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Section 3.4 Slope-Intercept Form

Subsection Writing a Linear Equation

In Lesson 3.3, we plotted data for the temperature inside an oven used to cure pottery.

Time, \(x\) 0 10 20 30 40 50 60
Temperature, \(y\) 70 74 78 82 86 90 94

We computed the slope of the graph, shown in the figure. The slope is

\begin{align*} \dfrac{\text{change in temperature}}{\text{change in time}} \amp= \dfrac{\Delta y}{\Delta x}\\ \amp= \dfrac{\text{4 degrees}}{\text{10 minutes}} \end{align*}
graph

or \(m=0.4\) degrees per minute.

You can see from either the table or the graph that the \(y\)-intercept is the point \((0,70)\text{.}\) This means that the initial temperature inside the drying oven was 70 degrees.

Look Ahead.

If we know two pieces of information about a line: its slope and its initial value or \(y\)-intercept, we can write its equation.

Example 3.19.

Write an equation for the temperature, \(H\text{,}\) inside the pottery drying oven \(t\) minutes after the oven is turned on.

Solution

The initial temperature in the oven was 70 degrees, so \(H=70\) when \(t=0\text{.}\) The temperature rose at a rate of 0.4 degrees per minute, so we add 0.4 degrees to \(H\) for each minute that passes. After \(t\) minutes, we have added \(0.4t\) degrees, giving us a temperature of

\begin{equation*} H = 70 + 0.4t \end{equation*}

We can also write the equation as \(H=0.4t+70\text{.}\)

Reading Questions Reading Questions

1.

What does the constant term in an equation tell us about the graph?

2.

What does the coefficient of the input variable tell us?

From Example 3.19, we see that the coefficients of a linear equation tell us something about its graph.

  1. The constant term tells us the vertical intercept of the graph.
  2. The coefficient of the input variable tells us the slope of the graph.
Slope-Intercept Form.

A linear equation written in the form

\begin{equation*} \blert{y=mx+b} \end{equation*}

is said to be in slope-intercept form. The coefficient \(m\) is the slope of the graph, and \(b\) is the \(y\)-intercept.

Reading Questions Reading Questions

4.

What is the slope-intercept form of an equation?

Subsection Slope-Intercept Method of Graphing

A linear equation has the form \(Ax+By=C\text{,}\) and its graph is a straight line. We have already studied two methods for graphing linear equations:

  1. Make a table of values and plot points
  2. Find and plot the intercepts (the intercept method)

There is a third graphing method that makes use of the slope of the line. We can use the slope-intercept form to sketch a graph quickly, without having to plot a lot of points.

Example 3.20.

Graph the equation \(~y=\dfrac{3}{4}x-2\)

Solution

The slope of the line is \(\dfrac{3}{4}\) and its \(y\)-intercept is the point \((0,-2)\text{.}\) We begin by plotting the \(y\)-intercept, as shown in the figure. Next, we use the slope to find another point on the line.

graph of line
The slope,

\begin{equation*} m = \dfrac{\Delta y}{\Delta x} = \dfrac{3}{4} \end{equation*}

gives the ratio of the change in \(y\)-coordinate to the change in \(x\)-coordinate as we move from any point on the line to another. Thus, starting at the point \((0,-2)\text{,}\) we move:

  • 3 units up (the positive \(y\)-direction),then
  • 4 units right (the positive \(x\)-direction )

to locate another point on the line. The coordinates of this new point are \((4,1)\text{.}\) Finally,we draw a line through the two points, as shown in the figure.

Look Closer.

To improve the accuracy of the graph in Example 3.20, we can find a third point on the line by writing the slope in an equivalent form. We change the sign of both numerator and denominator of the slope to get

\begin{equation*} m = \dfrac{\Delta y}{\Delta x} = \dfrac{-3}{-4} \end{equation*}

Starting again from the \(y\)-intercept \((0,-2)\text{,}\) we now move 3 units down and 4 units left, and find the point \((-4,-5)\) on the graph.

Reading Questions Reading Questions

5.

Name two methods of graphing that we have already studied.

6.

When using the slope-intercept method, what is the first point we plot?

