Precalculus Examples

$(- \frac{1}{5}, 1)$ , $(- \frac{2}{5}, \frac{9}{10})$

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.

$m = \frac{\frac{9}{10} - (1)}{- \frac{2}{5} - (- \frac{1}{5})}$

Step 5

Finding the slope $m$ .

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Step 5.1

Multiply the numerator and denominator of the fraction by $10$ .

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Step 5.1.1

Multiply $\frac{\frac{9}{10} - (1)}{- \frac{2}{5} - - \frac{1}{5}}$ by $\frac{10}{10}$ .

$m = \frac{10}{10} \cdot \frac{\frac{9}{10} - (1)}{- \frac{2}{5} + \frac{1}{5}}$

Step 5.1.2

Combine.

$m = \frac{10 (\frac{9}{10} - (1))}{10 (- \frac{2}{5} + \frac{1}{5})}$

Step 5.2

Apply the distributive property.

$m = \frac{10 (\frac{9}{10}) + 10 (- (1))}{10 (- \frac{2}{5}) + 10 (\frac{1}{5})}$

Step 5.3

Simplify by cancelling.

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Step 5.3.1

Cancel the common factor of $10$ .

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Step 5.3.1.1

Cancel the common factor.

$m = \frac{10 (\frac{9}{10}) + 10 (- (1))}{10 (- \frac{2}{5}) + 10 (\frac{1}{5})}$

Step 5.3.1.2

Rewrite the expression.

$m = \frac{9 + 10 (- (1))}{10 (- \frac{2}{5}) + 10 (\frac{1}{5})}$

Step 5.3.2

Cancel the common factor of $5$ .

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Step 5.3.2.1

Move the leading negative in $- \frac{2}{5}$ into the numerator.

$m = \frac{9 + 10 (- (1))}{10 (\frac{- 2}{5}) + 10 (\frac{1}{5})}$

Step 5.3.2.2

Factor $5$ out of $10$ .

$m = \frac{9 + 10 (- (1))}{5 (2) (\frac{- 2}{5}) + 10 (\frac{1}{5})}$

Step 5.3.2.3

Cancel the common factor.

$m = \frac{9 + 10 (- (1))}{5 \cdot (2 (\frac{- 2}{5})) + 10 (\frac{1}{5})}$

Step 5.3.2.4

Rewrite the expression.

$m = \frac{9 + 10 (- (1))}{2 \cdot - 2 + 10 (\frac{1}{5})}$

Step 5.3.3

Multiply $2$ by $- 2$ .

$m = \frac{9 + 10 (- (1))}{- 4 + 10 (\frac{1}{5})}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Multiply $10 (- (1))$ .

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Step 5.4.1.1

Multiply $- 1$ by $1$ .

$m = \frac{9 + 10 \cdot - 1}{- 4 + 10 (\frac{1}{5})}$

Step 5.4.1.2

Multiply $10$ by $- 1$ .

$m = \frac{9 - 10}{- 4 + 10 (\frac{1}{5})}$

Step 5.4.2

Subtract $10$ from $9$ .

$m = \frac{- 1}{- 4 + 10 (\frac{1}{5})}$

Step 5.5

Simplify the denominator.

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Step 5.5.1

Multiply $- - \frac{1}{5}$ .

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Step 5.5.1.1

Multiply $- 1$ by $- 1$ .

$m = \frac{- 1}{- 4 + 10 (1 (\frac{1}{5}))}$

Step 5.5.1.2

Multiply $\frac{1}{5}$ by $1$ .

$m = \frac{- 1}{- 4 + 10 (\frac{1}{5})}$

Step 5.5.2

Cancel the common factor of $5$ .

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Step 5.5.2.1

Factor $5$ out of $10$ .

$m = \frac{- 1}{- 4 + 5 (2) (\frac{1}{5})}$

Step 5.5.2.2

Cancel the common factor.

$m = \frac{- 1}{- 4 + 5 \cdot (2 (\frac{1}{5}))}$

Step 5.5.2.3

Rewrite the expression.

$m = \frac{- 1}{- 4 + 2}$

Step 5.5.3

Add $- 4$ and $2$ .

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

Step 5.6

Dividing two negative values results in a positive value.

$m = \frac{1}{2}$

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.

$y = (\frac{1}{2}) x + b$

Step 6.3

Substitute the value of $x$ into the equation.

$y = (\frac{1}{2}) \cdot (- \frac{1}{5}) + b$

Step 6.4

Substitute the value of $y$ into the equation.

$1 = (\frac{1}{2}) \cdot (- \frac{1}{5}) + b$

Step 6.5

Find the value of $b$ .

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Step 6.5.1

Rewrite the equation as $\frac{1}{2} \cdot (- 1 (\frac{1}{5})) + b = 1$ .

$\frac{1}{2} \cdot (- 1 (\frac{1}{5})) + b = 1$

Step 6.5.2

Simplify each term.

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Step 6.5.2.1

Rewrite $- 1 (\frac{1}{5})$ as $- (\frac{1}{5})$ .

$\frac{1}{2} \cdot (- (\frac{1}{5})) + b = 1$

Step 6.5.2.2

Multiply $\frac{1}{2} (- \frac{1}{5})$ .

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Step 6.5.2.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{5}$ .

$- \frac{1}{2 \cdot 5} + b = 1$

Step 6.5.2.2.2

Multiply $2$ by $5$ .

$- \frac{1}{10} + b = 1$

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 $\frac{1}{10}$ to both sides of the equation.

$b = 1 + \frac{1}{10}$

Step 6.5.3.2

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

$b = \frac{10}{10} + \frac{1}{10}$

Step 6.5.3.3

Combine the numerators over the common denominator.

$b = \frac{10 + 1}{10}$

Step 6.5.3.4

Add $10$ and $1$ .

$b = \frac{11}{10}$

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.

$y = \frac{1}{2} x + \frac{11}{10}$

Step 8