Finding Ordered Pair for Two Solutions of Equations

Finding Ordered Pair for Two Solutions of Equations

33 Graph Linear Equations in Two Variables

Hordern Hationes

33 Graph Linear Equations in Two Variables

Menu Halaman Statis

Learning Objectives

Recognize the Relationship Between the Solutions of an Equation and its Graph

Graph a Linear Equation by Plotting Points

Graph Vertical and Horizontal Lines

Key Concepts

Learning Objectives

Recognize the Relationship Between the Solutions of an Equation and its Graph

Graph a Linear Equation by Plotting Points

Graph Vertical and Horizontal Lines

Key Concepts

Practice Makes Perfect

Mixed Practice

Everyday Math

Writing Exercises

Self Check

Practice Makes Perfect

Mixed Practice

Everyday Math

Writing Exercises

Self Check

Graphs

By the end of this section, you will be able to:

Recognize the relationship between the solutions of an equation and its graph.
Graph a linear equation by plotting points.
Graph vertical and horizontal lines.

In the previous section, we found several solutions to the equation . They are listed in (Figure). So, the ordered pairs $\left(0,3\right)$ , $\left(2,0\right)$ , and $\left(1,\frac{3}{2}\right)$ are some solutions to the equation . We can plot these solutions in the rectangular coordinate system as shown in (Figure).


		$\left(x,y\right)$
0	3	$\left(0,3\right)$
2	0	$\left(2,0\right)$
1	$\frac{3}{2}$	$\left(1,\frac{3}{2}\right)$

The figure shows four points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the four points at (0, 3), (1, three halves), (2, 0), and (4, negative 3). The four points appear to line up along a straight line.

Notice how the points line up perfectly? We connect the points with a line to get the graph of the equation . See (Figure). Notice the arrows on the ends of each side of the line. These arrows indicate the line continues.

Every point on the line is a solution of the equation. Also, every solution of this equation is a point on this line. Points not on the line are not solutions.

Notice that the point whose coordinates are $\left(-2,6\right)$ is on the line shown in (Figure). If you substitute and into the equation, you find that it is a solution to the equation.

The figure shows a straight line and two points and on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the two points and are labeled by the coordinates

The figure shows a series of equations to check if the ordered pair (negative 2, 6) is a solution to the equation 3x plus 2y equals 6. The first line states

So the point $\left(-2,6\right)$ is a solution to the equation . (The phrase "the point whose coordinates are $\left(-2,6\right)$ " is often shortened to "the point $\left(-2,6\right)$ .")

The figure shows a series of equations to check if the ordered pair (4, 1) is a solution to the equation 3x plus 2y equals 6. The first line states

So $\left(4,1\right)$ is not a solution to the equation . Therefore, the point $\left(4,1\right)$ is not on the line. See (Figure). This is an example of the saying, "A picture is worth a thousand words." The line shows you all the solutions to the equation. Every point on the line is a solution of the equation. And, every solution of this equation is on this line. This line is called the graph of the equation .

Graph of a Linear Equation

The graph of a linear equation is a line.

Every point on the line is a solution of the equation.
Every solution of this equation is a point on this line.

The graph of is shown.

The figure shows a straight line on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line has a positive slope and goes through the y-axis at the (0, negative 3). The line is labeled with the equation y equals 2x negative 3.

For each ordered pair, decide:

ⓐ Is the ordered pair a solution to the equation?
ⓑ Is the point on the line?

A $\left(0,-3\right)$ B $\left(3,3\right)$ C $\left(2,-3\right)$ D $\left(-1,-5\right)$

Use the graph of to decide whether each ordered pair is:

a solution to the equation.
on the line.

ⓐ $\left(0,-1\right)$ ⓑ $\left(2,5\right)$

The figure shows a straight line on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the point (negative 2, negative 7) and for every 3 units it goes up, it goes one unit to the right. The line is labeled with the equation y equals 3x minus 1.

ⓐ yes, yesⓑ yes, yes

Use graph of to decide whether each ordered pair is:

a solution to the equation
on the line

ⓐ $\left(3,-1\right)$ ⓑ $\left(-1,-4\right)$

ⓐ no, noⓑ yes, yes

There are several methods that can be used to graph a linear equation. The method we used to graph is called plotting points, or the Point–Plotting Method.

How To Graph an Equation By Plotting Points

Graph the equation by plotting points.

