coordinate system. These axes divide a plane into four areas, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise.

Condensed: the full section is in OpenStax Prealgebra 2e.

Identify Points on a Graph

In algebra, being able to identify the coordinates of a point shown on a graph is just as important as being able to plot points. To identify the x-coordinate of a point on a graph, read the number on the x-axis directly above or below the point. To identify the y-coordinate of a point, read the number on the y-axis directly to the left or right of the point. Remember, to write the ordered pair using the correct order \((x,y).\)

Example

Try it.

Name the ordered pair of each point shown:

Solution

Point A is above \(-3\) on the \(x\text{-axis},\) so the \(x\text{-coordinate}\) of the point is \(-3.\) The point is to the left of \(3\) on the \(y\text{-axis},\) so the \(y\text{-coordinate}\) of the point is \(3.\) The coordinates of the point are \((-3,3).\)

Point B is below \(-1\) on the \(x\text{-axis},\) so the \(x\text{-coordinate}\) of the point is \(-1.\) The point is to the left of \(-3\) on the \(y\text{-axis},\) so the \(y\text{-coordinate}\) of the point is \(-3.\) The coordinates of the point are \((-1,-3).\)

Point C is above \(2\) on the \(x\text{-axis},\) so the \(x\text{-coordinate}\) of the point is \(2.\) The point is to the right of \(4\) on the \(y\text{-axis},\) so the \(y\text{-coordinate}\) of the point is \(4.\) The coordinates of the point are \((2,4).\)

Point D is below \(4\) on the \(x-\text{axis},\) so the \(x\text{-coordinate}\) of the point is \(4.\) The point is to the right of \(-4\) on the \(y\text{-axis},\) so the \(y\text{-coordinate}\) of the point is \(-4.\) The coordinates of the point are \((4,-4).\)

Example

Try it.

Name the ordered pair of each point shown:

Solution

Point A is on the x-axis at \(x=-4\).	The coordinates of point A are \((-4,0)\).
Point B is on the y-axis at \(y=-2\)	The coordinates of point B are \((0,-2)\).
Point C is on the x-axis at \(x=3\).	The coordinates of point C are \((3,0)\).
Point D is on the y-axis at \(y=1\).	The coordinates of point D are \((0,1)\).

Verify Solutions to an Equation in Two Variables

All the equations we solved so far have been equations with one variable. In almost every case, when we solved the equation we got exactly one solution. The process of solving an equation ended with a statement such as \(x=4.\) Then we checked the solution by substituting back into the equation.

Here’s an example of a linear equation in one variable, and its one solution.

\[\begin{array}{l}3x+5=17 \\ \\ 3x=12 \\ x=4\end{array}\]

But equations can have more than one variable. Equations with two variables can be written in the general form \(Ax+By=C.\) An equation of this form is called a linear equation in two variables.

Notice that the word “line” is in linear.

Here is an example of a linear equation in two variables, \(x\) and \(y\text{:}\)

Is \(y=-5x+1\) a linear equation? It does not appear to be in the form \(Ax+By=C.\) But we could rewrite it in this form.

Add \(5x\) to both sides.
Simplify.
Use the Commutative Property to put it in \(Ax+By=C.\)

By rewriting \(y=-5x+1\) as \(5x+y=1,\) we can see that it is a linear equation in two variables because it can be written in the form \(Ax+By=C.\)

Example

Try it.

Determine which ordered pairs are solutions of the equation \(x+4y=8\text{:}\)

1. ⓐ \(\ (0,2)\)
2. ⓑ \(\ (2,-4)\)
3. ⓒ \(\ (-4,3)\)

Solution

Substitute the \(x\text{- and}\ y\text{-values}\) from each ordered pair into the equation and determine if the result is a true statement.

ⓐ \(\ (0,2)\)	ⓑ \(\ (2,-4)\)	ⓒ \(\ (-4,3)\)
\((0,2)\) is a solution.	\((2,-4)\) is not a solution.	\((-4,3)\) is a solution.

Example

Try it.

Determine which ordered pairs are solutions of the equation. \(y=5x-1\text{:}\)

1. ⓐ \(\ (0,-1)\)
2. ⓑ \(\ (1,4)\)
3. ⓒ \(\ (-2,-7)\)

Solution

Substitute the \(x\text{-}\) and \(y\text{-values}\) from each ordered pair into the equation and determine if it results in a true statement.

