Precalculus Examples

$(- 5, 5)$ , $(k, 0)$

Step 1

Find the slope of the line between $(- 5, 5)$ and $(k, 0)$ 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.

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

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 $5$ .

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

Step 1.4.1.2

Subtract $5$ from $0$ .

$m = \frac{- 5}{k - (- 5)}$

Step 1.4.2

Simplify the expression.

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Step 1.4.2.1

Multiply $- 1$ by $- 5$ .

$m = \frac{- 5}{k + 5}$

Step 1.4.2.2

Move the negative in front of the fraction.

$m = - \frac{5}{k + 5}$

Step 2

Use the slope $- \frac{5}{k + 5}$ and a given point $(- 5, 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = - \frac{5}{k + 5} \cdot (x - (- 5))$

Step 3

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

$y - 5 = - \frac{5}{k + 5} \cdot (x + 5)$

Step 4

Solve for $y$ .

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Step 4.1

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

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Step 4.1.1

Rewrite.

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

Step 4.1.2

Simplify by adding zeros.

$y - 5 = - \frac{5}{k + 5} \cdot (x + 5)$

Step 4.1.3

Apply the distributive property.

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

Step 4.1.4

Combine $x$ and $\frac{5}{k + 5}$ .

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

Step 4.1.5

Multiply $- \frac{5}{k + 5} \cdot 5$ .

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Step 4.1.5.1

Multiply $5$ by $- 1$ .

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

Step 4.1.5.2

Combine $- 5$ and $\frac{5}{k + 5}$ .

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

Step 4.1.5.3

Multiply $- 5$ by $5$ .

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

Step 4.1.6

Combine the numerators over the common denominator.

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

Step 4.1.7

Multiply $5$ by $- 1$ .

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

Step 4.1.8

Factor $5$ out of $- 5 x - 25$ .

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Step 4.1.8.1

Factor $5$ out of $- 5 x$ .

$y - 5 = \frac{5 (- x) - 25}{k + 5}$

Step 4.1.8.2

Factor $5$ out of $- 25$ .

$y - 5 = \frac{5 (- x) + 5 (- 5)}{k + 5}$

Step 4.1.8.3

Factor $5$ out of $5 (- x) + 5 (- 5)$ .

$y - 5 = \frac{5 (- x - 5)}{k + 5}$

Step 4.1.9

Factor $- 1$ out of $- x$ .

$y - 5 = \frac{5 (- (x) - 5)}{k + 5}$

Step 4.1.10

Rewrite $- 5$ as $- 1 (5)$ .

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

Step 4.1.11

Factor $- 1$ out of $- (x) - 1 (5)$ .

$y - 5 = \frac{5 (- (x + 5))}{k + 5}$

Step 4.1.12

Simplify the expression.

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Step 4.1.12.1

Rewrite $- (x + 5)$ as $- 1 (x + 5)$ .

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

Step 4.1.12.2

Move the negative in front of the fraction.

$y - 5 = - \frac{5 (x + 5)}{k + 5}$

Step 4.2

Add $5$ to both sides of the equation.

$y = - \frac{5 (x + 5)}{k + 5} + 5$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = - \frac{5 (x + 5)}{k + 5} + 5$

Point-slope form:

$y - 5 = - \frac{5}{k + 5} \cdot (x + 5)$

Step 6