maths.free › Algebra › 4. Graphs › Use the Rectangular Coordinate System

Use the Rectangular Coordinate System

Plot points in a 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. See .

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, as shown in .

\(y=5x-1\)
\(x\)	\(y\)	\((x,y)\)
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.

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

Solution

Try it.

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

Try it.

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

Solution

Try it.

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

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

Try it.

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

Solution

Try it.

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

Try it.

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

Solution

Try it.

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

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

Try it.

Solution

A: \((-4,1)\) B: \((-3,-4)\) C: \((1,-3)\) D: \((4,3)\)

Try it.

Solution

A: \((0,-2)\) B: \((-2,0)\) C: \((0,5)\) D: \((5,0)\)

Try it.

Verify Solutions to an Equation in Two Variables

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

Try it.

\(2x+y=6\)

ⓐ \((1,4)\)
ⓑ \((3,0)\)
ⓒ \((2,3)\)

Solution

a, b

Try it.

\(x+3y=9\)

ⓐ \((0,3)\)
ⓑ \((6,1)\)
ⓒ \((-3,-3)\)

Try it.

\(4x-2y=8\)

ⓐ \((3,2)\)
ⓑ \((1,4)\)
ⓒ \((0,-4)\)

Solution

a, c

Try it.

\(3x-2y=12\)

ⓐ \((4,0)\)
ⓑ \((2,-3)\)
ⓒ \((1,6)\)

Try it.

\(y=4x+3\)

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

Solution

b, c

Try it.

\(y=2x-5\)

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

Try it.

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

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

Solution

a, b

Try it.

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

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

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.

\(y=5x-8\)

Solution

Answers will vary.

Try it.

\(y=3x-9\)

Try it.

\(y=-4x+5\)

Solution

Answers will vary.

Try it.

\(y=-2x+7\)

Try it.

\(x+y=8\)

Solution

Answers will vary.

Try it.

\(x+y=6\)

Try it.

\(x+y=-2\)

Solution

Answers will vary.

Try it.

\(x+y=-1\)

Try it.

\(3x+y=5\)

Solution

Answers will vary.

Try it.

\(2x+y=3\)

Try it.

\(4x-y=8\)

Solution

Answers will vary.

Try it.

\(5x-y=10\)

Try it.

\(2x+4y=8\)

Solution

Answers will vary.

Try it.

\(3x+2y=6\)

Try it.

\(5x-2y=10\)

Solution

Answers will vary.

Try it.

\(4x-3y=12\)

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

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Evaluate \(x+3\) when \(x=-1\).
If you missed this problem, review .

उत्तर उघडा

2
Evaluate \(2x-5y\) when \(x=3\) and \(y=-2.\)
If you missed this problem, review .

उत्तर उघडा

16
Solve for \(y\): \(40-4y=20\).
If you missed this problem, review .

उत्तर उघडा

5
Plot each point in the rectangular coordinate system and identify the quadrant in which the point is located:
ⓐ \((-5,4)\) ⓑ \((-3,-4)\) ⓒ \((2,-3)\) ⓓ \((-2,3)\) ⓔ \((3,\frac{5}{2})\).
उत्तर उघडा
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.
Plot each point in a rectangular coordinate system and identify the quadrant in which the point is located:
ⓐ \((-2,1)\) ⓑ \((-3,-1)\) ⓒ \((4,-4)\) ⓓ \((-4,4)\) ⓔ \((-4,\frac{3}{2})\).

उत्तर उघडा
Plot each point in a rectangular coordinate system and identify the quadrant in which the point is located:
ⓐ \((-4,1)\) ⓑ \((-2,3)\) ⓒ \((2,-5)\) ⓓ \((-2,5)\) ⓔ \((-3,\frac{5}{2})\).

उत्तर उघडा
Plot each point:
ⓐ \((0,5)\) ⓑ \((4,0)\) ⓒ \((-3,0)\) ⓓ \((0,0)\) ⓔ \((0,-1)\).
उत्तर उघडा
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.
Name the ordered pair of each point shown in the rectangular coordinate system.
उत्तर उघडा
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 \((3,0).\)
Name the ordered pair of each point shown in the rectangular coordinate system.

उत्तर उघडा

A: \((5,1)\) B: \((-2,4)\) C: \((-5,-1)\) D: \((3,-2)\) E: \((0,-5)\) F: \((4,0)\)
Name the ordered pair of each point shown in the rectangular coordinate system.

