Precalculus Examples

$(5, 4)$ , $(- 9, 7)$

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.

$m = \frac{7 - (4)}{- 9 - (5)}$

Step 5

Finding the slope $m$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Multiply $- 1$ by $4$ .

$m = \frac{7 - 4}{- 9 - (5)}$

Step 5.1.2

Subtract $4$ from $7$ .

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

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply $- 1$ by $5$ .

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

Step 5.2.2

Subtract $5$ from $- 9$ .

$m = \frac{3}{- 14}$

Step 5.3

Move the negative in front of the fraction.

$m = - \frac{3}{14}$

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.

$y = (- \frac{3}{14}) \cdot x + b$

Step 6.3

Substitute the value of $x$ into the equation.

$y = (- \frac{3}{14}) \cdot (5) + b$

Step 6.4

Substitute the value of $y$ into the equation.

$4 = (- \frac{3}{14}) \cdot (5) + b$

Step 6.5

Find the value of $b$ .

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Step 6.5.1

Rewrite the equation as $- \frac{3}{14} \cdot 5 + b = 4$ .

$- \frac{3}{14} \cdot 5 + b = 4$

Step 6.5.2

Simplify each term.

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Step 6.5.2.1

Multiply $- \frac{3}{14} \cdot 5$ .

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Step 6.5.2.1.1

Multiply $5$ by $- 1$ .

$- 5 (\frac{3}{14}) + b = 4$

Step 6.5.2.1.2

Combine $- 5$ and $\frac{3}{14}$ .

$\frac{- 5 \cdot 3}{14} + b = 4$

Step 6.5.2.1.3

Multiply $- 5$ by $3$ .

$\frac{- 15}{14} + b = 4$

Step 6.5.2.2

Move the negative in front of the fraction.

$- \frac{15}{14} + b = 4$

Step 6.5.3

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

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Step 6.5.3.1

Add $\frac{15}{14}$ to both sides of the equation.

$b = 4 + \frac{15}{14}$

Step 6.5.3.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{14}{14}$ .

$b = 4 \cdot \frac{14}{14} + \frac{15}{14}$

Step 6.5.3.3

Combine $4$ and $\frac{14}{14}$ .

$b = \frac{4 \cdot 14}{14} + \frac{15}{14}$

Step 6.5.3.4

Combine the numerators over the common denominator.

$b = \frac{4 \cdot 14 + 15}{14}$

Step 6.5.3.5

Simplify the numerator.

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Step 6.5.3.5.1

Multiply $4$ by $14$ .

$b = \frac{56 + 15}{14}$

Step 6.5.3.5.2

Add $56$ and $15$ .

$b = \frac{71}{14}$

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.

$y = - \frac{3}{14} x + \frac{71}{14}$

Step 8

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