ⓐ \(\ (0,-1)\)	ⓑ \(\ (1,4)\)	ⓒ \(\ (-2,-7)\)
\((0,-1)\) is a solution.	\((1,4)\) is a solution.	\((-2,-7)\) is not a solution.

Condensed: the full section is in OpenStax Prealgebra 2e.

Complete a Table of Solutions to a Linear Equation

In the previous examples, we substituted the \(x\text{- and}\ y\text{-values}\) of a given ordered pair to determine whether or not it was a solution to a linear equation. But how do we find the ordered pairs if they are not given? One way is to choose a value for \(x\) and then solve the equation for \(y.\) Or, choose a value for \(y\) and then solve for \(x.\)

We’ll start by looking at the solutions to the equation \(y=5x-1\) we found in . We can summarize this information in a table of solutions.

\(y=5x-1\)
\(x\)	\(y\)	\((x,y)\)
\(0\)	\(-1\)	\((0,-1)\)
\(1\)	\(4\)	\((1,4)\)

To find a third solution, we’ll let \(x=2\) and solve for \(y.\)

	\(y=5x-1\)
Multiply.	\(y=10-1\)
Simplify.	\(y=9\)

The ordered pair is a solution to \(y=5x-1\). We will add it to the table.

\(y=5x-1\)
\(x\)	\(y\)	\((x,y)\)
\(0\)	\(-1\)	\((0,-1)\)
\(1\)	\(4\)	\((1,4)\)
\(2\)	\(9\)	\((2,9)\)

We can find more solutions to the equation by substituting any value of \(x\) or any value of \(y\) and solving the resulting equation to get another ordered pair that is a solution. There are an infinite number of solutions for this equation.

Example

Try it.

Complete the table to find three solutions to the equation \(y=4x-2\text{:}\)

\(y=4x-2\)
\(x\)	\(y\)	\((x,y)\)
\(0\)
\(-1\)
\(2\)

Solution

Substitute \(x=0,x=-1,\) and \(x=2\) into \(y=4x-2.\)

\(y=4x-2\)	\(y=4x-2\)	\(y=4x-2\)
\(y=0-2\)	\(y=-4-2\)	\(y=8-2\)
\(y=-2\)	\(y=-6\)	\(y=6\)
\((0,-2)\)	\((-1,-6)\)	\((2,6)\)

The results are summarized in the table.

\(y=4x-2\)
\(x\)	\(y\)	\((x,y)\)
\(0\)	\(-2\)	\((0,-2)\)
\(-1\)	\(-6\)	\((-1,-6)\)
\(2\)	\(6\)	\((2,6)\)

Example

Try it.

Complete the table to find three solutions to the equation \(5x-4y=20\text{:}\)

\(5x-4y=20\)
\(x\)	\(y\)	\((x,y)\)
\(0\)
	\(0\)
	\(5\)

Solution

The results are summarized in the table.

\(5x-4y=20\)
\(x\)	\(y\)	\((x,y)\)
\(0\)	\(-5\)	\((0,-5)\)
\(4\)	\(0\)	\((4,0)\)
\(8\)	\(5\)	\((8,5)\)

Find Solutions to Linear Equations in Two Variables

To find a solution to a linear equation, we can choose any number we want to substitute into the equation for either \(x\) or \(y.\) We could choose \(1,100,1,000,\) or any other value we want. But it’s a good idea to choose a number that’s easy to work with. We’ll usually choose \(0\) as one of our values.

Example

Try it.

Find a solution to the equation \(3x+2y=6.\)

Solution

Step 1: Choose any value for one of the variables in the equation.		We can substitute any value we want for \(\ x\) or any value for \(\ y.\) Let's pick \(x=0.\) What is the value of \(\ y\) if \(\ x=0\)?
Step 2: Substitute that value into the equation. Solve for the other variable.	Substitute \(0\) for \(\ x.\) Simplify. Divide both sides by 2.
Step 3: Write the solution as an ordered pair.	So, when \(x=0,y=3.\)	This solution is represented by the ordered pair \((0,3).\)
Step 4: Check.	Is the result a true equation? Yes!

We said that linear equations in two variables have infinitely many solutions, and we’ve just found one of them. Let’s find some other solutions to the equation \(3x+2y=6.\)

Example

Try it.