7.

How do we use the slope to find a second point on the line?

The slope-intercept method can be used to graph any non-vertical line.

To Graph a Line Using the Slope-Intercept Method.
  1. Write the equation in the form \(y=mx+b\text{.}\)
  2. Plot the \(y\)-intercept, \((0,b)\text{.}\)
  3. Write the slope as a fraction, \(m = \dfrac{\Delta y}{\Delta x}\text{.}\)
  4. Use the slope to find a second point on the graph: Starting at the \(y\)-intercept, move \({\Delta y}\) units in the \(y\)-direction, then \({\Delta x}\) units in the \(x\)-direction.
  5. Find a third point by moving \({-\Delta y}\) units in the \(y\)-direction, then \({-\Delta x}\) units in the \(x\)-direction, starting from the \(y\)-intercept.
  6. Draw a line through the three plotted points.

Subsection Finding the Slope-Intercept Form

Not all linear equations appear in slope-intercept form. However, we can write the equation of any non-vertical line in slope-intercept form by solving the equation for \(y\) in terms of \(x\text{.}\)

Example 3.21.

Find the slope and \(y\)-intercept of the graph of \(~3x-4y=8\)

Solution

To write the equation in slope-intercept form, we solve for \(y\) in terms of \(x\text{.}\)

\begin{equation*} \begin{aligned} 3x-4y \amp = 8 \amp\amp \blert{\text{Subtract}~ 3x ~ \text{from both sides.}}\\ -4y \amp = -3x+8 \amp\amp \blert{\text{Divide both sides by} ~-4.}\\ \dfrac{-4y}{-4} \amp = \dfrac{-3x+8}{-4} \amp\amp \blert{\text{Divide each term of the right side by} ~-4.}\\ y \amp = \dfrac{-3x}{-4}+\dfrac{-8}{-4} \amp\amp \blert{\text{Simplify each quotient.}}\\ y \amp = \dfrac{3}{4}x-2 \end{aligned} \end{equation*}

The equation is now in slope-intercept form, with \(m= \dfrac{3}{4}\) and \(b=-2\text{.}\) Thus, the slope of the graph is \(\dfrac{3}{4}\) and the \(y\)-intercept is the point \((0,-2)\text{.}\)

Caution 3.22.

Do not confuse solving for \(y\) with finding the \(y\)-intercept. In the Example above, we do not set \(x=0\) before solving for \(y\text{.}\)

  • When we find the \(y\)-intercept, we are looking for a specific point, namely, the point with \(x\)-coordinate zero, so we replace \(x\) by 0.
  • When we "solve for \(y\text{,}\)" we are writing the equation in another form, so both variables, \(x\) and \(y\text{,}\) still appear in the equation.

Reading Questions Reading Questions

8.

How do we put an equation into slope-intercept form?

Subsection Skills Warm-Up

Exercises Exercises

Solve for the indicated variable.

1.

\(2q+p=10~~~~~~~~\) for \(p\)

2.

\(2l+2w=18~~~~~~~~\) for \(l\)

3.

\(3a+9=-6b~~~~~~~~\) for \(a\)

4.

\(2c=2d+22~~~~~~~~\) for \(d\)

5.

\(5r-4s=24~~~~~~~~\) for \(s\)

6.

\(2m=11-3n~~~~~~~~\) for \(n\)

Solutions Answers to Skills Warm-Up

Exercises Exercises

Exercises Homework 3.4

For Problems 1–4,

  1. Write the equation in slope-intercept form.
  2. State the slope and \(y\)-intercept of the graph.
1.
\(y=3x+4\)
2.
\(6x+3y=6\)
3.
\(2x-3y=6\)
4.
\(5x=4y\)

For Problems 5–8,

  1. Find the slope and \(y\)-intercept of the line.
  2. Write the equation for the line.
5.
graph of line
6.
graph of ine
7.
graph of line
8.
graph of line

For Problems 9–12,

  1. Fill in the \(y\)-values in the tables and graph the lines.
  2. Choose two points on each line and compute its slope.
  3. What is the \(y\)-intercept of each line?
9.
  1. \(y=2x-6\)

    \(x\) \(-1\) \(0\) \(1\) \(2\) \(3\)
    \(y\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\)

  2. \(y=2x+1\)

    \(x\) \(-1\) \(0\) \(1\) \(2\) \(3\)
    \(y\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\)

  3. \(y=2x+3\)

    \(x\) \(-1\) \(0\) \(1\) \(2\) \(3\)
    \(y\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\)

grid

All three lines have the same .

