Algebra Examples

$(- 3, 7)$ and $(6, - 8)$

Step 1

Find the slope of the line between $(- 3, 7)$ and $(6, - 8)$ 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.

$m = \frac{- 8 - (7)}{6 - (- 3)}$

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 $7$ .

$m = \frac{- 8 - 7}{6 - (- 3)}$

Step 1.4.1.2

Subtract $7$ from $- 8$ .

$m = \frac{- 15}{6 - (- 3)}$

Step 1.4.2

Simplify the denominator.

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Step 1.4.2.1

Multiply $- 1$ by $- 3$ .

$m = \frac{- 15}{6 + 3}$

Step 1.4.2.2

Add $6$ and $3$ .

$m = \frac{- 15}{9}$

Step 1.4.3

Reduce the expression by cancelling the common factors.

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Step 1.4.3.1

Cancel the common factor of $- 15$ and $9$ .

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Step 1.4.3.1.1

Factor $3$ out of $- 15$ .

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

Step 1.4.3.1.2

Cancel the common factors.

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Step 1.4.3.1.2.1

Factor $3$ out of $9$ .

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

Step 1.4.3.1.2.2

Cancel the common factor.

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

Step 1.4.3.1.2.3

Rewrite the expression.

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

Step 1.4.3.2

Move the negative in front of the fraction.

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

Step 2

Use the slope $- \frac{5}{3}$ and a given point $(- 3, 7)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 3

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

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

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

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Step 4.1.1.1

Rewrite.

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

Step 4.1.1.2

Simplify terms.

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Step 4.1.1.2.1

Apply the distributive property.

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

Step 4.1.1.2.2

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

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

Step 4.1.1.2.3

Cancel the common factor of $3$ .

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Step 4.1.1.2.3.1

Move the leading negative in $- \frac{5}{3}$ into the numerator.

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

Step 4.1.1.2.3.2

Cancel the common factor.

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

Step 4.1.1.2.3.3

Rewrite the expression.

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

Step 4.1.1.3

Move $5$ to the left of $x$ .

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

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 $7$ to both sides of the equation.

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

Step 4.1.2.2

Add $- 5$ and $7$ .

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

Step 4.2

Reorder terms.

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

Step 4.3

Remove parentheses.

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

Step 5