Find three more solutions to the equation \(3x+2y=6.\)

Solution

To find solutions to \(3x+2y=6,\) choose a value for \(x\) or \(y.\) Remember, we can choose any value we want for \(x\) or \(y.\) Here we chose \(1\) for \(x,\) and \(0\) and \(-3\) for \(y.\)

Substitute it into the equation.
Simplify. Solve.
Write the ordered pair.	\((2,0)\)	\((1,\frac{3}{2})\)	\((4,-3)\)

Check your answers.

\((2,0)\)	\((1,\frac{3}{2})\)	\((4,-3)\)

So \((2,0),(1,\frac{3}{2})\) and \((4,-3)\) are all solutions to the equation \(3x+2y=6.\) In the previous example, we found that \((0,3)\) is a solution, too. We can list these solutions in a table.

\(3x+2y=6\)
\(x\)	\(y\)	\((x,y)\)
\(0\)	\(3\)	\((0,3)\)
\(2\)	\(0\)	\((2,0)\)
\(1\)	\(\frac{3}{2}\)	\((1,\frac{3}{2})\)
\(4\)	\(-3\)	\((4,-3)\)

Let’s find some solutions to another equation now.

Example

Try it.

Find three solutions to the equation \(x-4y=8.\)

Solution

Choose a value for \(x\) or \(y.\)
Substitute it into the equation.
Solve.
Write the ordered pair.	\((0,-2)\)	\((8,0)\)	\((20,3)\)

So \((0,-2),(8,0),\) and \((20,3)\) are three solutions to the equation \(x-4y=8.\)

\(x-4y=8\)
\(x\)	\(y\)	\((x,y)\)
\(0\)	\(-2\)	\((0,-2)\)
\(8\)	\(0\)	\((8,0)\)
\(20\)	\(3\)	\((20,3)\)

Remember, there are an infinite number of solutions to each linear equation. Any point you find is a solution if it makes the equation true.

Condensed: the full section is in OpenStax Prealgebra 2e.

Key Concepts

• Sign Patterns of the Quadrants

  Quadrant I	Quadrant II	Quadrant III	Quadrant IV
  (x,y)	(x,y)	(x,y)	(x,y)
  (+,+)	(−,+)	(−,−)	(+,−)

• Coordinates of Zero
  • Points with a y-coordinate equal to 0 are on the x-axis, and have coordinates ( a, 0).
  • Points with a x-coordinate equal to 0 are on the y-axis, and have coordinates ( 0, b).
  • The point (0, 0) is called the origin. It is the point where the x-axis and y-axis intersect.

Use the Rectangular Coordinate System

Plot Points on a Rectangular Coordinate System

In the following exercises, plot each point on a coordinate grid.

Try it.

\((4,1),(1,4)\)

Solution

Try it.

\((3,2),(2,3)\)

Try it.

\((3,4),(4,3)\)

Solution

In the following exercises, plot each point on a coordinate grid and identify the quadrant in which the point is located.

Try it.

1. ⓐ \(\ (-4,2)\)
2. ⓑ \(\ (-1,-2)\)
3. ⓒ \(\ (3,-5)\)
4. ⓓ \(\ (2,\frac{5}{2})\)

Try it.

1. ⓐ \(\ (-2,-3)\)
2. ⓑ \(\ (3,-3)\)
3. ⓒ \(\ (-4,1)\)
4. ⓓ \(\ (1,\frac{3}{2})\)

Solution

Try it.

1. ⓐ \(\ (-1,1)\)
2. ⓑ \(\ (-2,-1)\)
3. ⓒ \(\ (1,-4)\)
4. ⓓ \(\ (3,\frac{7}{2})\)

In the following exercises, plot each point on a coordinate grid.

Try it.

1. ⓐ \(\ (3,-2)\)
2. ⓑ \(\ (-3,2)\)
3. ⓒ \(\ (-3,-2)\)
4. ⓓ \(\ (3,2)\)

Solution

Try it.

1. ⓐ \(\ (4,-1)\)
2. ⓑ \(\ (-4,1)\)
3. ⓒ \(\ (-4,-1)\)
4. ⓓ \(\ (4,1)\)

Try it.

1. ⓐ \(\ (-2,0)\)
2. ⓑ \(\ (-3,0)\)
3. ⓒ \(\ (0,4)\)
4. ⓓ \(\ (0,2)\)

Solution

Identify Points on a Graph

In the following exercises, name the ordered pair of each point shown.

