Precalculus Examples

$(\frac{3}{7}, \frac{9}{10})$ , $(- \frac{3}{7}, - \frac{3}{10})$

Step 1

Find the slope of the line between $(\frac{3}{7}, \frac{9}{10})$ and $(- \frac{3}{7}, - \frac{3}{10})$ 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.

$m = \frac{- \frac{3}{10} - (\frac{9}{10})}{- \frac{3}{7} - (\frac{3}{7})}$

Step 1.4

Simplify.

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Step 1.4.1

Multiply the numerator and denominator of the fraction by $70$ .

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Step 1.4.1.1

Multiply $\frac{- \frac{3}{10} - \frac{9}{10}}{- \frac{3}{7} - \frac{3}{7}}$ by $\frac{70}{70}$ .

$m = \frac{70}{70} \cdot \frac{- \frac{3}{10} - \frac{9}{10}}{- \frac{3}{7} - \frac{3}{7}}$

Step 1.4.1.2

Combine.

$m = \frac{70 (- \frac{3}{10} - \frac{9}{10})}{70 (- \frac{3}{7} - \frac{3}{7})}$

Step 1.4.2

Apply the distributive property.

$m = \frac{70 (- \frac{3}{10}) + 70 (- \frac{9}{10})}{70 (- \frac{3}{7}) + 70 (- \frac{3}{7})}$

Step 1.4.3

Simplify by cancelling.

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Step 1.4.3.1

Cancel the common factor of $10$ .

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Step 1.4.3.1.1

Move the leading negative in $- \frac{3}{10}$ into the numerator.

$m = \frac{70 (\frac{- 3}{10}) + 70 (- \frac{9}{10})}{70 (- \frac{3}{7}) + 70 (- \frac{3}{7})}$

Step 1.4.3.1.2

Factor $10$ out of $70$ .

$m = \frac{10 (7) (\frac{- 3}{10}) + 70 (- \frac{9}{10})}{70 (- \frac{3}{7}) + 70 (- \frac{3}{7})}$

Step 1.4.3.1.3

Cancel the common factor.

$m = \frac{10 \cdot (7 (\frac{- 3}{10})) + 70 (- \frac{9}{10})}{70 (- \frac{3}{7}) + 70 (- \frac{3}{7})}$

Step 1.4.3.1.4

Rewrite the expression.

$m = \frac{7 \cdot - 3 + 70 (- \frac{9}{10})}{70 (- \frac{3}{7}) + 70 (- \frac{3}{7})}$

Step 1.4.3.2

Multiply $7$ by $- 3$ .

$m = \frac{- 21 + 70 (- \frac{9}{10})}{70 (- \frac{3}{7}) + 70 (- \frac{3}{7})}$

Step 1.4.3.3

Cancel the common factor of $10$ .

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Step 1.4.3.3.1

Move the leading negative in $- \frac{9}{10}$ into the numerator.

$m = \frac{- 21 + 70 (\frac{- 9}{10})}{70 (- \frac{3}{7}) + 70 (- \frac{3}{7})}$

Step 1.4.3.3.2

Factor $10$ out of $70$ .

$m = \frac{- 21 + 10 (7) (\frac{- 9}{10})}{70 (- \frac{3}{7}) + 70 (- \frac{3}{7})}$

Step 1.4.3.3.3

Cancel the common factor.

$m = \frac{- 21 + 10 \cdot (7 (\frac{- 9}{10}))}{70 (- \frac{3}{7}) + 70 (- \frac{3}{7})}$

Step 1.4.3.3.4

Rewrite the expression.

$m = \frac{- 21 + 7 \cdot - 9}{70 (- \frac{3}{7}) + 70 (- \frac{3}{7})}$

Step 1.4.3.4

Multiply $7$ by $- 9$ .

$m = \frac{- 21 - 63}{70 (- \frac{3}{7}) + 70 (- \frac{3}{7})}$

Step 1.4.3.5

Cancel the common factor of $7$ .

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Step 1.4.3.5.1

Move the leading negative in $- \frac{3}{7}$ into the numerator.

