6.5 Slope y-Intercept Form: Part 1

Leanne Thompson

6.5 Slope y-Intercept Form: Part 1

As mentioned in previous lessons, linear equations will form a straight line when graphed. Many of the linear equations that we have worked with have been written in what we call slope y-intercept form. Slope y-intercept form is a way to represent the equation of a straight line, and it is written as y = mx+b. The m value and the b value in slope y-intercept form can tell us important information about the graph of the equation and help us graph it. Answer the following questions to determine the significance of the m and b values.

The graph of $y=-\dfrac{2}{5}x+3$ is shown below.
The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0,3) and (1,5).

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

a) Find the slope of the graph.

b) Find the y-intercept of the graph.

c) What are the values of m and b in the equation?

d) What do you think the m and b values represent on the graph?

Table 6.5.1
Slope y-Intercept Form
The slope y–intercept form of an equation of a line can be written as $y=mx+b$ , where m represents the slope of the graph, and (0, b) represents the y-intercept of the graph.

Practice 1

Write an equation in slope y-intercept form to represent the following graphs.

a)

The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The line goes through the points (0, negative 1) and (6, 3).

b)

Practice 2

Identify the slope and y-intercept of the following equations.

a) $y=-\dfrac{3}{5}x-10$

b) $y=-3x+7$

c) $y=-0.3x-6.5$

Sometimes a linear equation will not be written in slope y-intercept form. When this is the case and you want the equation to be in slope y-intercept form, you can rearrange the equation.

Practice 3

Rearrange the following equations so they are written in slope y-intercept form. Then identify the slope and y-intercept of the equations.

a) $y=-5-4x$

b) $-\dfrac{1}{2}x=y-6$

c) $2y=10x-6$

d) $3y=5x-2$

e) $-4x+5=-4y$

f) $2y=\dfrac{2}{3}x+8$

We can use the slope y-intercept form of an equation to create a graph of the relation.

**Table 6.5.2**
To graph an equation in slope y-intercept form:
Step 1: If needed, rearrange the equation so that it is in slope y-intercept form. Step 2: Identify the slope and y-intercept. Step 3: Plot the y-intercept. Step 4: Use the slope ratio $m=\dfrac{\text{rise}}{\text{run}}$ to identify the rise and the run. Step 5: Starting at the y-intercept, count out the rise and run to mark the second point. Step 6: Connect the points with a line.

Practice 4

Graph the following equations using the slope and y-intercept of the graph.

a) $y=-2x+3$

Blank coordinate plane. Both axes have a minimum of -6 and maximum of 6 and a scale of 1.

b) $y=\dfrac{3}{2}x-1$

Blank coordinate plane. Both axes have a minimum of -6 and maximum of 6 and a scale of 1.

c) $y-6=-\dfrac{5}{2}x$

Blank coordinate plane. Both axes have a minimum of -6 and maximum of 6 and a scale of 1.

d) $2y=8x-6$

Blank coordinate plane. Both axes have a minimum of -6 and maximum of 6 and a scale of 1.

Homework

Write an equation in slope y-intercept form to represent the following graphs.

a)
b)
c)

d)
e)
f)
Identify the slope and y-intercept of the following equations.

a) $y=\dfrac{5}{8}x+3$

b) $y=-x+\dfrac{7}{2}$

c) $y=9.9x-25$

d) $y+4=\dfrac{1}{2}x+3$

e) $-y=-4x+7$

f) $y-2x=-10$

g) $-2y=8x-2$

h) $5x=3y+5$

i) $-4y=\dfrac{2}{5}x-12$
Graph the following equations using the slope and y-intercept.

a) $y=x-2$

b) $-2+y=-\dfrac{2}{3}x$

c) $\dfrac{1}{2}x-3=-y$

d) $4x+2y=8$

e) $3y-x=\dfrac{1}{3}$

f) $-7y-21=-14x$

Which of the following statements about the slope y-intercept form of a linear equation is false?

a) The slope y-intercept form is written as y = mx + b, where m represents the slope of the line.
b) The slope y-intercept form can be used to represent vertical lines.
c) In the slope y-intercept form, b represents the y-intercept, which is where the line crosses the y-axis.
d) The slope of the line in the slope y-intercept form can be positive, negative, or zero.

Which of the following statements about the slope y-intercept form of linear equations is true?

a) The equation y = –2x + 5 represents a line with a slope of 2 and a y-intercept of 5.
b) The equation y = 0x + 3 represents a vertical line that crosses the y-axis at 3.
c) The equation y = x + 7 represents a line with a slope of 0 and a y-intercept of 7.
d) The equation 3y = 9x − 12 represents a line with a slope of 3 and a y-intercept of –4.

Which of the following statements about the slope y-intercept form of linear equations is/are true? Circle all that apply.

a) The equation y = $\dfrac{1}{2}$ x – 3 represents a line with a slope of $\dfrac{1}{2}$ and a y-intercept of –3.
b) The slope y-intercept form y = mx + b can be used to represent vertical lines when m = 0.
c) The equation y = 4x + 1 represents a line with a slope of 4 and a y-intercept of 1.
d) The slope of the line –2y = 10x − 16 is negative, and the line intersects the y-axis at 8.

Answers

1.

a) $y=\dfrac{1}{2}x+3$	b) $y=-\dfrac{4}{3}x+1$	c) $y=x$
d) $y=\dfrac{1}{2}x+2$	e) $y=4x-2$	f) $y=\dfrac{4}{5}x-3$

2.

a) m = $\dfrac{5}{8}$ y-int = 3	b) m = –1 y-int = $\dfrac{7}{2}$	c) m = 9.9 y-int = -25
d) m = $\dfrac{1}{2}$ y-int = –1	e) m = 4 y-int = –7	f) m = 2 y-int = –10
g) m = –4 y-int = 1	h) m = $\dfrac{5}{3}$ y-int = $-\dfrac{5}{3}$	i) m = $-\dfrac{1}{10}$ y-int = 3

3.

a) m = 1
y-int = –2
Graph of the equation y = x − 2. The x-intercept is the point (2, 0) and the y-intercept is the point (0, −2)

b) m = $-\dfrac{2}{3}$
y-int = 2
Graph of the equation y = − 2 thirds x + 2 and the x-intercept is the point (3, 0) and the y-intercept is the point (0, 2).

c) m = $-\dfrac{1}{2}$
y-int = 3
Graph of the equation y = − 1 half x + 3. The x-intercept is the point (6, 0) and the y-intercept is the point (0, 3).