Verify Solutions to an Equation in Two Variables

In the following exercises, determine which ordered pairs are solutions to the given equation.

Try it.

\(2x+y=6\)

1. ⓐ \(\ (1,4)\)
2. ⓑ \(\ (3,0)\)
3. ⓒ \(\ (2,3)\)

Solution

ⓐ , ⓑ

Try it.

\(x+3y=9\)

1. ⓐ \(\ (0,3)\)
2. ⓑ \(\ (6,1)\)
3. ⓒ \(\ (-3,-3)\)

Try it.

\(4x-2y=8\)

1. ⓐ \(\ (3,2)\)
2. ⓑ \(\ (1,4)\)
3. ⓒ \(\ (0,-4)\)

Solution

ⓐ , ⓒ

Try it.

\(3x-2y=12\)

1. ⓐ \(\ (4,0)\)
2. ⓑ \(\ (2,-3)\)
3. ⓒ \(\ (1,6)\)

Try it.

\(y=4x+3\)

1. ⓐ \(\ (4,3)\)
2. ⓑ \(\ (-1,-1)\)
3. ⓒ \(\ (\frac{1}{2},5)\)

Solution

ⓑ , ⓒ

Try it.

\(y=2x-5\)

1. ⓐ \(\ (0,-5)\)
2. ⓑ \(\ (2,1)\)
3. ⓒ \(\ (\frac{1}{2},-4)\)

Try it.

\(y=\frac{1}{2}x-1\)

1. ⓐ \(\ (2,0)\)
2. ⓑ \(\ (-6,-4)\)
3. ⓒ \(\ (-4,-1)\)

Solution

ⓐ , ⓑ

Try it.

\(y=\frac{1}{3}x+1\)

1. ⓐ \(\ (-3,0)\)
2. ⓑ \(\ (9,4)\)
3. ⓒ \(\ (-6,-1)\)

Find Solutions to Linear Equations in Two Variables

In the following exercises, complete the table to find solutions to each linear equation.

Try it.

\(y=2x-4\)

\(x\)	\(y\)	\((x,y)\)
\(-1\)
\(0\)
\(2\)

Solution

\(x\)	\(y\)	\((x,y)\)
\(-1\)	\(-6\)	\((-1,-6)\)
\(0\)	\(-4\)	\((0,-4)\)
\(2\)	\(0\)	\((2,0)\)

Try it.

\(y=3x-1\)

\(x\)	\(y\)	\((x,y)\)
\(-1\)
\(0\)
\(2\)

Try it.

\(y=-x+5\)

\(x\)	\(y\)	\((x,y)\)
\(-2\)
\(0\)
\(3\)

Solution

\(x\)	\(y\)	\((x,y)\)
\(-2\)	\(7\)	\((-2,7)\)
\(0\)	\(5\)	\((0,5)\)
\(3\)	\(2\)	\((3,2)\)

Try it.

\(y=\frac{1}{3}x+1\)

\(x\)	\(y\)	\((x,y)\)
\(0\)
\(3\)
\(6\)

Try it.

\(y=-\frac{3}{2}x-2\)

\(x\)	\(y\)	\((x,y)\)
\(-2\)
\(0\)
\(2\)

Solution

\(x\)	\(y\)	\((x,y)\)
\(-2\)	\(1\)	\((-2,1)\)
\(0\)	\(-2\)	\((0,-2)\)
\(2\)	\(-5\)	\((2,-5)\)

Try it.

\(x+2y=8\)

\(x\)	\(y\)	\((x,y)\)
\(0\)
\(4\)
	\(0\)

Try it.

Have you ever used a map with a rectangular coordinate system? Describe the map and how you used it.

Solution

Answers may vary.

Try it.

How do you determine if an ordered pair is a solution to a given equation?

Condensed: the full section is in OpenStax Prealgebra 2e.

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π Разрешете

Практика (40)

Опитайте всеки на хартия първо. Открийте отговора за проверка; потвърдени могат да бъдат отворени в решавача за всяка стъпка.

