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Use the Rectangular Coordinate System

Plot points on a rectangular coordinate system

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 .

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

Solution

ⓐ Read the number below the Residence Halls, \(4,\) and the letter to the side, A. So the Residence Halls are in grid section 4A.
ⓑ 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 \(0.\) See . 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. See .

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 \(x+4y=8\text{:}\)

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

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. \(y=5x-1\text{:}\)

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

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.

\((3,2)\)

Solution

Try it.

\((4,1)\)

Try it.

\((1,5)\)

Solution

Try it.

\((3,4)\)

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.

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

Try it.

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

Solution

Try it.

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

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

Try it.

ⓐ \(\ (3,-2)\)
ⓑ \(\ (-3,2)\)
ⓒ \(\ (-3,-2)\)
ⓓ \(\ (3,2)\)

Solution

Try it.

ⓐ \(\ (4,-1)\)
ⓑ \(\ (-4,1)\)
ⓒ \(\ (-4,-1)\)
ⓓ \(\ (4,1)\)

Try it.

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

Solution

Identify Points on a Graph

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

Try it.

Solution

C(1, -3) D(4, 3)

Try it.

Solution

S(-2, 4) T(-4, -2)

Try it.

Solution

C(0, -1) D(-1, 0)

Try it.

Verify Solutions to an Equation in Two Variables

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

Try it.

\(2x+y=6\)

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

Solution

ⓐ , ⓑ

Try it.

\(x+3y=9\)

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

Try it.

\(4x-2y=8\)

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

Solution

ⓐ , ⓒ

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

ⓑ , ⓒ

Try it.

\(y=2x-5\)

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

Try it.

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

ⓐ \(\ (2,0)\)
ⓑ \(\ (-6,-4)\)

Solution

ⓐ , ⓑ

Try it.

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

ⓐ \(\ (-3,0)\)
ⓑ \(\ (9,4)\)

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.

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 .

Jawaby görkez

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

Jawaby görkez

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

Jawaby görkez

\(5\)
Use the map in .
1. ⓐ Find the grid section of the Residence Halls.
2. ⓑ What is located in grid section 4C?
Jawaby görkez
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.
Use the map in .
1. ⓐ Find the grid section of Taylor Hall.
2. ⓑ What is located in section 3B?
Jawaby görkez
1. ⓐ 1C
2. ⓑ Engineering Building
Use the map in .
1. ⓐ Find the grid section of the Parking Garage.
2. ⓑ What is located in section 2C?
Jawaby görkez
1. ⓐ 1A
2. ⓑ Library
Plot \((1,3)\) and \((3,1)\) in the same rectangular coordinate system.

Jawaby görkez

The coordinate values are the same for both points, but the \(x\) and \(y\) values are reversed. Let’s begin with point \((1,3).\) The \(x\text{-coordinate}\) is \(1\) so find \(1\) on the \(x\text{-axis}\) and sketch a vertical line through \(x=1.\) The \(y\text{-coordinate}\) is \(3\) so we find \(3\) on the \(y\text{-axis}\) and sketch a horizontal line through \(y=3.\) Where the two lines meet, we plot the point \((1,3).\)

To plot the point \((3,1),\) we start by locating \(3\) on the \(x\text{-axis}\) and sketch a vertical line through \(x=3.\) Then we find \(1\) on the \(y\text{-axis}\) and sketch a horizontal line through \(y=1.\) Where the two lines meet, we plot the point \((3,1).\)

Notice that the order of the coordinates does matter, so, \((1,3)\) is not the same point as \((3,1).\)
Plot each point on the same rectangular coordinate system: \((5,2),(2,5).\)

Jawaby görkez
Plot each point on the same rectangular coordinate system: \((4,2),(2,4).\)

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

The first number of the coordinate pair is the \(x\text{-coordinate},\) and the second number is the \(y\text{-coordinate}.\)
ⓐ Since \(x=-1,y=3,\) the point \((-1,3)\) is in Quadrant II.
ⓑ Since \(x=-3,y=-4,\) the point \((-3,-4)\) is in Quadrant III.
ⓒ Since \(x=2,y=-1,\) the point \((2,-1)\) is in Quadrant lV.
ⓓ Since \(x=3,y=\frac{5}{2},\) the point \((3,\frac{5}{2})\) is in Quadrant I. It may be helpful to write \(\frac{5}{2}\) as the mixed number, \(2\frac{1}{2},\) or decimal, \(2.5.\) Then we know that the point is halfway between \(2\) and \(3\) on the \(y\text{-axis}.\)
Plot each point on a rectangular coordinate system and identify the quadrant in which the point is located.
1. ⓐ \(\ (-2,1)\\)
2. ⓑ \(\ (-3,-1)\)
3. ⓒ \(\ (4,-4)\)
4. ⓓ \(\ (-4,\frac{3}{2})\)
Jawaby görkez

(a) Quadrant II, (b) Quadrant III, (c) Quadrant IV, (d) Quadrant II
Plot each point on a rectangular coordinate system and identify the quadrant in which the point is located.
1. ⓐ \(\ (-4,1)\)
2. ⓑ \(\ (-2,3)\)
3. ⓒ \(\ (2,-5)\)
4. ⓓ \(\ (-3,\frac{5}{2})\)
Jawaby görkez

(a) Quadrant II, (b) Quadrant II, (c) Quadrant IV, (d) Quadrant II
Plot each point:
1. ⓐ \(\ (-5,2)\)
2. ⓑ \(\ (-5,-2)\)
3. ⓒ \(\ (5,2)\)
4. ⓓ \(\ (5,-2)\)
Jawaby görkez