$m = \frac{- 21 - 63}{70 (\frac{- 3}{7}) + 70 (- \frac{3}{7})}$

Step 1.4.3.5.2

Factor $7$ out of $70$ .

$m = \frac{- 21 - 63}{7 (10) (\frac{- 3}{7}) + 70 (- \frac{3}{7})}$

Step 1.4.3.5.3

Cancel the common factor.

$m = \frac{- 21 - 63}{7 \cdot (10 (\frac{- 3}{7})) + 70 (- \frac{3}{7})}$

Step 1.4.3.5.4

Rewrite the expression.

$m = \frac{- 21 - 63}{10 \cdot - 3 + 70 (- \frac{3}{7})}$

Step 1.4.3.6

Multiply $10$ by $- 3$ .

$m = \frac{- 21 - 63}{- 30 + 70 (- \frac{3}{7})}$

Step 1.4.3.7

Cancel the common factor of $7$ .

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Step 1.4.3.7.1

Move the leading negative in $- \frac{3}{7}$ into the numerator.

$m = \frac{- 21 - 63}{- 30 + 70 (\frac{- 3}{7})}$

Step 1.4.3.7.2

Factor $7$ out of $70$ .

$m = \frac{- 21 - 63}{- 30 + 7 (10) (\frac{- 3}{7})}$

Step 1.4.3.7.3

Cancel the common factor.

$m = \frac{- 21 - 63}{- 30 + 7 \cdot (10 (\frac{- 3}{7}))}$

Step 1.4.3.7.4

Rewrite the expression.

$m = \frac{- 21 - 63}{- 30 + 10 \cdot - 3}$

Step 1.4.3.8

Multiply $10$ by $- 3$ .

$m = \frac{- 21 - 63}{- 30 - 30}$

Step 1.4.4

Reduce the expression by cancelling the common factors.

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Step 1.4.4.1

Cancel the common factor of $- 21 - 63$ and $- 30 - 30$ .

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Step 1.4.4.1.1

Factor $3$ out of $- 21$ .

$m = \frac{3 (- 7) - 63}{- 30 - 30}$

Step 1.4.4.1.2

Factor $3$ out of $- 63$ .

$m = \frac{3 \cdot - 7 + 3 \cdot - 21}{- 30 - 30}$

Step 1.4.4.1.3

Factor $3$ out of $3 \cdot - 7 + 3 \cdot - 21$ .

$m = \frac{3 \cdot (- 7 - 21)}{- 30 - 30}$

Step 1.4.4.1.4

Cancel the common factors.

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Step 1.4.4.1.4.1

Factor $3$ out of $- 30$ .

$m = \frac{3 \cdot (- 7 - 21)}{3 (- 10) - 30}$

Step 1.4.4.1.4.2

Factor $3$ out of $- 30$ .

$m = \frac{3 \cdot (- 7 - 21)}{3 \cdot - 10 + 3 \cdot - 10}$

Step 1.4.4.1.4.3

Factor $3$ out of $3 \cdot - 10 + 3 \cdot - 10$ .

$m = \frac{3 \cdot (- 7 - 21)}{3 \cdot (- 10 - 10)}$

Step 1.4.4.1.4.4

Cancel the common factor.

$m = \frac{3 \cdot (- 7 - 21)}{3 \cdot (- 10 - 10)}$

Step 1.4.4.1.4.5

Rewrite the expression.

$m = \frac{- 7 - 21}{- 10 - 10}$

Step 1.4.4.2

Simplify by subtracting numbers.

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Step 1.4.4.2.1

Subtract $21$ from $- 7$ .

$m = \frac{- 28}{- 10 - 10}$

Step 1.4.4.2.2

Subtract $10$ from $- 10$ .

$m = \frac{- 28}{- 20}$

Step 1.4.4.3

Cancel the common factor of $- 28$ and $- 20$ .

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Step 1.4.4.3.1

Factor $- 4$ out of $- 28$ .

$m = \frac{- 4 \cdot 7}{- 20}$

Step 1.4.4.3.2

Cancel the common factors.