1. Evaluate \(x+3\) when \(x=-1\).

   Процедете го. Подсказка

   Разкрийте отговора

   2

2. Evaluate \(2x-5y\) when \(x=3\) and \(y=-2.\)

   Процедете го. Подсказка

   Разкрийте отговора

   16

3. Solve for \(y\): \(40-4y=20\).

   Процедете го. Подсказка

   Разкрийте отговора

   5

4. Plot each point in the rectangular coordinate system and identify the quadrant in which the point is located:

   ⓐ \((-5,4)\) ⓑ \((-3,-4)\) ⓒ \((2,-3)\) ⓓ \((-2,3)\) ⓔ \((3,\frac{5}{2})\).

   Процедете го. Подсказка

   Разкрийте отговора

   The first number of the coordinate pair is the x-coordinate, and the second number is the y-coordinate.

   1. ⓐ Since \(x=-5\), the point is to the left of the y-axis. Also, since \(y=4\), the point is above the x-axis. The point \((-5,4)\) is in Quadrant II.
   2. ⓑ Since \(x=-3\), the point is to the left of the y-axis. Also, since \(y=-4\), the point is below the x-axis. The point \((-3,-4)\) is in Quadrant III.
   3. ⓒ Since \(x=2\), the point is to the right of the y-axis. Since \(y=-3\), the point is below the x-axis. The point \((2,-3)\) is in Quadrant lV.
   4. ⓓ Since \(x=-2\), the point is to the left of the y-axis. Since \(y=3\), the point is above the x-axis. The point \((-2,3)\) is in Quadrant II.
   5. ⓔ Since \(x=3\), the point is to the right of the y-axis. Since \(y=\frac{5}{2}\), the point is above the x-axis. (It may be helpful to write \(\frac{5}{2}\) as a mixed number or decimal.) The point \((3,\frac{5}{2})\) is in Quadrant I.

5. Plot each point in a rectangular coordinate system and identify the quadrant in which the point is located:

   ⓐ \((-2,1)\) ⓑ \((-3,-1)\) ⓒ \((4,-4)\) ⓓ \((-4,4)\) ⓔ \((-4,\frac{3}{2})\).

   Процедете го. Подсказка

6. Plot each point in a rectangular coordinate system and identify the quadrant in which the point is located:

   ⓐ \((-4,1)\) ⓑ \((-2,3)\) ⓒ \((2,-5)\) ⓓ \((-2,5)\) ⓔ \((-3,\frac{5}{2})\).

   Процедете го. Подсказка

7. Plot each point:

   ⓐ \((0,5)\) ⓑ \((4,0)\) ⓒ \((-3,0)\) ⓓ \((0,0)\) ⓔ \((0,-1)\).

   Процедете го. Подсказка

   Разкрийте отговора

   1. ⓐ Since \(x=0\), the point whose coordinates are \((0,5)\) is on the y-axis.
   2. ⓑ Since \(y=0\), the point whose coordinates are \((4,0)\) is on the x-axis.
   3. ⓒ Since \(y=0\), the point whose coordinates are \((-3,0)\) is on the x-axis.
   4. ⓓ Since \(x=0\) and \(y=0\), the point whose coordinates are \((0,0)\) is the origin.
   5. ⓔ Since \(x=0\), the point whose coordinates are \((0,-1)\) is on the y-axis.

8. Plot each point:

   ⓐ \((4,0)\) ⓑ \((-2,0)\) ⓒ \((0,0)\) ⓓ \((0,2)\) ⓔ \((0,-3)\).

   Процедете го. Подсказка

9. Plot each point:

   ⓐ \((-5,0)\) ⓑ \((3,0)\) ⓒ \((0,0)\) ⓓ \((0,-1)\) ⓔ \((0,4)\).

   Процедете го. Подсказка

10. Name the ordered pair of each point shown in the rectangular coordinate system.

    Процедете го. Подсказка

    Разкрийте отговора

    Point A is above \(-3\) on the x-axis, so the x-coordinate of the point is \(-3\).

    • The point is to the left of 3 on the y-axis, so the y-coordinate of the point is 3.
    • The coordinates of the point are \((-3,3)\).

    Point B is below \(-1\) on the x-axis, so the x-coordinate of the point is \(-1\).

    • The point is to the left of \(-3\) on the y-axis, so the y-coordinate of the point is \(-3\).
    • The coordinates of the point are \((-1,-3)\).

    Point C is above 2 on the x-axis, so the x-coordinate of the point is 2.

    • The point is to the right of 4 on the y-axis, so the y-coordinate of the point is 4.
    • The coordinates of the point are \((2,4)\).

