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Writing an Equation Given the Slope and y -intercept

1.12.2 Writing an Equation Given the Slope and y -intercept

1.12.2 β€’ Writing an Equation Given the Slope and y -intercept

Activity

Recall that the slope-intercept form of the equation of a line is y = m x + b y = m x + b , where m m is the slope and b b is the 𝑦 𝑦 -intercept. The 𝑦 𝑦 -intercept is the point where the graph crosses the 𝑦 𝑦 -axis. For 𝑦 𝑦 -intercept 𝑏 𝑏 , the coordinates of the point are ( 0 , 𝑏 ) ( 0 , 𝑏 ) .

In the following activity, you will complete a series of questions regarding this information.

For questions 1 – 3, determine the slope, m m , and the y y -intercept, b b .

1.

y = 2 x + 5 y = 2 x + 5

m = 2 m = 2 ; b = 5 b = 5

2.

y = βˆ’ 3 x + 2 y = βˆ’ 3 x + 2

m = βˆ’ 3 m = βˆ’ 3 ; b = 2 b = 2

3.

y = βˆ’ 1 5 x βˆ’ 1 y = βˆ’ 1 5 x βˆ’ 1

m = βˆ’ 1 5 m = βˆ’ 1 5 ; b = βˆ’ 1 b = βˆ’ 1

For questions 4 – 7, match the correct slope and y y -intercept to the given equations.

  1. slope: βˆ’ 1 2 βˆ’ 1 2 , y y -intercept: βˆ’ 3 βˆ’ 3
  2. slope: 2 2 , y y -intercept: βˆ’ 3 βˆ’ 3
  3. slope: βˆ’ 2 βˆ’ 2 , y y -intercept: 3 3
  4. slope: 3 3 , y y -intercept: βˆ’ 2 βˆ’ 2
4.

What is the correct slope and y y -intercept of y = βˆ’ 2 x + 3 y = βˆ’ 2 x + 3 ?

c. slope: βˆ’ 2 βˆ’ 2 , y y -intercept: 3 3

5.

What is the correct slope and y y -intercept of y = 3 x βˆ’ 2 y = 3 x βˆ’ 2 ?

d. slope: 3 3 , y y -intercept: βˆ’ 2 βˆ’ 2

6.

What is the correct slope and y y -intercept of y = 2 x βˆ’ 3 y = 2 x βˆ’ 3 ?

b. slope: 2 2 , y y -intercept: βˆ’ 3 βˆ’ 3

7.

What is the correct slope and y y -intercept of y = βˆ’ 1 2 x βˆ’ 3 y = βˆ’ 1 2 x βˆ’ 3 ?

a. slope: βˆ’ 1 2 βˆ’ 1 2 , y y -intercept: βˆ’ 3 βˆ’ 3

8.

Write the equation of a line, in slope-intercept form, that has a slope of βˆ’9 βˆ’9 and a y y -intercept at ( 0, βˆ’4 ) ( 0, βˆ’4 ) .

y = βˆ’ 9 x βˆ’ 4 y = βˆ’ 9 x βˆ’ 4

9.

Write the equation of a line, in slope-intercept form, that has a slope of 2 5 2 5 and a y y -intercept at ( 0, βˆ’4 ) ( 0, βˆ’4 ) .

y = 2 5 x + 4 y = 2 5 x + 4

Self Check

Write the equation of a line in slope-intercept form with a slope ofΒ  βˆ’ 3 4 and a y -intercept of βˆ’ 2 .
  1. y = 3 4 x βˆ’ 2
  2. 3 x + 4 y = βˆ’ 8
  3. y = βˆ’ 3 4 x + 2
  4. y = βˆ’ 3 4 x βˆ’ 2

Additional Resources

Write Equations Given a Slope and y y -Intercept

We can determine the slope and intercept of a line if the equation is written in slope-intercept form, y = m x + b y = m x + b . Now we will do the reverseβ€”we will start with the slope and y y -intercept and use them to find the equation of the line.

Example 1

Find the equation of a line with slope βˆ’ 1 βˆ’ 1 and y y -intercept ( 0 , βˆ’ 3 ) ( 0 , βˆ’ 3 ) .

Solution

Step 1 - Name the slope.

m = βˆ’ 1 m = βˆ’ 1

Step 2 - Name the y y -intercept.

b = βˆ’ 3 b = βˆ’ 3

Step 3 - Substitute the values into y = m x + b y = m x + b .

y = βˆ’ 1 x + βˆ’ 3 y = βˆ’ 1 x + βˆ’ 3

y = βˆ’ x βˆ’ 3 y = βˆ’ x βˆ’ 3

The same information can be picked up from a graph of a line.

Example 2

Find the equation of the line shown in the graph.

A line with positive slope on a coordinate grid. A teal point is marked on the line at (3, -2), and both axes range from -8 to 8.

Solution

Step 1 - Find the slope by counting the rise and run.

m = r i s e r u n m = r i s e r u n

From ( 0 , βˆ’ 4 ) ( 0 , βˆ’ 4 ) to ( 3 , βˆ’ 2 ) ( 3 , βˆ’ 2 ) , the slope is m = 2 3 m = 2 3 .

Step 2 - Find the y y -intercept.

y y -intercept ( 0 , βˆ’ 4 ) ( 0 , βˆ’ 4 )

b = βˆ’ 4 b = βˆ’ 4

Step 3 - Substitute the values into y = m x + b y = m x + b .

y = m x + b y = m x + b

y = 2 3 x βˆ’ 4 y = 2 3 x βˆ’ 4

Try it

Write Equations Given a Slope and y y -Intercept

Write the equation of the line graphed below in slope-intercept form:

This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8.

Compare your answer: Here is how to write the equation of the line in slope-intercept form.

Step 1 - Find the slope by counting the rise and run.

m = r i s e r u n m = r i s e r u n

From ( 0 , 1 ) ( 0 , 1 ) to ( 5 , 4 ) ( 5 , 4 ) , the slope is m = 3 5 m = 3 5 .

Step 2 - Find the y y -intercept.

y y -intercept ( 0 , 1 ) ( 0 , 1 )

b = 1 b = 1

Step 3 - Substitute the values into y = m x + b y = m x + b .

y = m x + b y = m x + b

y = 3 5 x + 1 y = 3 5 x + 1