Precalculus Examples

$(- 5, 9)$ , $(3, 0)$

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.

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

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

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).

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

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

$m = \frac{0 - (9)}{3 - (- 5)}$

Finding the slope $m$ .

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Simplify the numerator.

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Multiply $- 1$ by $9$ .

$m = \frac{0 - 9}{3 - (- 5)}$

Subtract $9$ from $0$ .

$m = \frac{- 9}{3 - (- 5)}$

Simplify the denominator.

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Multiply $- 1$ by $- 5$ .

$m = \frac{- 9}{3 + 5}$

Add $3$ and $5$ .

$m = \frac{- 9}{8}$

Move the negative in front of the fraction.

$m = - \frac{9}{8}$

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

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

$y = m x + b$

Substitute the value of $m$ into the equation.

$y = (- \frac{9}{8}) \cdot x + b$

Substitute the value of $x$ into the equation.

$y = (- \frac{9}{8}) \cdot (- 5) + b$

Substitute the value of $y$ into the equation.

$9 = (- \frac{9}{8}) \cdot (- 5) + b$

Find the value of $b$ .

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Rewrite the equation as $(- \frac{9}{8}) \cdot (- 5) + b = 9$ .

$(- \frac{9}{8}) \cdot (- 5) + b = 9$

Multiply $(- \frac{9}{8}) (- 5)$ .

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Multiply $- 5$ by $- 1$ .

$5 (\frac{9}{8}) + b = 9$

Combine $5$ and $\frac{9}{8}$ .

$\frac{5 \cdot 9}{8} + b = 9$

Multiply $5$ by $9$ .

$\frac{45}{8} + b = 9$

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

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Subtract $\frac{45}{8}$ from both sides of the equation.

$b = 9 - \frac{45}{8}$

Simplify the right side of the equation.

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To write $9$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$b = 9 \cdot \frac{8}{8} - \frac{45}{8}$

Combine $9$ and $\frac{8}{8}$ .

$b = \frac{9 \cdot 8}{8} - \frac{45}{8}$

Combine the numerators over the common denominator.

$b = \frac{9 \cdot 8 - 45}{8}$

Simplify the numerator.

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Multiply $9$ by $8$ .

$b = \frac{72 - 45}{8}$

Subtract $45$ from $72$ .

$b = \frac{27}{8}$

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.

$y = - \frac{9}{8} x + \frac{27}{8}$

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