    Point D is below 4 on the x-axis, so the x-coordinate of the point is 4.

    • The point is to the right of \(-4\) on the y-axis, so the y-coordinate of the point is \(-4.\)
    • The coordinates of the point are \((4,-4)\).

    Point E is on the y-axis at \(y=-2\). The coordinates of point E are \((0,-2).\)

    Point F is on the x-axis at \(x=3\). The coordinates of point F are \((3,0).\)

11. Name the ordered pair of each point shown in the rectangular coordinate system.

    Процедете го. Подсказка

    Разкрийте отговора

    A: \((5,1)\) B: \((-2,4)\) C: \((-5,-1)\) D: \((3,-2)\) E: \((0,-5)\) F: \((4,0)\)

12. Name the ordered pair of each point shown in the rectangular coordinate system.

    Процедете го. Подсказка

    Разкрийте отговора

    A: \((4,2)\) B: \((-2,3)\) C: \((-4,-4)\) D: \((3,-5)\) E: \((-3,0)\) F: \((0,2)\)

13. Determine which ordered pairs are solutions to the equation \(x+4y=8\).

    ⓐ \((0,2)\) ⓑ \((2,-4)\) ⓒ \((-4,3)\)

    Процедете го. Подсказка

    Разкрийте отговора

    Substitute the x- and y-values from each ordered pair into the equation and determine if the result is a true statement.

14. Which of the following ordered pairs are solutions to \(2x+3y=6\)?

    ⓐ \((3,0)\) ⓑ \((2,0)\) ⓒ \((6,-2)\)

    Процедете го. Подсказка

    Разкрийте отговора

    a, c

15. Which of the following ordered pairs are solutions to the equation \(4x-y=8\)?

    ⓐ \((0,8)\) ⓑ \((2,0)\) ⓒ \((1,-4)\)

    Процедете го. Подсказка

    Разкрийте отговора

    b, c

16. Which of the following ordered pairs are solutions to the equation \(y=5x-1\)?

    ⓐ \((0,-1)\) ⓑ \((1,4)\) ⓒ \((-2,-7)\)

    Процедете го. Подсказка

    Разкрийте отговора

    Substitute the x- and y-values from each ordered pair into the equation and determine if it results in a true statement.

17. Which of the following ordered pairs are solutions to the equation \(y=4x-3\)?

    ⓐ \((0,3)\) ⓑ \((1,1)\) ⓒ \((-1,-1)\)

    Процедете го. Подсказка

    Разкрийте отговора

    b

18. Which of the following ordered pairs are solutions to the equation \(y=-2x+6\)?

    ⓐ \((0,6)\) ⓑ \((1,4)\) ⓒ \((-2,-2)\)

    Процедете го. Подсказка

    Разкрийте отговора

    a, b

19. Complete to find three solutions to the equation \(y=4x-2\).

    \(y=4x-2\)
    \(x\)	\(y\)	\((x,y)\)
    0
    \(-1\)
    2

    Процедете го. Подсказка

    Разкрийте отговора

    Substitute \(x=0\), \(x=-1\), and \(x=2\) into \(y=4x-2\).

    The results are summarized in .

    \(y=4x-2\)
    \(x\)	\(y\)	\((x,y)\)
    0	\(-2\)	\((0,-2)\)
    \(-1\)	\(-6\)	\((-1,-6)\)
    2	6	\((2,6)\)

20. Complete the table to find three solutions to this equation: \(y=3x-1\).

    \(y=3x-1\)
    \(x\)	\(y\)	\((x,y)\)
    0
    \(-1\)
    2

    Процедете го. Подсказка

    Разкрийте отговора

    \(y=3x-1\)
    \(x\)	\(y\)	\((x,y)\)
    0	\(-1\)	\((0,-1)\)
    \(-1\)	\(-4\)	\((-1,-4)\)
    2	5	\((2,5)\)

21. Complete the table to find three solutions to this equation: \(y=6x+1\).

    \(y=6x+1\)
    \(x\)	\(y\)	\((x,y)\)
    0
    1
    \(-2\)

    Процедете го. Подсказка

    Разкрийте отговора

    \(y=6x+1\)
    \(x\)	\(y\)	\((x,y)\)
    0	1	\((0,1)\)
    1	7	\((1,7)\)
    \(-2\)	\(-11\)	\((-2,-11)\)

22. Complete to find three solutions to the equation \(5x-4y=20\).

    \(5x-4y=20\)
    \(x\)	\(y\)	\((x,y)\)
    0
    	0
    	5

    Процедете го. Подсказка

    Разкрийте отговора

    Substitute the given value into the equation \(5x-4y=20\) and solve for the other variable. Then, fill in the values in the table.

