Precalculus Examples

$(5, y)$ , $(- 4, 4)$

Step 1

Use $y = m x + b$ to calculate the equation of the line, where $m$ represents the slope and $b$ represents the y-intercept.

To calculate the equation of the line, use the $y = m x + b$ format.

Step 2

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{(change in y)}{(change in x)}$

Step 3

The change in $x$ is equal to the difference in x-coordinates (also called run), and the change in $y$ is equal to the difference in y-coordinates (also called rise).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Step 4

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{4 - (y)}{- 4 - (5)}$

Step 5

Finding the slope $m$ .

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Step 5.1

Simplify the denominator.

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Step 5.1.1

Multiply $- 1$ by $5$ .

$m = \frac{4 - y}{- 4 - 5}$

Step 5.1.2

Subtract $5$ from $- 4$ .

$m = \frac{4 - y}{- 9}$

Step 5.2

Move the negative in front of the fraction.

$m = - \frac{4 - y}{9}$

Step 6

Find the value of $b$ using the formula for the equation of a line.

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Step 6.1

Use the formula for the equation of a line to find $b$ .

$y = m x + b$

Step 6.2

Substitute the value of $m$ into the equation.

$y = (- \frac{4 - y}{9}) \cdot x + b$

Step 6.3

Substitute the value of $x$ into the equation.

$y = (- \frac{4 - y}{9}) \cdot (5) + b$

Step 6.4

Substitute the value of $y$ into the equation.

$y = (- \frac{4 - (y)}{9}) \cdot (5) + b$

Step 6.5

Find the value of $b$ .

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Step 6.5.1

Rewrite the equation as $- \frac{4 - y}{9} \cdot 5 + b = y$ .

$- \frac{4 - y}{9} \cdot 5 + b = y$

Step 6.5.2

Simplify $- \frac{4 - y}{9} \cdot 5 + b$ .

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Step 6.5.2.1

Simplify each term.

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Step 6.5.2.1.1

Multiply $- \frac{4 - y}{9} \cdot 5$ .

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Step 6.5.2.1.1.1

Multiply $5$ by $- 1$ .

$- 5 \frac{4 - y}{9} + b = y$

Step 6.5.2.1.1.2

Combine $- 5$ and $\frac{4 - y}{9}$ .

$\frac{- 5 (4 - y)}{9} + b = y$

Step 6.5.2.1.2

Move the negative in front of the fraction.

$- \frac{5 (4 - y)}{9} + b = y$

Step 6.5.2.2

To write $b$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$- \frac{5 (4 - y)}{9} + b \cdot \frac{9}{9} = y$

Step 6.5.2.3

Simplify terms.

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Step 6.5.2.3.1

Combine $b$ and $\frac{9}{9}$ .

$- \frac{5 (4 - y)}{9} + \frac{b \cdot 9}{9} = y$

Step 6.5.2.3.2

Combine the numerators over the common denominator.

$\frac{- 5 (4 - y) + b \cdot 9}{9} = y$

Step 6.5.2.4

Simplify the numerator.

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Step 6.5.2.4.1

Apply the distributive property.

$\frac{- 5 \cdot 4 - 5 (- y) + b \cdot 9}{9} = y$

Step 6.5.2.4.2

Multiply $- 5$ by $4$ .

$\frac{- 20 - 5 (- y) + b \cdot 9}{9} = y$

Step 6.5.2.4.3

Multiply $- 1$ by $- 5$ .

$\frac{- 20 + 5 y + b \cdot 9}{9} = y$

Step 6.5.2.4.4

Move $9$ to the left of $b$ .

$\frac{- 20 + 5 y + 9 b}{9} = y$

Step 6.5.2.5

Simplify with factoring out.

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Step 6.5.2.5.1

Rewrite $- 20$ as $- 1 (20)$ .

$\frac{- 1 (20) + 5 y + 9 b}{9} = y$

Step 6.5.2.5.2

Factor $- 1$ out of $5 y$ .

