Algebra Examples

$(- 10, 3)$ and $(- 8, - 8)$

Step 1

Find the slope of the line between $(- 10, 3)$ and $(- 8, - 8)$ 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.

$m = \frac{- 8 - (3)}{- 8 - (- 10)}$

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 $3$ .

$m = \frac{- 8 - 3}{- 8 - (- 10)}$

Step 1.4.1.2

Subtract $3$ from $- 8$ .

$m = \frac{- 11}{- 8 - (- 10)}$

Step 1.4.2

Simplify the denominator.

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Step 1.4.2.1

Multiply $- 1$ by $- 10$ .

$m = \frac{- 11}{- 8 + 10}$

Step 1.4.2.2

Add $- 8$ and $10$ .

$m = \frac{- 11}{2}$

Step 1.4.3

Move the negative in front of the fraction.

$m = - \frac{11}{2}$

Step 2

Use the slope $- \frac{11}{2}$ and a given point $(- 10, 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (3) = - \frac{11}{2} \cdot (x - (- 10))$

Step 3

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

$y - 3 = - \frac{11}{2} \cdot (x + 10)$

Step 4

Write the equation in $y = m x + b$ form.

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Step 4.1

Solve for $y$ .

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Step 4.1.1

Simplify $- \frac{11}{2} \cdot (x + 10)$ .

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Step 4.1.1.1

Rewrite.

$y - 3 = 0 + 0 - \frac{11}{2} \cdot (x + 10)$

Step 4.1.1.2

Simplify terms.

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Step 4.1.1.2.1

Apply the distributive property.

$y - 3 = - \frac{11}{2} x - \frac{11}{2} \cdot 10$

Step 4.1.1.2.2

Combine $x$ and $\frac{11}{2}$ .

$y - 3 = - \frac{x \cdot 11}{2} - \frac{11}{2} \cdot 10$

Step 4.1.1.2.3

Cancel the common factor of $2$ .

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Step 4.1.1.2.3.1

Move the leading negative in $- \frac{11}{2}$ into the numerator.

$y - 3 = - \frac{x \cdot 11}{2} + \frac{- 11}{2} \cdot 10$

Step 4.1.1.2.3.2

Factor $2$ out of $10$ .

$y - 3 = - \frac{x \cdot 11}{2} + \frac{- 11}{2} \cdot (2 (5))$

Step 4.1.1.2.3.3

Cancel the common factor.

$y - 3 = - \frac{x \cdot 11}{2} + \frac{- 11}{2} \cdot (2 \cdot 5)$

Step 4.1.1.2.3.4

Rewrite the expression.

$y - 3 = - \frac{x \cdot 11}{2} - 11 \cdot 5$

Step 4.1.1.2.4

Multiply $- 11$ by $5$ .

$y - 3 = - \frac{x \cdot 11}{2} - 55$

Step 4.1.1.3

Move $11$ to the left of $x$ .

$y - 3 = - \frac{11 x}{2} - 55$

Step 4.1.2

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

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Step 4.1.2.1

Add $3$ to both sides of the equation.

$y = - \frac{11 x}{2} - 55 + 3$

Step 4.1.2.2

Add $- 55$ and $3$ .

$y = - \frac{11 x}{2} - 52$

Step 4.2

Reorder terms.

$y = - (\frac{11}{2} x) - 52$

Step 4.3

Remove parentheses.

$y = - \frac{11}{2} x - 52$

Step 5