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Step 1.4.4.3.2.1

Factor $- 4$ out of $- 20$ .

$m = \frac{- 4 \cdot 7}{- 4 \cdot 5}$

Step 1.4.4.3.2.2

Cancel the common factor.

$m = \frac{- 4 \cdot 7}{- 4 \cdot 5}$

Step 1.4.4.3.2.3

Rewrite the expression.

$m = \frac{7}{5}$

Step 2

Use the slope $\frac{7}{5}$ and a given point $(\frac{3}{7}, \frac{9}{10})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{9}{10}) = \frac{7}{5} \cdot (x - (\frac{3}{7}))$

Step 3

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

$y - \frac{9}{10} = \frac{7}{5} \cdot (x - \frac{3}{7})$

Step 4

Solve for $y$ .

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Step 4.1

Simplify $\frac{7}{5} \cdot (x - \frac{3}{7})$ .

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Step 4.1.1

Rewrite.

$y - \frac{9}{10} = 0 + 0 + \frac{7}{5} \cdot (x - \frac{3}{7})$

Step 4.1.2

Simplify by adding zeros.

$y - \frac{9}{10} = \frac{7}{5} \cdot (x - \frac{3}{7})$

Step 4.1.3

Apply the distributive property.

$y - \frac{9}{10} = \frac{7}{5} x + \frac{7}{5} (- \frac{3}{7})$

Step 4.1.4

Combine $\frac{7}{5}$ and $x$ .

$y - \frac{9}{10} = \frac{7 x}{5} + \frac{7}{5} (- \frac{3}{7})$

Step 4.1.5

Cancel the common factor of $7$ .

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Step 4.1.5.1

Move the leading negative in $- \frac{3}{7}$ into the numerator.

$y - \frac{9}{10} = \frac{7 x}{5} + \frac{7}{5} \cdot \frac{- 3}{7}$

Step 4.1.5.2

Cancel the common factor.

$y - \frac{9}{10} = \frac{7 x}{5} + \frac{7}{5} \cdot \frac{- 3}{7}$

Step 4.1.5.3

Rewrite the expression.

$y - \frac{9}{10} = \frac{7 x}{5} + \frac{1}{5} \cdot - 3$

Step 4.1.6

Combine $\frac{1}{5}$ and $- 3$ .

$y - \frac{9}{10} = \frac{7 x}{5} + \frac{- 3}{5}$

Step 4.1.7

Move the negative in front of the fraction.

$y - \frac{9}{10} = \frac{7 x}{5} - \frac{3}{5}$

Step 4.2

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

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Step 4.2.1

Add $\frac{9}{10}$ to both sides of the equation.

$y = \frac{7 x}{5} - \frac{3}{5} + \frac{9}{10}$

Step 4.2.2

To write $- \frac{3}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{7 x}{5} - \frac{3}{5} \cdot \frac{2}{2} + \frac{9}{10}$

Step 4.2.3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.3.1

Multiply $\frac{3}{5}$ by $\frac{2}{2}$ .

$y = \frac{7 x}{5} - \frac{3 \cdot 2}{5 \cdot 2} + \frac{9}{10}$

Step 4.2.3.2

Multiply $5$ by $2$ .

$y = \frac{7 x}{5} - \frac{3 \cdot 2}{10} + \frac{9}{10}$

Step 4.2.4

Combine the numerators over the common denominator.

$y = \frac{7 x}{5} + \frac{- 3 \cdot 2 + 9}{10}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $- 3$ by $2$ .

$y = \frac{7 x}{5} + \frac{- 6 + 9}{10}$

Step 4.2.5.2

Add $- 6$ and $9$ .

$y = \frac{7 x}{5} + \frac{3}{10}$

Step 5

Reorder terms.

$y = \frac{7}{5} x + \frac{3}{10}$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{7}{5} x + \frac{3}{10}$

Point-slope form:

$y - \frac{9}{10} = \frac{7}{5} \cdot (x - \frac{3}{7})$

Step 7