10.
  1. \(y=\dfrac{-3}{2}x-4\)

    \(x\) \(-6\) \(-4\) \(-2\) \(0\) \(2\)
    \(y\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\)

  2. \(y=\dfrac{-3}{2}x+2\)

    \(x\) \(-6\) \(-4\) \(-2\) \(0\) \(2\)
    \(y\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\)

  3. \(y=\dfrac{-3}{2}x+6\)

    \(x\) \(-6\) \(-4\) \(-2\) \(0\) \(2\)
    \(y\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\)

grid

All three lines have the same .

11.
  1. \(y=\dfrac{1}{4}x+2\)

    \(x\) \(-4\) \(-2\) \(0\) \(2\) \(4\)
    \(y\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\)

  2. \(y=\dfrac{1}{2}x+2\)

    \(x\) \(-4\) \(-2\) \(0\) \(2\) \(4\)
    \(y\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\)

  3. \(y=x+2\)

    \(x\) \(-4\) \(-2\) \(0\) \(2\) \(4\)
    \(y\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\)

grid

All three lines have the same .

12.
  1. \(y=-3x-2\)

    \(x\) \(-6\) \(-3\) \(0\) \(3\) \(6\)
    \(y\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\)

  2. \(y=-2x-2\)

    \(x\) \(-6\) \(-3\) \(0\) \(3\) \(6\)
    \(y\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\)

  3. \(y=\dfrac{-5}{3}x-2\)

    \(x\) \(-6\) \(-3\) \(0\) \(3\) \(6\)
    \(y\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\)

grid

All three lines have the same .

For Problems 13–14,

  1. Find the intercepts of the graph and graph the line.
  2. Compute the slope of the line.
  3. Put the equation in slope-intercept form.
13.

\(3x+4y=12\)

\(x\) \(0\) \(\hphantom{0000}\)
\(y\) \(\hphantom{0000}\) \(0\)

grid
14.

\(y+3x-8=0\)

\(x\) \(0\) \(\hphantom{0000}\)
\(y\) \(\hphantom{0000}\) \(0\)

grid

For Problems 15–16,

  1. Put the equation in slope-intercept form.
  2. What is the \(y\)-intercept of each line? What is its slope?
  3. Use the slope to find two more points on the line.
  4. Graph the line.
15.

\(3x-5y=0\)

grid
16.

\(5x+4y=0\)

grid

For Problems 17–20, graph the equation by using the slope-intercept method.

17.

\(y=3-x\)

grid
18.

\(y=3x-1\)

grid
19.

\(y=\dfrac{3}{4}x+2\)

grid
20.

\(y=-2-\dfrac{4}{3}x\)

grid
21.

Robin opened a yogurt smoothie shop near campus. The graph shows Robin's profit \(P\) after selling \(s\) smoothies.

graph of line
  1. What is the \(P\)-intercept of the line?
  2. Calculate the slope of the line.
  3. Write an equation for the line in slope-intercept form.
  4. What do the slope and the \(P\)-intercept tell us about the problem?

For Problems 22–24,

  1. Graph the line by the slope-intercept method.
  2. Explain what the slope and the vertical intercept tell us about the problem.
22.

Serda's score on her driving test is computed by the equation \(S=120-4n\text{,}\) where \(n\) is the number of wrong answers she gives.

grid
23.

Greg is monitoring the growth of a new variety of string beans. The height of the vine each day is given in inches by \(h=18+3d\) where \(d=0\) represents today.

grid
24.

Cliff's score was negative at the end of the first round of College Quiz, but in Double Quiz his score improved according the equation \(S=-400+20q\text{,}\) where \(q\) is the number of questions he answered.

grid