Precalculus Examples

$3 x - 7 y = 5$ , $(2, 1)$

Step 1

Rewrite in slope-intercept form.

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Step 1.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 1.2

Subtract $3 x$ from both sides of the equation.

$- 7 y = 5 - 3 x$

Step 1.3

Divide each term in $- 7 y = 5 - 3 x$ by $- 7$ and simplify.

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Step 1.3.1

Divide each term in $- 7 y = 5 - 3 x$ by $- 7$ .

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

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $- 7$ .

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Step 1.3.2.1.1

Cancel the common factor.

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

Step 1.3.2.1.2

Divide $y$ by $1$ .

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

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Move the negative in front of the fraction.

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

Step 1.3.3.1.2

Dividing two negative values results in a positive value.

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

Step 1.4

Write in $y = m x + b$ form.

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Step 1.4.1

Reorder $- \frac{5}{7}$ and $\frac{3 x}{7}$ .

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

Step 1.4.2

Reorder terms.

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

Step 2

Using the slope-intercept form, the slope is $\frac{3}{7}$ .

$m = \frac{3}{7}$

Step 3

To find an equation that is parallel, the slopes must be equal. Find the parallel line using the point-slope formula.

Step 4

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

$y - (1) = \frac{3}{7} \cdot (x - (2))$

Step 5

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

$y - 1 = \frac{3}{7} \cdot (x - 2)$

Step 6

Solve for $y$ .

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Step 6.1

Simplify $\frac{3}{7} \cdot (x - 2)$ .

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Step 6.1.1

Rewrite.

$y - 1 = 0 + 0 + \frac{3}{7} \cdot (x - 2)$

Step 6.1.2

Simplify by adding zeros.

$y - 1 = \frac{3}{7} \cdot (x - 2)$

Step 6.1.3

Apply the distributive property.

$y - 1 = \frac{3}{7} x + \frac{3}{7} \cdot - 2$

Step 6.1.4

Combine $\frac{3}{7}$ and $x$ .

$y - 1 = \frac{3 x}{7} + \frac{3}{7} \cdot - 2$

Step 6.1.5

Multiply $\frac{3}{7} \cdot - 2$ .

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Step 6.1.5.1

Combine $\frac{3}{7}$ and $- 2$ .

$y - 1 = \frac{3 x}{7} + \frac{3 \cdot - 2}{7}$

Step 6.1.5.2

Multiply $3$ by $- 2$ .

$y - 1 = \frac{3 x}{7} + \frac{- 6}{7}$

Step 6.1.6

Move the negative in front of the fraction.

$y - 1 = \frac{3 x}{7} - \frac{6}{7}$

Step 6.2

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

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Step 6.2.1

Add $1$ to both sides of the equation.

$y = \frac{3 x}{7} - \frac{6}{7} + 1$

Step 6.2.2

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

$y = \frac{3 x}{7} - \frac{6}{7} + \frac{7}{7}$

Step 6.2.3

Combine the numerators over the common denominator.

$y = \frac{3 x}{7} + \frac{- 6 + 7}{7}$

Step 6.2.4

Add $- 6$ and $7$ .

$y = \frac{3 x}{7} + \frac{1}{7}$

Step 6.3

Reorder terms.

$y = \frac{3}{7} x + \frac{1}{7}$

Step 7