Graph the equation by plotting points: .

The figure shows a straight line on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 1, 6), (0, 4), (1, 2), (2, 0), (3, negative 2), (4, negative 4), and (5, negative 6). There are arrows at the ends of the line pointing to the outside of the figure.

The steps to take when graphing a linear equation by plotting points are summarized below.

Graph a linear equation by plotting points.

Find three points whose coordinates are solutions to the equation. Organize them in a table.
Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.

It is true that it only takes two points to determine a line, but it is a good habit to use three points. If you only plot two points and one of them is incorrect, you can still draw a line but it will not represent the solutions to the equation. It will be the wrong line.

If you use three points, and one is incorrect, the points will not line up. This tells you something is wrong and you need to check your work. Look at the difference between part (a) and part (b) in (Figure).

Figure a shows three points with a straight line going through them. Figure b shows three points that do not lie on the same line.

Let's do another example. This time, we'll show the last two steps all on one grid.

Graph the equation .

Solution

Find three points that are solutions to the equation. Here, again, it's easier to choose values for . Do you see why?

We list the points in (Figure).


		$\left(x,y\right)$
0	0	$\left(0,0\right)$
1		$\left(1,-3\right)$
	6	$\left(-2,6\right)$

Plot the points, check that they line up, and draw the line.

Graph the equation by plotting points: .

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 2, 8), (0, 0), and (2, negative 8). The line has arrows on both ends pointing to the outside of the figure.

Graph the equation by plotting points: .

When an equation includes a fraction as the coefficient of , we can still substitute any numbers for . But the math is easier if we make 'good' choices for the values of . This way we will avoid fraction answers, which are hard to graph precisely.

Graph the equation $y=\frac{1}{2}x+3$ .

Graph the equation $y=\frac{1}{3}x-1$ .

Graph the equation $y=\frac{1}{4}x+2$ .

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 12, negative 1), (negative 8, 0), (negative 4, 1), (0, 2), (4, 3), (8, 4), and (12, 5). The line has arrows on both ends pointing to the outside of the figure.

So far, all the equations we graphed had given in terms of . Now we'll graph an equation with and on the same side. Let's see what happens in the equation . If what is the value of ?

This point has a fraction for the x– coordinate and, while we could graph this point, it is hard to be precise graphing fractions. Remember in the example $y=\frac{1}{2}x+3$ , we carefully chose values for so as not to graph fractions at all. If we solve the equation for , it will be easier to find three solutions to the equation.

$\begin{array}{ccc}\hfill 2x+y& =\hfill & 3\hfill \\ \hfill y& =\hfill & -2x+3\hfill \end{array}$

The solutions for , , and are shown in the (Figure). The graph is shown in (Figure).


		$\left(x,y\right)$
0	3	$\left(0,3\right)$
1	1	$\left(1,1\right)$
	5	$\left(-1,5\right)$

Can you locate the point $\left(\frac{3}{2},0\right)$ , which we found by letting , on the line?

Graph the equation .

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 4, 10), (negative 2, 6), (0, 2), (2, negative 2), (4, negative 6), and (6, negative 10). The line has arrows on both ends pointing to the outside of the figure.

Graph the equation .

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 3, 9), (negative 2, 5), (negative 1, 1), (0, negative 3), (1, negative 7), and (2, negative 10). The line has arrows on both ends pointing to the outside of the figure.

If you can choose any three points to graph a line, how will you know if your graph matches the one shown in the answers in the book? If the points where the graphs cross the x– and y-axis are the same, the graphs match!

The equation in (Figure) was written in standard form, with both and on the same side. We solved that equation for in just one step. But for other equations in standard form it is not that easy to solve for , so we will leave them in standard form. We can still find a first point to plot by letting and solving for . We can plot a second point by letting and then solving for . Then we will plot a third point by using some other value for or .

Graph the equation .

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 1, 6), (0, 4), (1, 2), (2, 0), (3, negative 2), and (4, negative 4). The line has arrows on both ends pointing to the outside of the figure.

Graph the equation .

Can we graph an equation with only one variable? Just and no , or just without an ? How will we make a table of values to get the points to plot?

Let's consider the equation . This equation has only one variable, . The equation says that is always equal to , so its value does not depend on . No matter what is, the value of is always .