    The results are summarized in .

    \(5x-4y=20\)
    \(x\)	\(y\)	\((x,y)\)
    0	\(-5\)	\((0,-5)\)
    4	0	\((4,0)\)
    8	5	\((8,5)\)

23. Complete the table to find three solutions to this equation: \(2x-5y=20\).

    \(2x-5y=20\)
    \(x\)	\(y\)	\((x,y)\)
    0
    	0
    \(-5\)

    Процедете го. Подсказка

    Разкрийте отговора

    \(2x-5y=20\)
    \(x\)	\(y\)	\((x,y)\)
    0	\(-4\)	\((0,-4)\)
    10	0	\((10,0)\)
    \(-5\)	\(-6\)	\((-5,-6)\)

24. Complete the table to find three solutions to this equation: \(3x-4y=12\).

    \(3x-4y=12\)
    \(x\)	\(y\)	\((x,y)\)
    0
    	0
    \(-4\)

    Процедете го. Подсказка

    Разкрийте отговора

    \(3x-4y=12\)
    \(x\)	\(y\)	\((x,y)\)
    0	\(-3\)	\((0,-3)\)
    4	0	\((4,0)\)
    \(-4\)	\(-6\)	\((-4,-6)\)

25. Find three solutions to the equation \(y=-3x+2\).

    Процедете го. Подсказка

    Разкрийте отговора

    We can substitute any value we want for \(x\) or any value for \(y\). Since the equation is in y-form, it will be easier to substitute in values of \(x\). Let’s pick \(x=0\), \(x=1\), and \(x=-1\).
    

    Substitute the value into the equation.
    Simplify.
    Simplify.
    Write the ordered pair.	(0, 2)	(1, −1)	(−1, 5)
    Check.
    \(\ y=-3x+2\)	\(\ y=-3x+2\)	\(\ y=-3x+2\)
    \(2≟-3⋅0+2\)	\(-1≟-3⋅1+2\)	\(5≟-3(-1)+2\)
    \(2≟0+2\)	\(-1≟-3+2\)	\(5≟3+2\)
    \(2=2✓\)	\(-1=-1✓\)	\(5=5✓\)

    So, \((0,2)\), \((1,-1)\) and \((-1,5)\) are all solutions to \(y=-3x+2\). We show them in .
    

    \(y=-3x+2\)
    \(x\)	\(y\)	\((x,y)\)
    0	2	\((0,2)\)
    1	\(-1\)	\((1,-1)\)
    \(-1\)	5	\((-1,5)\)

26. Find three solutions to this equation: \(y=-2x+3\).

    Процедете го. Подсказка

    Разкрийте отговора

    Answers will vary.

27. Find three solutions to this equation: \(y=-4x+1\).

    Процедете го. Подсказка

    Разкрийте отговора

    Answers will vary.

28. Find three solutions to the equation \(3x+2y=6\).

    Процедете го. Подсказка

    Разкрийте отговора

    We can substitute any value we want for \(x\) or any value for \(y\). Since the equation is in standard form, let’s pick first \(x=0\), then \(y=0\), and then find a third point.
    

    Substitute the value into the equation.
    Simplify.
    Solve.
    Write the ordered pair.	(0, 3)	(2, 0)	\((1,\frac{3}{2})\)

    Check.
    \(3x+2y=6\)	\(3x+2y=6\)	\(3x+2y=6\)
    \(3⋅0+2⋅3≟6\)	\(3⋅2+2⋅0≟6\)	\(3⋅1+2⋅\frac{3}{2}≟6\)
    \(0+6≟6\)	\(6+0≟6\)	\(3+3≟6\)
    \(6=6✓\)	\(6=6✓\)	\(6=6✓\)

    So \((0,3)\), \((2,0)\), and \((1,\frac{3}{2})\) are all solutions to the equation \(3x+2y=6\). We can list these three solutions in .
    