$\frac{- 1 (20) - (- 5 y) + 9 b}{9} = y$

Step 6.5.2.5.3

Factor $- 1$ out of $- 1 (20) - (- 5 y)$ .

$\frac{- 1 (20 - 5 y) + 9 b}{9} = y$

Step 6.5.2.5.4

Factor $- 1$ out of $9 b$ .

$\frac{- 1 (20 - 5 y) - (- 9 b)}{9} = y$

Step 6.5.2.5.5

Factor $- 1$ out of $- 1 (20 - 5 y) - (- 9 b)$ .

$\frac{- 1 (20 - 5 y - 9 b)}{9} = y$

Step 6.5.2.5.6

Move the negative in front of the fraction.

$- \frac{20 - 5 y - 9 b}{9} = y$

Step 6.5.3

Multiply both sides of the equation by $- 9$ .

$- 9 (- \frac{20 - 5 y - 9 b}{9}) = - 9 y$

Step 6.5.4

Simplify the left side.

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Step 6.5.4.1

Simplify $- 9 (- \frac{20 - 5 y - 9 b}{9})$ .

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Step 6.5.4.1.1

Cancel the common factor of $9$ .

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Step 6.5.4.1.1.1

Move the leading negative in $- \frac{20 - 5 y - 9 b}{9}$ into the numerator.

$- 9 \frac{- (20 - 5 y - 9 b)}{9} = - 9 y$

Step 6.5.4.1.1.2

Factor $9$ out of $- 9$ .

$9 (- 1) \frac{- (20 - 5 y - 9 b)}{9} = - 9 y$

Step 6.5.4.1.1.3

Cancel the common factor.

$9 \cdot - 1 \frac{- (20 - 5 y - 9 b)}{9} = - 9 y$

Step 6.5.4.1.1.4

Rewrite the expression.

$- - (20 - 5 y - 9 b) = - 9 y$

Step 6.5.4.1.2

Multiply.

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Step 6.5.4.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (20 - 5 y - 9 b) = - 9 y$

Step 6.5.4.1.2.2

Multiply $20 - 5 y - 9 b$ by $1$ .

$20 - 5 y - 9 b = - 9 y$

Step 6.5.5

Move all terms not containing $b$ to the right side of the equation.

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Step 6.5.5.1

Subtract $20$ from both sides of the equation.

$- 5 y - 9 b = - 9 y - 20$

Step 6.5.5.2

Add $5 y$ to both sides of the equation.

$- 9 b = - 9 y - 20 + 5 y$

Step 6.5.5.3

Add $- 9 y$ and $5 y$ .

$- 9 b = - 4 y - 20$

Step 6.5.6

Divide each term in $- 9 b = - 4 y - 20$ by $- 9$ and simplify.

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Step 6.5.6.1

Divide each term in $- 9 b = - 4 y - 20$ by $- 9$ .

$\frac{- 9 b}{- 9} = \frac{- 4 y}{- 9} + \frac{- 20}{- 9}$

Step 6.5.6.2

Simplify the left side.

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Step 6.5.6.2.1

Cancel the common factor of $- 9$ .

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Step 6.5.6.2.1.1

Cancel the common factor.

$\frac{- 9 b}{- 9} = \frac{- 4 y}{- 9} + \frac{- 20}{- 9}$

Step 6.5.6.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 4 y}{- 9} + \frac{- 20}{- 9}$

Step 6.5.6.3

Simplify the right side.

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Step 6.5.6.3.1

Simplify each term.

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Step 6.5.6.3.1.1

Dividing two negative values results in a positive value.

$b = \frac{4 y}{9} + \frac{- 20}{- 9}$

Step 6.5.6.3.1.2

Dividing two negative values results in a positive value.

$b = \frac{4 y}{9} + \frac{20}{9}$

Step 7

Now that the values of $m$ (slope) and $b$ (y-intercept) are known, substitute them into $y = m x + b$ to find the equation of the line.

$y = - \frac{1}{9} x (4 - y) + \frac{4}{9} y + \frac{20}{9}$

Step 8