So to make a table of values, write in for all the values. Then choose any values for . Since does not depend on , you can choose any numbers you like. But to fit the points on our coordinate graph, we'll use 1, 2, and 3 for the y-coordinates. See (Figure).


		$\left(x,y\right)$
	1	$\left(-3,1\right)$
	2	$\left(-3,2\right)$
	3	$\left(-3,3\right)$

Plot the points from (Figure) and connect them with a straight line. Notice in (Figure) that we have graphed a vertical line.

Vertical Line

A vertical line is the graph of an equation of the form .

The line passes through the x-axis at $\left(a,0\right)$ .

Graph the equation .

The figure shows a straight vertical line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (5, 1), (5, 2), (5, 3), and all other points with first coordinate 5. The line has arrows on both ends pointing to the outside of the figure.

Graph the equation .

What if the equation has but no ? Let's graph the equation . This time the y– value is a constant, so in this equation, does not depend on . Fill in 4 for all the 's in (Figure) and then choose any values for . We'll use 0, 2, and 4 for the x-coordinates.


		$\left(x,y\right)$
0	4	$\left(0,4\right)$
2	4	$\left(2,4\right)$
4	4	$\left(4,4\right)$

The graph is a horizontal line passing through the y-axis at 4. See (Figure).

Horizontal Line

A horizontal line is the graph of an equation of the form .

The line passes through the y-axis at $\left(0,b\right)$ .

Graph the equation

Graph the equation .

The figure shows a straight horizontal line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 4, 3), (0, 3), (4, 3), and all other points with second coordinate 3. The line has arrows on both ends pointing to the outside of the figure.

The equations for vertical and horizontal lines look very similar to equations like What is the difference between the equations and ?

The equation has both and . The value of depends on the value of . The y-coordinate changes according to the value of . The equation has only one variable. The value of is constant. The y-coordinate is always 4. It does not depend on the value of . See (Figure).

The figure shows a two straight lines drawn on the same x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. One line is a straight horizontal line labeled with the equation y equals 4. The other line is a slanted line labeled with the equation y equals 4x.

Notice, in (Figure), the equation gives a slanted line, while gives a horizontal line.

Graph and in the same rectangular coordinate system.

Graph a Linear Equation by Plotting Points
1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
2. Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work!
3. Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.

Recognize the Relationship Between the Solutions of an Equation and its Graph

In the following exercises, for each ordered pair, decide:

ⓐ Is the ordered pair a solution to the equation?ⓑ Is the point on the line?

ⓐ yes; noⓑ no; noⓒ yes; yesⓓ yes; yes

ⓐ yes; yesⓑ yes; yesⓒ yes; yesⓓ no; no

Graph a Linear Equation by Plotting Points

In the following exercises, graph by plotting points.

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 3, negative 10), (negative 2, negative 7), (negative 1, negative 4), (0, negative 1), (1, 2), (2, 5), and (3, 8).

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 4, 10), (negative 3, 8), (negative 2, 6), (negative 1, 4), (0, 2), (1, 0), (2, negative 2), (3, negative 4), (4, negative 6), and (5, negative 8).

$y=\text{−}x-3$

$y=\text{−}x-2$

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 5, negative 10), (negative 4, negative 8), (negative 3, negative 6), (negative 2, negative 4), (negative 1, negative 2), (0, 0), (1, 2), (2, 4), (3, 6), (4, 8), and (5, 10).

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 3, 12), (negative 2, 8), (negative 1, 4), (0, 0), (1, negative 4), (2, negative 8), and (3, negative 12).

$y=\frac{1}{2}x+2$

$y=\frac{1}{3}x-1$

$y=\frac{4}{3}x-5$

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 3, negative 9), (0, negative 5), (3, negative 1), (6, 3), and (9, 7).

$y=\frac{3}{2}x-3$

$y=-\frac{2}{5}x+1$

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 10, 5), (negative 5, 3), (0, 1), (5, negative 1), and (10, negative 3).

$y=-\frac{4}{5}x-1$

$y=-\frac{3}{2}x+2$

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 6, 11), (negative 4, 8), (negative 2, 5), (0, 2), (2, negative 1), (4, negative 4), (6, negative 7), and (8, negative 10).

$y=-\frac{5}{3}x+4$

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to -7. The equation 3 x plus y equals 7 is graphed.

