Precalculus Examples

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

Step 1

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.

Step 2

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 3

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 4

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

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

Step 5

Finding the slope $m$ .

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Combine.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Simplify by cancelling.

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Step 5.3.1

Cancel the common factor of $7$ .

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Step 5.3.1.1

Move the leading negative in $- \frac{4}{7}$ into the numerator.

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

Step 5.3.1.2

Factor $7$ out of $21$ .

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

Step 5.3.1.3

Cancel the common factor.

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

Step 5.3.1.4

Rewrite the expression.

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

Step 5.3.2

Multiply $3$ by $- 4$ .

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

Step 5.3.3

Cancel the common factor of $3$ .

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Step 5.3.3.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

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

Step 5.3.3.2

Factor $3$ out of $21$ .

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

Step 5.3.3.3

Cancel the common factor.

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

Step 5.3.3.4

Rewrite the expression.

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

Step 5.3.4

Multiply $7$ by $- 1$ .

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

Step 5.4

Simplify the numerator.

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Step 5.4.1

Multiply $21$ by $0$ .

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

Step 5.4.2

Subtract $12$ from $0$ .

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

Step 5.5

Simplify the denominator.

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Step 5.5.1

Multiply $21$ by $1$ .

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

Step 5.5.2

Subtract $7$ from $21$ .

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

Step 5.6

Reduce the expression by cancelling the common factors.

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Step 5.6.1

Cancel the common factor of $- 12$ and $14$ .

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Step 5.6.1.1

Factor $2$ out of $- 12$ .

$m = \frac{2 (- 6)}{14}$

Step 5.6.1.2

Cancel the common factors.

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Step 5.6.1.2.1

Factor $2$ out of $14$ .

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

Step 5.6.1.2.2

Cancel the common factor.

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

Step 5.6.1.2.3

Rewrite the expression.

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

Step 5.6.2

Move the negative in front of the fraction.

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

Step 6

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

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Step 6.1

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

$y = m x + b$

Step 6.2

Substitute the value of $m$ into the equation.

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

Step 6.3

Substitute the value of $x$ into the equation.

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

Step 6.4

Substitute the value of $y$ into the equation.

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

Step 6.5

Find the value of $b$ .

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Step 6.5.1

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

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

Step 6.5.2

Simplify each term.

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Step 6.5.2.1

Cancel the common factor of $3$ .

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Step 6.5.2.1.1

Move the leading negative in $- \frac{6}{7}$ into the numerator.

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

Step 6.5.2.1.2

Factor $3$ out of $- 6$ .

$\frac{3 (- 2)}{7} \cdot \frac{1}{3} + b = \frac{4}{7}$

Step 6.5.2.1.3

Cancel the common factor.

$\frac{3 \cdot - 2}{7} \cdot \frac{1}{3} + b = \frac{4}{7}$

Step 6.5.2.1.4

Rewrite the expression.

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

Step 6.5.2.2

Move the negative in front of the fraction.

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

Step 6.5.3

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

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Step 6.5.3.1

Add $\frac{2}{7}$ to both sides of the equation.

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

Step 6.5.3.2

Combine the numerators over the common denominator.

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

Step 6.5.3.3

Add $4$ and $2$ .

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

Step 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.

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

Step 8

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