Precalculus Examples

$(5, 7)$ , $(- 1, - 3)$

Step 1

Find the slope of the line between $(5, 7)$ and $(- 1, - 3)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

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Step 1.1

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 1.2

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 1.3

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

$m = \frac{- 3 - (7)}{- 1 - (5)}$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Multiply $- 1$ by $7$ .

$m = \frac{- 3 - 7}{- 1 - (5)}$

Step 1.4.1.2

Subtract $7$ from $- 3$ .

$m = \frac{- 10}{- 1 - (5)}$

Step 1.4.2

Simplify the denominator.

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Step 1.4.2.1

Multiply $- 1$ by $5$ .

$m = \frac{- 10}{- 1 - 5}$

Step 1.4.2.2

Subtract $5$ from $- 1$ .

$m = \frac{- 10}{- 6}$

Step 1.4.3

Cancel the common factor of $- 10$ and $- 6$ .

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Step 1.4.3.1

Factor $- 2$ out of $- 10$ .

$m = \frac{- 2 \cdot 5}{- 6}$

Step 1.4.3.2

Cancel the common factors.

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Step 1.4.3.2.1

Factor $- 2$ out of $- 6$ .

$m = \frac{- 2 \cdot 5}{- 2 \cdot 3}$

Step 1.4.3.2.2

Cancel the common factor.

$m = \frac{- 2 \cdot 5}{- 2 \cdot 3}$

Step 1.4.3.2.3

Rewrite the expression.

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

Step 2

Use the slope $\frac{5}{3}$ and a given point $(5, 7)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (7) = \frac{5}{3} \cdot (x - (5))$

Step 3

Simplify the equation and keep it in point-slope form.

$y - 7 = \frac{5}{3} \cdot (x - 5)$

Step 4

Solve for $y$ .

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Step 4.1

Simplify $\frac{5}{3} \cdot (x - 5)$ .

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Step 4.1.1

Rewrite.

$y - 7 = 0 + 0 + \frac{5}{3} \cdot (x - 5)$

Step 4.1.2

Simplify by adding zeros.

$y - 7 = \frac{5}{3} \cdot (x - 5)$

Step 4.1.3

Apply the distributive property.

$y - 7 = \frac{5}{3} x + \frac{5}{3} \cdot - 5$

Step 4.1.4

Combine $\frac{5}{3}$ and $x$ .

$y - 7 = \frac{5 x}{3} + \frac{5}{3} \cdot - 5$

Step 4.1.5

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

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Step 4.1.5.1

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

$y - 7 = \frac{5 x}{3} + \frac{5 \cdot - 5}{3}$

Step 4.1.5.2

Multiply $5$ by $- 5$ .

$y - 7 = \frac{5 x}{3} + \frac{- 25}{3}$

Step 4.1.6

Move the negative in front of the fraction.

$y - 7 = \frac{5 x}{3} - \frac{25}{3}$

Step 4.2

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

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Step 4.2.1

Add $7$ to both sides of the equation.

$y = \frac{5 x}{3} - \frac{25}{3} + 7$

Step 4.2.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{5 x}{3} - \frac{25}{3} + 7 \cdot \frac{3}{3}$

Step 4.2.3

Combine $7$ and $\frac{3}{3}$ .

$y = \frac{5 x}{3} - \frac{25}{3} + \frac{7 \cdot 3}{3}$

Step 4.2.4

Combine the numerators over the common denominator.

$y = \frac{5 x}{3} + \frac{- 25 + 7 \cdot 3}{3}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $7$ by $3$ .

$y = \frac{5 x}{3} + \frac{- 25 + 21}{3}$

Step 4.2.5.2

Add $- 25$ and $21$ .

$y = \frac{5 x}{3} + \frac{- 4}{3}$

Step 4.2.6

Move the negative in front of the fraction.

$y = \frac{5 x}{3} - \frac{4}{3}$

Step 5

Reorder terms.

$y = \frac{5}{3} x - \frac{4}{3}$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{5}{3} x - \frac{4}{3}$

Point-slope form:

$y - 7 = \frac{5}{3} \cdot (x - 5)$

Step 7