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Algebra 1

4.7.3 Writing Linear Equations

Algebra 14.7.3 Writing Linear Equations
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Student Activity

In Algebra, equations are commonly written in three different ways: slope-intercept form, point-slope form, and standard form. They are equivalent ways to write the same equation, but each one is useful for different reasons.

Slope-intercept form Point-slope form Standard form

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

Given:

m = m = slope

b = y b = y -intercept

y − y 1 = m ( x − x 1 ) y − y 1 = m ( x − x 1 )

Given:

a point ( x 1 , y 1 ) ( x 1 , y 1 )

m = m = slope

A x + B y = C A x + B y = C

m = − A B = m = − A B = slope

Where:

There are no fractions and A A cannot be negative.

For slope-intercept form, the slope and y y -intercept are needed:

1. Write an equation in slope-intercept form for a line whose slope is -2 and y y -intercept is ( 0 , 4 ) ( 0 , 4 ) .

For point-slope form, the slope and one point are needed:

2. Write an equation in point-slope form for a line whose slope is 1 2 1 2 and goes through the point ( 3 , − 5 ) ( 3 , − 5 )

Standard form of a linear equation is often used to find x x - and y y -intercepts of graphs.

Often, an equation starts in another form and gets put into standard form.

Standard form does not have fractions or decimals and cannot lead with a negative number.

3. What is the slope in the equation y = 2 x + 3 y = 2 x + 3 ?

4. Write the equation y = 2 x + 3 y = 2 x + 3 in standard form.

Work with a partner to write equations in different forms.

5. What is the slope of y − 3 = 2 ( x + 4 ) y − 3 = 2 ( x + 4 )

6. Write the equation y − 3 = 2 ( x + 4 ) y − 3 = 2 ( x + 4 ) in slope-intercept form.

7. Write the equation y + 5 = − 3 ( x − 2 ) y + 5 = − 3 ( x − 2 ) into standard form.

8. Write the equation y = 1 3 x − 4 y = 1 3 x − 4 in standard form.

9. What is the slope of 3 x + 12 y = 24 3 x + 12 y = 24 ?

Video: Writing Equations in Different Forms

Watch the following video to learn more about writing equations in different forms.

Access multimedia content

Self Check

Write the equation of the line in point-slope form that has a slope of 7 and goes through ( − 1 , 2 ) .
  1. 2 − y = 7 ( − 1 − x )
  2. y − 2 = 7 ( x − 1 )
  3. y − 2 = 7 ( x + 1 )
  4. y = 7 x + 9

Additional Resources

Writing Linear Equations in Different Forms

Example 1

When you look at equations in these forms, it's easy to identify the slope of m m .

Equation Equation formula Slope formula Slope value
y = 5 x − 15 y = 5 x − 15 y = m x + b y = m x + b m m 5
y = 6 x + 8 y = 6 x + 8 y = m x + b y = m x + b m m 6
y − 7 = 9 ( x + 1 ) y − 7 = 9 ( x + 1 ) y − y 1 = m ( x − x 1 ) y − y 1 = m ( x − x 1 ) m m 9
y − 20 = 3 ( x + 16 ) y − 20 = 3 ( x + 16 ) y − y 1 = m ( x − x 1 ) y − y 1 = m ( x − x 1 ) m m 3
18 x − 6 y = 24 18 x − 6 y = 24 A x + B y = C A x + B y = C m = − A B m = − A B − ( 18 − 6 ) = 3 − ( 18 − 6 ) = 3
30 x + 15 y = 24 30 x + 15 y = 24 A x + B y = C A x + B y = C m = − A B m = − A B − 30 15 = − 2 − 30 15 = − 2

Example 2

Find the equation of a line with slope −9 and y y -intercept ( 0 , − 4 ) ( 0 , − 4 ) in slope-intercept form.

Step 1 - Identify the m m and b b .

m = − 9 m = − 9 , b = − 4 b = − 4

Step 2 - Substitute into the formula.

y = − 9 x − 4 y = − 9 x − 4

Example 3

Find the equation of a line in point-slope form given a slope of − 1 3 − 1 3 and the point ( 6 , − 4 ) ( 6 , − 4 )

Step 1 - Identify the m m and ( x 1 , y 1 ) ( x 1 , y 1 ) .

m = − 1 3 x 1 = 6 y 1 = − 4 m = − 1 3 x 1 = 6 y 1 = − 4

Step 2 - Substitute into the formula.

y − ( − 4 ) = − 1 3 ( x − 6 ) y − ( − 4 ) = − 1 3 ( x − 6 )

Step 3 - Simplify

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

Example 4

Write the equation y + 4 = − 1 3 ( x − 6 ) y + 4 = − 1 3 ( x − 6 ) in standard form.

Step 1 - Multiply to get rid of the fraction.

Remember, standard form cannot have decimals or fractions.

Multiply every term by 3.

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

3 y + 12 = − ( x − 6 ) 3 y + 12 = − ( x − 6 )

Step 2 - Distribute.

3 y + 12 = − x + 6 3 y + 12 = − x + 6

Step 3 - Rearrange terms so the x x term and the y y term are both on the left side of the equation.

Add x x to both sides.

3 y + 12 + x = − x + x + 6 3 y + 12 + x = − x + x + 6

Step 4 - Simplify.

3 y + 12 + x = 6 3 y + 12 + x = 6

Step 5 - Bring the constants to the right side.

Subtract 12 from both sides.

3 y + 12 + x − 12 = 6 − 12 3 y + 12 + x − 12 = 6 − 12

Step 6 - Simplify.

3 y + x = − 6 3 y + x = − 6

Step 7 - Write the x x term before the y y term.

3 x + y = − 6 3 x + y = − 6

Try it

Try It: Writing Linear Equations in Different Forms

Write the equation y − 3 = 2 ( x + 5 ) y − 3 = 2 ( x + 5 ) in standard form.

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