    \(3x+2y=6\)
    \(x\)	\(y\)	\((x,y)\)
    0	3	\((0,3)\)
    2	0	\((2,0)\)
    1	\(\frac{3}{2}\)	\((1,\frac{3}{2})\)

29. Find three solutions to the equation \(2x+3y=6\).

    Процедете го. Подсказка

    Разкрийте отговора

    Answers will vary.

30. Find three solutions to the equation \(4x+2y=8\).

    Процедете го. Подсказка

    Разкрийте отговора

    Answers will vary.

31. 1. ⓐ \((-4,2)\)
    2. ⓑ \((-1,-2)\)
    3. ⓒ \((3,-5)\)
    4. ⓓ \((-3,5)\)
       ⓔ \((\frac{5}{3},2)\)
    Процедете го. Подсказка
32. 1. ⓐ \((-2,-3)\)
    2. ⓑ \((3,-3)\)
    3. ⓒ \((-4,1)\)
    4. ⓓ \((4,-1)\)
    5. ⓔ \((\frac{3}{2},1)\)
    Процедете го. Подсказка
33. 1. ⓐ \((3,-1)\)
    2. ⓑ \((-3,1)\)
    3. ⓒ \((-2,2)\)
    4. ⓓ \((-4,-3)\)
    5. ⓔ \((1,\frac{14}{5})\)
    Процедете го. Подсказка
34. 1. ⓐ \((-1,1)\)
    2. ⓑ \((-2,-1)\)
    3. ⓒ \((2,1)\)
    4. ⓓ \((1,-4)\)
    5. ⓔ \((3,\frac{7}{2})\)
    Процедете го. Подсказка
35. 1. ⓐ \((-2,0)\)
    2. ⓑ \((-3,0)\)
    3. ⓒ \((0,0)\)
    4. ⓓ \((0,4)\)
    5. ⓔ \((0,2)\)
    Процедете го. Подсказка
36. 1. ⓐ \((0,1)\)
    2. ⓑ \((0,-4)\)
    3. ⓒ \((-1,0)\)
    4. ⓓ \((0,0)\)
    5. ⓔ \((5,0)\)
    Процедете го. Подсказка
37. 1. ⓐ \((0,0)\)
    2. ⓑ \((0,-3)\)
    3. ⓒ \((-4,0)\)
    4. ⓓ \((1,0)\)
    5. ⓔ \((0,-2)\)
    Процедете го. Подсказка
38. 1. ⓐ \((-3,0)\)
    2. ⓑ \((0,5)\)
    3. ⓒ \((0,-2)\)
    4. ⓓ \((2,0)\)
    5. ⓔ \((0,0)\)
    Процедете го. Подсказка

39. \(2x+y=6\)

    1. ⓐ \((1,4)\)
    2. ⓑ \((3,0)\)
    3. ⓒ \((2,3)\)
    Процедете го. Подсказка

    Разкрийте отговора

    a, b

40. \(x+3y=9\)

    1. ⓐ \((0,3)\)
    2. ⓑ \((6,1)\)
    3. ⓒ \((-3,-3)\)
    Процедете го. Подсказка

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Записване Вход

Символи, използвани тук

Докоснете всеки символ за пълното определение, снимка и какво означава всяка буква в него.

\leqless/greater than or equal
Inequalities that allow equality; < and > exclude it.\pmplus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.\neqnot equal
The two sides are different.\sqrt{x}square root, n-th root
The non-negative number whose square (n-th power) is x.|x|absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.iimaginary unit
i² = −1.\log_b xlogarithm, natural log
The exponent b must be raised to for x; ln uses base e.\mathbb{N}number sets
Naturals, integers, rationals, reals, complex numbers.

Как да се: Use the Rectangular Coordinate System

1. Plot points on a rectangular coordinate system
2. Identify points on a graph
3. Verify solutions to an equation in two variables
4. Complete a table of solutions to a linear equation
5. Find solutions to linear equations in two variables
6. A: (5,1)
7. B: (−2,4)
8. C: (−5,−1)

Въпроси, които хората задават

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Части на тази страница са адаптирани от OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0), OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Тук е обяснено и обяснява, грешките са наши. Тъй като източникът му е лицензиран за некоммерчески употреба, този урок е безплатен, не продава нищо и се споделя по същата лицензия.

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Linear equationsQuadratic equationsSystems of equationsInequalitiesFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value