Algebra Examples

find the equation of the line $7 x - 4 y = - 4$ and passes through the point $(- 2, 4)$

Step 1

Solve $7 x - 4 y = - 4$ .

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Step 1.1

Subtract $7 x$ from both sides of the equation.

$- 4 y = - 4 - 7 x$

Step 1.2

Divide each term in $- 4 y = - 4 - 7 x$ by $- 4$ and simplify.

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Step 1.2.1

Divide each term in $- 4 y = - 4 - 7 x$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{- 4}{- 4} + \frac{- 7 x}{- 4}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{- 4}{- 4} + \frac{- 7 x}{- 4}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 4}{- 4} + \frac{- 7 x}{- 4}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Divide $- 4$ by $- 4$ .

$y = 1 + \frac{- 7 x}{- 4}$

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

$y = 1 + \frac{7 x}{4}$

Step 2

Find the slope when $y = 1 + \frac{7 x}{4}$ .

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Step 2.1

Rewrite in slope-intercept form.

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Step 2.1.1

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 2.1.2

Reorder $1$ and $\frac{7 x}{4}$ .

$y = \frac{7 x}{4} + 1$

Step 2.1.3

Reorder terms.

$y = \frac{7}{4} x + 1$

Step 2.2

Using the slope-intercept form, the slope is $\frac{7}{4}$ .

$m = \frac{7}{4}$

Step 3

The equation of a perpendicular line must have a slope that is the negative reciprocal of the original slope.

$m_{perpendicular} = - \frac{1}{\frac{7}{4}}$

Step 4

Simplify $- \frac{1}{\frac{7}{4}}$ to find the slope of the perpendicular line.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{4}{7}))$

Step 4.2

Multiply $\frac{4}{7}$ by $1$ .

$m_{perpendicular} = - \frac{4}{7}$

Step 5

Find the equation of the perpendicular line using the point-slope formula.

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Step 5.1

Use the slope $- \frac{4}{7}$ and a given point $(- 2, 4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (4) = - \frac{4}{7} \cdot (x - (- 2))$

Step 5.2

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

$y - 4 = - \frac{4}{7} \cdot (x + 2)$

Step 6

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

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Step 6.1

Solve for $y$ .

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Step 6.1.1

Simplify $- \frac{4}{7} \cdot (x + 2)$ .

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Step 6.1.1.1

Rewrite.

$y - 4 = 0 + 0 - \frac{4}{7} \cdot (x + 2)$

Step 6.1.1.2

Simplify by adding zeros.

$y - 4 = - \frac{4}{7} \cdot (x + 2)$

Step 6.1.1.3

Apply the distributive property.

$y - 4 = - \frac{4}{7} x - \frac{4}{7} \cdot 2$

Step 6.1.1.4

Combine $x$ and $\frac{4}{7}$ .

$y - 4 = - \frac{x \cdot 4}{7} - \frac{4}{7} \cdot 2$

Step 6.1.1.5

Multiply $- \frac{4}{7} \cdot 2$ .

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Step 6.1.1.5.1

Multiply $2$ by $- 1$ .

$y - 4 = - \frac{x \cdot 4}{7} - 2 (\frac{4}{7})$

Step 6.1.1.5.2

Combine $- 2$ and $\frac{4}{7}$ .

$y - 4 = - \frac{x \cdot 4}{7} + \frac{- 2 \cdot 4}{7}$

Step 6.1.1.5.3

Multiply $- 2$ by $4$ .

$y - 4 = - \frac{x \cdot 4}{7} + \frac{- 8}{7}$

Step 6.1.1.6

Simplify each term.

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Step 6.1.1.6.1

Move $4$ to the left of $x$ .

$y - 4 = - \frac{4 \cdot x}{7} + \frac{- 8}{7}$

Step 6.1.1.6.2

Move the negative in front of the fraction.

$y - 4 = - \frac{4 x}{7} - \frac{8}{7}$

Step 6.1.2

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

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Step 6.1.2.1

Add $4$ to both sides of the equation.

$y = - \frac{4 x}{7} - \frac{8}{7} + 4$

Step 6.1.2.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$y = - \frac{4 x}{7} - \frac{8}{7} + 4 \cdot \frac{7}{7}$

Step 6.1.2.3

Combine $4$ and $\frac{7}{7}$ .

$y = - \frac{4 x}{7} - \frac{8}{7} + \frac{4 \cdot 7}{7}$

Step 6.1.2.4

Combine the numerators over the common denominator.

$y = - \frac{4 x}{7} + \frac{- 8 + 4 \cdot 7}{7}$

Step 6.1.2.5

Simplify the numerator.

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Step 6.1.2.5.1

Multiply $4$ by $7$ .

$y = - \frac{4 x}{7} + \frac{- 8 + 28}{7}$

Step 6.1.2.5.2

Add $- 8$ and $28$ .

$y = - \frac{4 x}{7} + \frac{20}{7}$

Step 6.2

Reorder terms.

$y = - (\frac{4}{7} x) + \frac{20}{7}$

Step 6.3

Remove parentheses.

$y = - \frac{4}{7} x + \frac{20}{7}$

Step 7