Precalculus Examples

$(\frac{1}{3}, \frac{4}{7})$ , $(1, 0)$

Use $y = m x + b$ to calculate the equation of the line, where $m$ represents the slope and $b$ represents the y-intercept.

To calculate the equation of the line, use the $y = m x + b$ format.

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.

$m = \frac{0 - (\frac{4}{7})}{1 - (\frac{1}{3})}$

Finding the slope m.

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Multiply the numerator and denominator of the complex fraction by $21$ .

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Multiply $\frac{0 - (\frac{4}{7})}{1 - (\frac{1}{3})}$ by $\frac{21}{21}$ .

$m = \frac{21}{21} \cdot \frac{0 - (\frac{4}{7})}{1 - (\frac{1}{3})}$

Combine.

$m = \frac{21 (0 - (\frac{4}{7}))}{21 (1 - (\frac{1}{3}))}$

Apply the distributive property.

$m = \frac{21 \cdot 0 + 21 (- (\frac{4}{7}))}{21 \cdot 1 + 21 (- (\frac{1}{3}))}$

Simplify by cancelling.

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Cancel the common factor of $7$ .

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Rewrite.

$m = \frac{21 \cdot 0 + 21 (\frac{- 1 \cdot 4}{7})}{21 \cdot 1 + 21 (- (\frac{1}{3}))}$

Write $21$ as a fraction with denominator $1$ .

$m = \frac{21 \cdot 0 + \frac{21}{1} \cdot \frac{- 1 \cdot 4}{7}}{21 \cdot 1 + 21 (- (\frac{1}{3}))}$

Factor out the greatest common factor $7$ .

$m = \frac{21 \cdot 0 + \frac{7 \cdot 3}{1} \cdot \frac{- 1 \cdot 4}{7 \cdot 1}}{21 \cdot 1 + 21 (- (\frac{1}{3}))}$

Cancel the common factor.

$m = \frac{21 \cdot 0 + \frac{7 \cdot 3}{1} \cdot \frac{- 1 \cdot 4}{7 \cdot 1}}{21 \cdot 1 + 21 (- (\frac{1}{3}))}$

Rewrite the expression.

$m = \frac{21 \cdot 0 + \frac{3}{1} \cdot \frac{- 1 \cdot 4}{1}}{21 \cdot 1 + 21 (- (\frac{1}{3}))}$

Simplify.

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Multiply $- 1$ by $4$ to get $- 4$ .

$m = \frac{21 \cdot 0 + \frac{3}{1} \cdot \frac{- 4}{1}}{21 \cdot 1 + 21 (- (\frac{1}{3}))}$

Multiply $\frac{3}{1}$ and $\frac{- 4}{1}$ to get $\frac{3 \cdot - 4}{1}$ .

$m = \frac{21 \cdot 0 + \frac{3 \cdot - 4}{1}}{21 \cdot 1 + 21 (- (\frac{1}{3}))}$

Multiply $3$ by $- 4$ to get $- 12$ .

$m = \frac{21 \cdot 0 + \frac{- 12}{1}}{21 \cdot 1 + 21 (- (\frac{1}{3}))}$

Divide $- 12$ by $1$ to get $- 12$ .

$m = \frac{21 \cdot 0 - 12}{21 \cdot 1 + 21 (- (\frac{1}{3}))}$

Cancel the common factor of $3$ .

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Rewrite.

$m = \frac{21 \cdot 0 - 12}{21 \cdot 1 + 21 (\frac{- 1 \cdot 1}{3})}$

Write $21$ as a fraction with denominator $1$ .

$m = \frac{21 \cdot 0 - 12}{21 \cdot 1 + \frac{21}{1} \cdot \frac{- 1 \cdot 1}{3}}$

Factor out the greatest common factor $3$ .

$m = \frac{21 \cdot 0 - 12}{21 \cdot 1 + \frac{3 \cdot 7}{1} \cdot \frac{- 1 \cdot 1}{3 \cdot 1}}$

Cancel the common factor.

$m = \frac{21 \cdot 0 - 12}{21 \cdot 1 + \frac{3 \cdot 7}{1} \cdot \frac{- 1 \cdot 1}{3 \cdot 1}}$

Rewrite the expression.

$m = \frac{21 \cdot 0 - 12}{21 \cdot 1 + \frac{7}{1} \cdot \frac{- 1 \cdot 1}{1}}$

Reduce the expression $\frac{- 1 \cdot 1}{1}$ by cancelling the common factors.

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Cancel the common factor.

$m = \frac{21 \cdot 0 - 12}{21 \cdot 1 + \frac{7}{1} \cdot \frac{- 1 \cdot 1}{1}}$

Rewrite the expression.

$m = \frac{21 \cdot 0 - 12}{21 \cdot 1 + \frac{7}{1} \cdot \frac{- 1}{1}}$

Simplify.