As we locate the \(x\text{-coordinate}\) and the \(y\text{-coordinate},\) we must be careful with the signs.
Plot each point on a coordinate grid:
1. ⓐ \(\ (0,5)\)
2. ⓑ \(\ (4,0)\)
3. ⓒ \(\ (-3,0)\)
4. ⓓ \(\ (0,0)\)
5. ⓔ \(\ (0,-1)\)
Jawaby görkez
1. ⓐ Since \(x=0,\) the point whose coordinates are \((0,5)\) is on the \(y\text{-axis}.\)
2. ⓑ Since \(y=0,\) the point whose coordinates are \((4,0)\) is on the \(x\text{-axis}.\)
3. ⓒ Since \(y=0,\) the point whose coordinates are \((-3,0)\) is on the \(x\text{-axis}.\)
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 \(y\text{-axis}.\)
Plot each point on a coordinate grid:
1. ⓐ \(\ (4,0)\)
2. ⓑ \(\ (-2,0)\)
3. ⓒ \(\ (0,0)\)
4. ⓓ \(\ (0,2)\)
5. ⓔ \(\ (0,-3)\)
Jawaby görkez
Plot each point on a coordinate grid:
1. ⓐ \(\ (-5,0)\)
2. ⓑ \(\ (3,0)\)
3. ⓒ \(\ (0,0)\)
4. ⓓ \(\ (0,-1)\)
5. ⓔ \(\ (0,4)\)
Jawaby görkez
Name the ordered pair of each point shown:

Jawaby görkez

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 \((4,-4).\)
Name the ordered pair of each point shown:
Jawaby görkez
1. A: (5,1)
2. B: (−2,4)
3. C: (−5,−1)
4. D: (3,−2)
Name the ordered pair of each point shown:
Jawaby görkez
1. A: (4,2)
2. B: (−2,3)
3. C: (−4,−4)
4. D: (3,−5)

Name the ordered pair of each point shown:

Jawaby görkez

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 \((0,1)\).

Name the ordered pair of each point shown:
Jawaby görkez
1. A: (4,0)
2. B: (0,3)
3. C: (−3,0)
4. D: (0,−5)
Name the ordered pair of each point shown:
Jawaby görkez
1. A: (−3,0)
2. B: (0,−3)
3. C: (5,0)
4. D: (0,2)
Determine which ordered pairs are solutions of the equation \(x+4y=8\text{:}\)
1. ⓐ \(\ (0,2)\)
2. ⓑ \(\ (2,-4)\)
3. ⓒ \(\ (-4,3)\)
Jawaby görkez

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.
Determine which ordered pairs are solutions to the given equation: \(2x+3y=6\)
1. ⓐ \(\ (3,0)\)
2. ⓑ \(\ (2,0)\)
3. ⓒ \(\ (6,-2)\)
Jawaby görkez

ⓐ , ⓒ
Determine which ordered pairs are solutions to the given equation: \(4x-y=8\)
1. ⓐ \(\ (0,8)\)
2. ⓑ \(\ (2,0)\)
3. ⓒ \(\ (1,-4)\)
Jawaby görkez

ⓑ , ⓒ
Determine which ordered pairs are solutions of the equation. \(y=5x-1\text{:}\)
1. ⓐ \(\ (0,-1)\)
2. ⓑ \(\ (1,4)\)
3. ⓒ \(\ (-2,-7)\)
Jawaby görkez

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.
Determine which ordered pairs are solutions of the given equation: \(y=4x-3\)
1. ⓐ \(\ (0,3)\)
2. ⓑ \(\ (1,1)\)
3. ⓒ \(\ (1,0)\)
Jawaby görkez

ⓑ
Determine which ordered pairs are solutions of the given equation: \(y=-2x+6\)
1. ⓐ \(\ (0,6)\)
2. ⓑ \(\ (1,4)\)
3. ⓒ \(\ (-2,-2)\)
Jawaby görkez

ⓐ , ⓑ
Complete the table to find three solutions to the equation \(y=4x-2\text{:}\)
\(y=4x-2\)
\(x\) \(y\) \((x,y)\)
\(0\)
\(-1\)
\(2\)

Jawaby görkez

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

Jawaby görkez

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

Jawaby görkez

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

Jawaby görkez

The results are summarized in the table.
\(5x-4y=20\)
\(x\) \(y\) \((x,y)\)
\(0\) \(-5\) \((0,-5)\)
\(4\) \(0\) \((4,0)\)
\(8\) \(5\) \((8,5)\)
Complete the table to find three solutions to the equation: \(2x-5y=20.\)
\(2x-5y=20\)
\(x\) \(y\) \((x,y)\)
\(0\)
\(0\)
\(-5\)

Jawaby görkez

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

Jawaby görkez

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

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

Jawaby görkez

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!

Find a solution to the equation: \(4x+3y=12.\)

Jawaby görkez

Answers will vary.
Find a solution to the equation: \(2x+4y=8.\)

Jawaby görkez

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

Jawaby görkez

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

Symbols used here

\pm

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

\neq

not equal
The two sides are different.

\leq,\ \geq

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

\approx

approximately equal
Equal to the precision shown, not exactly.

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

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Use the Rectangular Coordinate System

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

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Özüňi synla

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

_Ýaşa Arithmetic

Order of operations Fractions Percentages Powers and roots

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

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

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

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