Writing Linear Equations

4.7.3 Writing Linear Equations

4.7.3 • Writing Linear Equations

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
$y = m x + b$ Given: $m =$ slope $b = y$ -intercept	$y - y_{1} = m (x - x_{1})$ Given: a point $(x_{1}, y_{1})$ $m =$ slope	$A x + B y = C$ $m = - \frac{A}{B} =$ slope Where: There are no fractions and $A$ cannot be negative.

Slope-intercept form

Point-slope form

Standard form

$y = m x + b$

Given:

$m =$ slope

$b = y$ -intercept

$y - y_{1} = m (x - x_{1})$

Given:

a point $(x_{1}, y_{1})$

$m =$ slope

$A x + B y = C$

$m = - \frac{A}{B} =$ slope

Where:

There are no fractions and $A$ cannot be negative.

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

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

Compare your answer:

$y = - 2 x + 4$

Substitute the slope in for $m$ and the $y$ -value of the $y$ -intercept in for $b$ in the formula $y = m x + b$ .

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 $\frac{1}{2}$ and goes through the point $(3, - 5)$

Compare your answer:

$y + 5 = \frac{1}{2} (x - 3)$

Substitute in the formula for $(x_{1}, y_{1})$ and slope.

$y - y_{1} = m (x - x_{1})$

$y - (- 5) = \frac{1}{2} (x - 3)$

$y + 5 = \frac{1}{2} (x - 3)$

Standard form of a linear equation is often used to find $x$ - and $y$ -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 $y = 2 x + 3$ ?

Compare your answer:

The slope is 2.

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

Compare your answer:

$2 x - y = - 3$

Write the equation in the form $A x + B y = C$

Step 1 - Rearrange terms so the $x$ term and the $y$ term are both on the left side of the equation. Subtract $2 x$ from both sides:

$y - 2 x = 2 x - 2 x + 3$

Step 2 - Write the $x$ term first and simplify.

$- 2 x + y = 3$

Step 3 - Multiply both sides by -1, so as not to lead with a negative.

$- 1 (- 2 x + y = 3)$

$2 x - y = - 3$

Work with a partner to write equations in different forms.

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

Compare your answer:

The slope is 2.

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

Compare your answer:

$y = 2 x + 11$

$y - 3 = 2 (x + 4)$

Step 1 - Distribute.

$y - 3 = 2 x + 8$

Step 2 - Add 3 to each side to solve for $y$ .

$y - 3 + 3 = 2 x + 8 + 3$

Step 3 - Simplify.

$y = 2 x + 11$

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

Compare your answer:

$3 x + y = 1$

Step 1 - Distribute. $y + 5 = - 3 x + 6$

Step 2 - Rearrange terms so the $x$ term and the $y$ term are both on the left side of the equation.

Add $3 x$ to both sides.

$y + 5 + 3 x = - 3 x + 3 x + 6$

Step 3 - Bring the constants to the right side of the equation.

Subtract 5 from each side.

$y + 5 - 5 + 3 x = - 3 x + 3 x + 6 - 5$

Step 4 - Simplify.

Make sure that the $x$ term is in front of the $y$ term on the left side.

$3 x + y = 1$

8. Write the equation $y = \frac{1}{3} x - 4$ in standard form.

Compare your answer:

$x - 3 y = 12$

Step 1 - Get rid of the fractions in the equation.

Multiply every term by 3.

$3 y = 3 (\frac{1}{3} x - 4)$

Step 2 - Distribute and simplify.

$3 y = x - 12$

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

Subtract $x$ from both sides.

$3 y - x = x - x - 12$

Step 4 - Simplify.

$3 y - x = - 12$

Step 5 - Write the equation so the $x$ term is first.

$- x + 3 y = - 12$

Step 6 - Multiply each term by -1, so as not to lead with a negative.

$- 1 (- x + 3 y = - 12)$

$x - 3 y = 12$

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

Compare your answer:

The slope is $m = - \frac{A}{B}$ . $m = - \frac{3}{12}$ $m = - \frac{1}{4}$

Video: Writing Equations in Different Forms

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

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Self Check

Write the equation of the line in point-slope form that has a slope of 7 and goes through

(-1,2)

$2-y=7(-1-x)$
$y-2=7(x-1)$
$y-2=7(x+1)$
$y = 7x + 9$

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 $m$ .

Equation	Equation formula	Slope formula	Slope value
$y = 5 x - 15$	$y = m x + b$	$m$	5
$y = 6 x + 8$	$y = m x + b$	$m$	6
$y - 7 = 9 (x + 1)$	$y - y_{1} = m (x - x_{1})$	$m$	9
$y - 20 = 3 (x + 16)$	$y - y_{1} = m (x - x_{1})$	$m$	3
$18 x - 6 y = 24$	$A x + B y = C$	$m = - \frac{A}{B}$	$- (\frac{18}{- 6}) = 3$
$30 x + 15 y = 24$	$A x + B y = C$	$m = - \frac{A}{B}$	$- \frac{30}{15} = - 2$

Example 2

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

Step 1 - Identify the $m$ and $b$ .

$m = - 9$ , $b = - 4$

Step 2 - Substitute into the formula.

$y = - 9 x - 4$

Example 3

Find the equation of a line in point-slope form given a slope of $- \frac{1}{3}$ and the point $(6, - 4)$

Step 1 - Identify the $m$ and $(x_{1}, y_{1})$ .

$m = - \frac{1}{3} x_{1} = 6 y_{1} = - 4$

Step 2 - Substitute into the formula.

$y - (- 4) = - \frac{1}{3} (x - 6)$

Step 3 - Simplify

$y + 4 = - \frac{1}{3} (x - 6)$

Example 4

Write the equation $y + 4 = - \frac{1}{3} (x - 6)$ 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.

$3 (y + 4) = 3 (- \frac{1}{3} (x - 6))$

$3 y + 12 = - (x - 6)$

Step 2 - Distribute.

$3 y + 12 = - x + 6$

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

Add $x$ to both sides.

$3 y + 12 + x = - x + x + 6$

Step 4 - Simplify.

$3 y + 12 + x = 6$

Step 5 - Bring the constants to the right side.

Subtract 12 from both sides.

$3 y + 12 + x - 12 = 6 - 12$

Step 6 - Simplify.

$3 y + x = - 6$

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

$3 x + y = - 6$

Try it

Writing Linear Equations in Different Forms

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

Compare your answer: $2 x - y = - 13$

Step 1 - Distribute.

$y - 3 = 2 x + 10$

Step 2 - Rearrange terms so the $x$ term and the $y$ term are both on the left side of the equation.

Subtract $2 x$ from both sides.

$y - 3 - 2 x = 2 x - 2 x + 10$

Step 3 - Simplify.

$y - 3 - 2 x = 10$

Step 4 - Bring the constants to the right side.

Add 3 to both sides.

$y - 3 - 2 x + 3 = 10 + 3$

Step 5 - Simplify.

$y - 2 x = 13$

Step 6 - Write the $x$ term before the $y$ term.

$- 2 x + y = 13$

Step 7 - Multiply each term by -1.

Standard form cannot lead with a negative.

$- 1 (- 2 x + y = 13)$

Step 8 - Simplify.

$2 x - y = - 13$