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Algebra 1

1.12.3 Writing an Equation Given the Slope and a Point

Algebra 11.12.3 Writing an Equation Given the Slope and a Point
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Activity

What if the point you are given isn’t a y y -intercept? Point-slope is a great form to use when you are given the slope and any point.

First, work to derive the point-slope formula by completing the following steps:

Start with the slope formula as m = y − y 1 x − x 1 m = y − y 1 x − x 1 .

Step 1 - Multiply both sides of the equation by x − x 1 x − x 1 .

Step 2 - Simplify.

Step 3 - Rewrite the equation with the y y terms on the left.

With your partner, you will find the equation of the line in point-slope form that goes through the point ( 3 , 4 ) ( 3 , 4 ) and has a slope of 2 by answering the following questions.

1. Use the point and slope to graph the line on a coordinate plane.

2. Write the point-slope form.

3. Name the point ( x 1 , y 1 ) ( x 1 , y 1 ) and the slope, m m .

4. Substitute the point and slope into the point-slope form.

5. Now, with your partner, each solve for y y to write the equation in slope-intercept form. Compare your answers for any possible errors. Come to an agreement on your equation. You may also choose to graph the line on a coordinate plane using a graphing calculator or Desmos with your final equation to check your answer.

What is the simplified equation in slope-intercept form ( y = m x + b ) ( y = m x + b ) ?

6. Convert slope-intercept form to standard form.

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Self Check

Which equation represents the slope-intercept form of the equation of a line with slope  m = − 3 4 that contains the point ( 4 , − 7 ) ?
  1. y = − 3 4 x + 10
  2. y = − 3 4 x − 11
  3. y = − 3 4 x − 5 4
  4. y = − 3 4 x − 4

Additional Resources

Write the Equation of a Line Given the Slope and a Point

Writing an equation using the slope-intercept form works well when you are given the slope and y y -intercept. If you are given the slope and another point that is not the y y -intercept, we need another form of the equation of a line.

The point-slope form of an equation of a line can be derived from the formula for slope. Suppose you have a line that has slope m m and that contains some specific point ( x 1 , y 1 ) ( x 1 , y 1 ) as point 1 and some other point, ( x , y ) ( x , y ) , as point 2. You can write the slope of this line and then change it to a different form.

The point-slope form of an equation of a line with slope m m and containing the point ( x 1 , y 1 ) ( x 1 , y 1 ) is: y − y 1 = m ( x − x 1 ) y − y 1 = m ( x − x 1 ) . You can use this form to find the equation of a line when you are given the slope and a point.

Example 1

Write the equation of a line with slope m = − 1 3 m = − 1 3 that contains the point ( 6 , − 4 ) ( 6 , − 4 ) . Write the equation in slope-intercept form and standard form.

Step 1 - Identify the slope.

The slope is given.

m = 1 3 m = 1 3

Step 2 - Identify the point.

The point is given.

( x 1 , y 1 ) ( x 1 , y 1 )

( 6 , − 4 ) ( 6 , − 4 )

Step 3 - Substitute the values into the point-slope form, y − y 1 = m ( x − x 1 ) y − y 1 = m ( x − x 1 ) . Simplify.

y − y 1 = m ( x − x 1 ) y − ( − 4 ) = 1 3 ( x − 6 ) y + 4 = 1 3 x + 2 y − y 1 = m ( x − x 1 ) y − ( − 4 ) = 1 3 ( x − 6 ) y + 4 = 1 3 x + 2

Step 4 - Write the equation in slope-intercept form.

y = 1 3 x − 2 y = 1 3 x − 2

Step 5 - Convert slope-intercept form to standard form.

y = 1 3 x − 2 y = 1 3 x − 2

3 y = 3 1 3 x − ( 3 ) 2 3 y = 3 1 3 x − ( 3 ) 2

3 y = x − 6 3 y = x − 6

− x + 3 y = − 6 − x + 3 y = − 6

( − 1 ) ( − x + 3 y = − 6 ( − 1 ) ( − x + 3 y = − 6 )

x − 3 y = 6 x − 3 y = 6

To find an equation of a line given the slope and a point, follow these steps:

Step 1 - Identify the slope.

Step 2 - Identify the point.

Step 3 - Substitute the values into the point-slope form, y − y 1 = m ( x − x 1 ) y − y 1 = m ( x − x 1 ) .

Step 4 - Write the equation in slope-intercept form.

Step 5 - Convert slope-intercept form to standard form.

Example 2

Find the equation of a horizontal line that contains the point ( – 2 , – 6 ) ( – 2 , – 6 ) . Write the equation in slope-intercept form.

Step 1 - Identify the slope. Every horizontal line has slope 0.

m = 0 m = 0

Step 2 - Identify the point.

( x 1 , y 1 ) ( x 1 , y 1 )

( – 2 , – 6 ) ( – 2 , – 6 )

Step 3 - Substitute the values into the point-slope form, y − y 1 = m ( x − x 1 ) y − y 1 = m ( x − x 1 ) . Simplify.

y − y 1 = m ( x − x 1 ) y − ( − 6 ) = 0 ( x − ( − 2 ) ) y + 6 = 0 y − y 1 = m ( x − x 1 ) y − ( − 6 ) = 0 ( x − ( − 2 ) ) y + 6 = 0

Step 4 - Write the equation in slope-intercept form.

y = − 6 y = − 6

y = 0 x − 6 y = 0 x − 6

Step 5 - Convert slope-intercept form to standard form.

y = 0 x − 6 y = 0 x − 6

0 x + y = − 6 0 x + y = − 6

Try it

Try It: Writing the Equation of a Line Given the Slope and Point

Write the equation of a line with slope m = − 2 5 m = − 2 5 that contains the point ( 10 , − 5 ) ( 10 , − 5 ) . Write the equation in slope-intercept form.

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