# 4.5 Use the slope–intercept form of an equation of a line  (Page 3/12)

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Graph the line of the equation $2x-y=6$ using its slope and y -intercept.

Graph the line of the equation $3x-2y=8$ using its slope and y -intercept.

We have used a grid with $x$ and $y$ both going from about $-10$ to 10 for all the equations we’ve graphed so far. Not all linear equations can be graphed on this small grid. Often, especially in applications with real-world data, we’ll need to extend the axes to bigger positive or smaller negative numbers.

Graph the line of the equation $y=0.2x+45$ using its slope and y -intercept.

## Solution

We’ll use a grid with the axes going from about $-80$ to 80.

 $y=mx+b$ The equation is in slope–intercept form. $y=0.2x+45$ Identify the slope and y -intercept. $m=0.2$ The y -intercept is (0, 45) Plot the y -intercept. See graph below. Count out the rise and run to mark the second point. The slope is $m=0.2$ ; in fraction form this means $m=\frac{2}{10}$ . Given the scale of our graph, it would be easier to use the equivalent fraction $m=\frac{10}{50}$ . Draw the line.

Graph the line of the equation $y=0.5x+25$ using its slope and y -intercept.

Graph the line of the equation $y=0.1x-30$ using its slope and y -intercept.

Now that we have graphed lines by using the slope and y -intercept, let’s summarize all the methods we have used to graph lines. See [link] .

## Choose the most convenient method to graph a line

Now that we have seen several methods we can use to graph lines, how do we know which method to use for a given equation?

While we could plot points, use the slope–intercept form, or find the intercepts for any equation, if we recognize the most convenient way to graph a certain type of equation, our work will be easier. Generally, plotting points is not the most efficient way to graph a line. We saw better methods in sections 4.3, 4.4, and earlier in this section. Let’s look for some patterns to help determine the most convenient method to graph a line.

Here are six equations we graphed in this chapter, and the method we used to graph each of them.

$\begin{array}{cccccccc}& & & \mathbf{\text{Equation}}\hfill & & \phantom{\rule{5em}{0ex}}& & \mathbf{\text{Method}}\hfill \\ \text{#1}\hfill & & & x=2\hfill & & \phantom{\rule{5em}{0ex}}& & \text{Vertical line}\hfill \\ \text{#2}\hfill & & & y=4\hfill & & \phantom{\rule{5em}{0ex}}& & \text{Horizontal line}\hfill \\ \text{#3}\hfill & & & \text{−}x+2y=6\hfill & & \phantom{\rule{5em}{0ex}}& & \text{Intercepts}\hfill \\ \text{#4}\hfill & & & 4x-3y=12\hfill & & \phantom{\rule{5em}{0ex}}& & \text{Intercepts}\hfill \\ \text{#5}\hfill & & & y=4x-2\hfill & & \phantom{\rule{5em}{0ex}}& & \text{Slope–intercept}\hfill \\ \text{#6}\hfill & & & y=\text{−}x+4\hfill & & \phantom{\rule{5em}{0ex}}& & \text{Slope–intercept}\hfill \end{array}$

Equations #1 and #2 each have just one variable. Remember, in equations of this form the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines.

In equations #3 and #4, both $x$ and $y$ are on the same side of the equation. These two equations are of the form $Ax+By=C$ . We substituted $y=0$ to find the x -intercept and $x=0$ to find the y -intercept, and then found a third point by choosing another value for $x$ or $y$ .

Equations #5 and #6 are written in slope–intercept form. After identifying the slope and y -intercept from the equation we used them to graph the line.

This leads to the following strategy.

## Strategy for choosing the most convenient method to graph a line

Consider the form of the equation.

• If it only has one variable, it is a vertical or horizontal line.
• $x=a$ is a vertical line passing through the x -axis at $a$ .
• $y=b$ is a horizontal line passing through the y -axis at $b$ .
• If $y$ is isolated on one side of the equation, in the form $y=mx+b$ , graph by using the slope and y-intercept.
• Identify the slope and y -intercept and then graph.
• If the equation is of the form $Ax+By=C$ , find the intercepts.
• Find the x - and y -intercepts, a third point, and then graph.

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315
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In 10 years, the population of Detroit fell from 950,000 to about 712,500. Find the percent decrease.
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25%
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25 percent
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Subtract one from the other to get the difference. Then take that difference and divided by 950000 and you will get .25 aka 25%
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one way is to set as ratio: 100%/950000 = x% / 712500, which yields that 712500 is 75% of the initial 950000. therefore, the decrease is 25%.
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