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 5, 7), (negative 4, 5), (negative 3, 3), (negative 2, 1), (negative 1, negative 1), (0, negative 3), (1, negative 5), and (2, negative 7).

$\frac{1}{3}x+y=2$

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, 4), (negative 3, 3), (0, 2), (3, 1), and (6, 0).

$\frac{1}{2}x+y=3$

$\frac{2}{5}x-y=4$

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 5, negative 2), (0, negative 4), and (5, negative 6).

$\frac{3}{4}x-y=6$

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 4, negative 6), (0, negative 3), (4, 0), and (8, 3).

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, negative three halves), (negative 3, negative 1), (0, negative one half), (3, 0), and (6, one half).

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 2, 7), (0, 2), (2, negative 3), and (4, negative 8).

Graph Vertical and Horizontal Lines

In the following exercises, graph each equation.

The figure shows a straight vertical line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The vertical line goes through the points (4, 0), (4, 1), (4, 2) and all points with first coordinate 4.

The figure shows a straight vertical line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The vertical line goes through the points (negative 2, 0), (negative 2, 1), (negative 2, 2) and all points with first coordinate negative 2.

The figure shows a straight horizontal line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The horizontal line goes through the points (0, 3), (1, 3), (2, 3) and all points with second coordinate 3.

The figure shows a straight horizontal line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The horizontal line goes through the points (0, negative 5), (1, negative 5), (2, negative 5) and all points with second coordinate negative 5.

$x=\frac{7}{3}$

The figure shows a straight vertical line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The vertical line goes through the points (7/3, 0), (7/3, 1), (7/3, 2) and all points with first coordinate 7/3.

$x=\frac{5}{4}$

$y=-\frac{15}{4}$

The figure shows a straight horizontal line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The horizontal line goes through the points (0, negative 15/4), (1, negative 15/4), (2, negative 15/4) and all points with second coordinate negative 15/4.

$y=-\frac{5}{3}$

In the following exercises, graph each pair of equations in the same rectangular coordinate system.

and

$y=-\frac{1}{3}x$ and $y=-\frac{1}{3}$

In the following exercises, graph each equation.

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 2, negative 8), (negative 1, negative 4), (0, 0), (1, 4), and (2, 8).

$y=-\frac{1}{2}x+3$

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, 6), (negative 4, 5), (negative 2, 4), (0, 3), (2, 2), (4, 1), and (6, 0).

$y=\frac{1}{4}x-2$

$y=\text{−}x$

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 3, negative 7), (negative 2, negative 6), (negative 1, negative 4), (0, negative 3), (1, negative 2), (2, negative 1), (3, 0), (4, 1), (5, 2), and (6, 3).

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 2, 6), (negative 1, 4), (0, 2), (1, negative 2), and (2, negative 6).

The figure shows a straight horizontal line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The horizontal line goes through the points (0, negative 1), (1, negative 1), (2, negative 1) and all points with second coordinate negative 1.

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, 4), (negative 3, 3), (0, 2), (3, 1), and (6, 0).

The figure shows a straight vertical line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The vertical line goes through the points (3, 0), (3, 1), (3, 2) and all points with first coordinate 3.

Motor home cost. The Robinsons rented a motor home for one week to go on vacation. It cost them ?594 plus ?0.32 per mile to rent the motor home, so the linear equation gives the cost, , for driving miles. Calculate the rental cost for driving 400, 800, and 1200 miles, and then graph the line.

?722, ?850, ?978

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from 0 to 1200 in increments of 100. The y-axis of the plane runs from 0 to 1000 in increments of 100. The straight line starts at the point (0, 594) and goes through the points (400, 722), (800, 850), and (1200, 978). The right end of the line has an arrow pointing up and to the right.

Explain how you would choose three x– values to make a table to graph the line $y=\frac{1}{5}x-2$ .

Answers will vary.

What is the difference between the equations of a vertical and a horizontal line?

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is

ⓑ After reviewing this checklist, what will you do to become confident for all goals?

Finding Ordered Pair for Two Solutions of Equations

Source: https://opentextbc.ca/elementaryalgebraopenstax/chapter/graph-linear-equations-in-two-variables/

$\left(x,y\right)$

$\left(x,y\right)$

$\left(0,0\right)$

$\left(0,4\right)$

$\left(1,4\right)$

$\left(1,4\right)$

$\left(2,8\right)$

$\left(2,4\right)$