Finite Math Examples

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

Find the slope of the line between $(5, 4)$ and $(- 9, 7)$ 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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Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

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).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

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

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

Simplify.

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Simplify the numerator.

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Multiply $- 1$ by $4$ .

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

Subtract $4$ from $7$ .

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

Simplify the denominator.

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Multiply $- 1$ by $5$ .

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

Subtract $5$ from $- 9$ .

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

Move the negative in front of the fraction.

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

Use the slope $m = - \frac{3}{14}$ and one of the given points such as $(5, 4)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m \cdot (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (4) = - \frac{3}{14} \cdot (x - (5))$

Solve for $y$ to get the equation.

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After finding the slope between the points, use point-slope form to set up the equation. Point-slope is derived from the equation for slope $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (4) = - \frac{3}{14} \cdot (x - (5))$

Multiply $- 1$ by $4$ .

$y - 4 = - \frac{3}{14} \cdot (x - (5))$

Simplify $- \frac{3}{14} \cdot (x - (5))$ .

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Simplify the expression.

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Multiply $- 1$ by $5$ .

$y - 4 = - \frac{3}{14} \cdot (x - 5)$

Apply the distributive property.

$y - 4 = - \frac{3}{14} x - \frac{3}{14} \cdot - 5$

Simplify $- \frac{3}{14} x$ .

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Write $x$ as a fraction with denominator $1$ .

$y - 4 = - (\frac{x}{1} \cdot \frac{3}{14}) - \frac{3}{14} \cdot - 5$

Multiply $\frac{x}{1}$ and $\frac{3}{14}$ .

$y - 4 = - \frac{x \cdot 3}{14} - \frac{3}{14} \cdot - 5$

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

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Multiply $- 5$ by $- 1$ .

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

Write $5$ as a fraction with denominator $1$ .

$y - 4 = - \frac{x \cdot 3}{14} + \frac{5}{1} \cdot \frac{3}{14}$

Multiply $\frac{5}{1}$ and $\frac{3}{14}$ .

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

Multiply $5$ by $3$ .

$y - 4 = - \frac{x \cdot 3}{14} + \frac{15}{14}$

Simplify each term.

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Move $3$ to the left of the expression $x \cdot 3$ .

$y - 4 = - \frac{3 \cdot x}{14} + \frac{15}{14}$

Multiply $3$ by $x$ .

$y - 4 = - \frac{3 x}{14} + \frac{15}{14}$

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

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Add $4$ to both sides of the equation.

$y = - \frac{3 x}{14} + \frac{15}{14} + 4$

Simplify the right side of the equation.

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To write $\frac{4}{1}$ as a fraction with a common denominator, multiply by $\frac{14}{14}$ .

$y = - \frac{3 x}{14} + (\frac{15}{14} + \frac{4}{1} \cdot \frac{14}{14})$

Write each expression with a common denominator of $14$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$y = - \frac{3 x}{14} + (\frac{15}{14} + \frac{4 \cdot 14}{1 \cdot 14})$

Multiply $14$ by $1$ .

$y = - \frac{3 x}{14} + (\frac{15}{14} + \frac{4 \cdot 14}{14})$

Combine the numerators over the common denominator.

$y = - \frac{3 x}{14} + \frac{15 + 4 \cdot 14}{14}$

Simplify the numerator.

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Multiply $4$ by $14$ .

$y = - \frac{3 x}{14} + \frac{15 + 56}{14}$

Add $15$ and $56$ .

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

The final answer is the equation in slope-intercept form.

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

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