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Multiply $\frac{7}{1}$ and $\frac{- 1}{1}$ to get $\frac{7 \cdot - 1}{1}$ .

$m = \frac{21 \cdot 0 - 12}{21 \cdot 1 + \frac{7 \cdot - 1}{1}}$

Multiply $7$ by $- 1$ to get $- 7$ .

$m = \frac{21 \cdot 0 - 12}{21 \cdot 1 + \frac{- 7}{1}}$

Divide $- 7$ by $1$ to get $- 7$ .

$m = \frac{21 \cdot 0 - 12}{21 \cdot 1 - 7}$

Simplify the numerator.

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Multiply $21$ by $0$ to get $0$ .

$m = \frac{0 - 12}{21 \cdot 1 - 7}$

Subtract $12$ from $0$ to get $- 12$ .

$m = \frac{- 12}{21 \cdot 1 - 7}$

Simplify the denominator.

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Rewrite.

$m = \frac{- 12}{21 (1 - (\frac{1}{3}))}$

Apply the distributive property.

$m = \frac{- 12}{21 \cdot 1 + 21 (- (\frac{1}{3}))}$

Cancel the common factor of $3$ .

$m = \frac{- 12}{21 \cdot 1 + \frac{7}{1} \cdot \frac{- 1 \cdot 1}{1}}$

Reduce the expression $\frac{- 1 \cdot 1}{1}$ by cancelling the common factors.

$m = \frac{- 12}{21 \cdot 1 + \frac{7}{1} \cdot \frac{- 1}{1}}$

Simplify.

$m = \frac{- 12}{21 \cdot 1 - 7}$

Multiply $21$ by $1$ to get $21$ .

$m = \frac{- 12}{21 - 7}$

Subtract $7$ from $21$ to get $14$ .

$m = \frac{- 12}{14}$

Reduce the expression by cancelling the common factors.

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Reduce the expression by cancelling the common factors.

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Factor $2$ out of $14$ .

$m = \frac{2 \cdot - 6}{2 \cdot 7}$

Cancel the common factor.

$m = \frac{2 \cdot - 6}{2 \cdot 7}$

Rewrite the expression.

$m = \frac{- 6}{7}$

Move the negative in front of the fraction.

$m = - \frac{6}{7}$

Find the value of $b$ using the formula for the equation of a line.

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Use the formula for the equation of a line to find $b$ .

$y = m x + b$

Substitute the value of $m$ into the equation.

$y = (- \frac{6}{7}) \cdot x + b$

Substitute the value of $x$ into the equation.

$y = (- \frac{6}{7}) \cdot (\frac{1}{3}) + b$

Substitute the value of $y$ into the equation.

$(\frac{4}{7}) = (- \frac{6}{7}) \cdot (\frac{1}{3}) + b$

Find the value of $b$ .

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Rewrite the equation as $(- \frac{6}{7}) \cdot (\frac{1}{3}) + b = \frac{4}{7}$ .

$(- \frac{6}{7}) \cdot (\frac{1}{3}) + b = \frac{4}{7}$

Simplify each term.

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Multiply $- \frac{6}{7}$ by $\frac{1}{3}$ to get $(- \frac{6}{7}) (\frac{1}{3})$ .

$(- \frac{6}{7}) (\frac{1}{3}) + b = \frac{4}{7}$

Simplify $(- \frac{6}{7}) (\frac{1}{3})$ .

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Multiply $\frac{1}{3}$ and $\frac{6}{7}$ to get $\frac{6}{3 \cdot 7}$ .

$- \frac{6}{3 \cdot 7} + b = \frac{4}{7}$

Multiply $3$ by $7$ to get $21$ .

$- \frac{6}{21} + b = \frac{4}{7}$

Reduce the expression by cancelling the common factors.

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Factor $3$ out of $21$ .

$- \frac{3 \cdot 2}{3 \cdot 7} + b = \frac{4}{7}$

Cancel the common factor.

$- \frac{3 \cdot 2}{3 \cdot 7} + b = \frac{4}{7}$

Rewrite the expression.

$- \frac{2}{7} + b = \frac{4}{7}$

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

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Since $- \frac{2}{7}$ does not contain the variable to solve for, move it to the right side of the equation by adding $\frac{2}{7}$ to both sides.

$b = \frac{2}{7} + \frac{4}{7}$

Simplify the right side of the equation.

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Combine the numerators over the common denominator.

$b = \frac{2 + 4}{7}$

Add $2$ and $4$ to get $6$ .

$b = \frac{6}{7}$

Now that the values of $m$ (slope) and $b$ (y-intercept) are known, substitute them into $y = m x + b$ to find the equation of the line.

$y = - \frac{6}{7} x + \frac{6}{7}$

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Precalculus Examples

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