उत्तर उघडा

A: \((4,2)\) B: \((-2,3)\) C: \((-4,-4)\) D: \((3,-5)\) E: \((-3,0)\) F: \((0,2)\)
Determine which ordered pairs are solutions to the equation \(x+4y=8\).
ⓐ \((0,2)\) ⓑ \((2,-4)\) ⓒ \((-4,3)\)

उत्तर उघडा

Substitute the x- and y-values from each ordered pair into the equation and determine if the result is a true statement.
Which of the following ordered pairs are solutions to \(2x+3y=6\)?
ⓐ \((3,0)\) ⓑ \((2,0)\) ⓒ \((6,-2)\)

उत्तर उघडा

a, c
Which of the following ordered pairs are solutions to the equation \(4x-y=8\)?
ⓐ \((0,8)\) ⓑ \((2,0)\) ⓒ \((1,-4)\)

उत्तर उघडा

b, c
Which of the following ordered pairs are solutions to the equation \(y=5x-1\)?
ⓐ \((0,-1)\) ⓑ \((1,4)\) ⓒ \((-2,-7)\)

उत्तर उघडा

Substitute the x- and y-values from each ordered pair into the equation and determine if it results in a true statement.
Which of the following ordered pairs are solutions to the equation \(y=4x-3\)?
ⓐ \((0,3)\) ⓑ \((1,1)\) ⓒ \((-1,-1)\)

उत्तर उघडा

b
Which of the following ordered pairs are solutions to the equation \(y=-2x+6\)?
ⓐ \((0,6)\) ⓑ \((1,4)\) ⓒ \((-2,-2)\)

उत्तर उघडा

a, b
Complete to find three solutions to the equation \(y=4x-2\).
\(y=4x-2\)
\(x\) \(y\) \((x,y)\)
0
\(-1\)
2

उत्तर उघडा

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 \((2,6)\)
Complete the table to find three solutions to this equation: \(y=3x-1\).
\(y=3x-1\)
\(x\) \(y\) \((x,y)\)
0
\(-1\)
2

उत्तर उघडा

\(y=3x-1\)
\(x\) \(y\) \((x,y)\)
0 \(-1\) \((0,-1)\)
\(-1\) \(-4\) \((-1,-4)\)
2 5 \((2,5)\)
Complete the table to find three solutions to this equation: \(y=6x+1\).
\(y=6x+1\)
\(x\) \(y\) \((x,y)\)
0
1
\(-2\)

उत्तर उघडा

\(y=6x+1\)
\(x\) \(y\) \((x,y)\)
0 1 \((0,1)\)
1 7 \((1,7)\)
\(-2\) \(-11\) \((-2,-11)\)
Complete to find three solutions to the equation \(5x-4y=20\).
\(5x-4y=20\)
\(x\) \(y\) \((x,y)\)
0
0
5

उत्तर उघडा

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 \((8,5)\)
Complete the table to find three solutions to this equation: \(2x-5y=20\).
\(2x-5y=20\)
\(x\) \(y\) \((x,y)\)
0
0
\(-5\)

उत्तर उघडा

\(2x-5y=20\)
\(x\) \(y\) \((x,y)\)
0 \(-4\) \((0,-4)\)
10 0 \((10,0)\)
\(-5\) \(-6\) \((-5,-6)\)
Complete the table to find three solutions to this equation: \(3x-4y=12\).
\(3x-4y=12\)
\(x\) \(y\) \((x,y)\)
0
0
\(-4\)

उत्तर उघडा

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

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

उत्तर उघडा

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	\((-1,5)\)

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

उत्तर उघडा

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

उत्तर उघडा

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

उत्तर उघडा

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 \(\frac{3}{2}\) \((1,\frac{3}{2})\)
Find three solutions to the equation \(2x+3y=6\).

उत्तर उघडा

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

उत्तर उघडा

Answers will vary.

Symbols used here

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\sqrt{x},\ \sqrt[n]{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.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Use the Rectangular Coordinate System

Plot points in a rectangular coordinate system
Verify solutions to an equation in two variables
Complete a table of solutions to a linear equation
Find solutions to a linear equation in two variables
The point is to the left of 3 on the
The coordinates of the point are
The point is to the left of
The coordinates of the point are

Questions people ask

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.

स्वतःचा प्रयत्न करा

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

अधिक माहिती Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value

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

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

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

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