Algebra Examples

$(8, 1)$ and is perpendicular to the line $2 y + 4 x = 12$

Step 1

Solve $2 y + 4 x = 12$ .

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Step 1.1

Subtract $4 x$ from both sides of the equation.

$2 y = 12 - 4 x$

Step 1.2

Divide each term in $2 y = 12 - 4 x$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 y = 12 - 4 x$ by $2$ .

$\frac{2 y}{2} = \frac{12}{2} + \frac{- 4 x}{2}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{12}{2} + \frac{- 4 x}{2}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{12}{2} + \frac{- 4 x}{2}$

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 $12$ by $2$ .

$y = 6 + \frac{- 4 x}{2}$

Step 1.2.3.1.2

Cancel the common factor of $- 4$ and $2$ .

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Step 1.2.3.1.2.1

Factor $2$ out of $- 4 x$ .

$y = 6 + \frac{2 (- 2 x)}{2}$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $2$ out of $2$ .

$y = 6 + \frac{2 (- 2 x)}{2 (1)}$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$y = 6 + \frac{2 (- 2 x)}{2 \cdot 1}$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$y = 6 + \frac{- 2 x}{1}$

Step 1.2.3.1.2.2.4

Divide $- 2 x$ by $1$ .

$y = 6 - 2 x$

Step 2

Find the slope when $y = 6 - 2 x$ .

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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 $6$ and $- 2 x$ .

$y = - 2 x + 6$

Step 2.2

Using the slope-intercept form, the slope is $- 2$ .

$m = - 2$

Step 3

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

$m_{perpendicular} = - \frac{1}{- 2}$

Step 4

Simplify $- \frac{1}{- 2}$ to find the slope of the perpendicular line.

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Step 4.1

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{2}$

Step 4.2

Multiply $- - \frac{1}{2}$ .

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Step 4.2.1

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{1}{2})$

Step 4.2.2

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

$m_{perpendicular} = \frac{1}{2}$

Step 5

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

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Step 5.1

Use the slope $\frac{1}{2}$ and a given point $(8, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = \frac{1}{2} \cdot (x - (8))$

Step 5.2

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

$y - 1 = \frac{1}{2} \cdot (x - 8)$

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 $\frac{1}{2} \cdot (x - 8)$ .

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Step 6.1.1.1

Rewrite.

$y - 1 = 0 + 0 + \frac{1}{2} \cdot (x - 8)$

Step 6.1.1.2

Simplify by adding zeros.

$y - 1 = \frac{1}{2} \cdot (x - 8)$

Step 6.1.1.3

Apply the distributive property.

$y - 1 = \frac{1}{2} x + \frac{1}{2} \cdot - 8$

Step 6.1.1.4

Combine $\frac{1}{2}$ and $x$ .

$y - 1 = \frac{x}{2} + \frac{1}{2} \cdot - 8$

Step 6.1.1.5

Cancel the common factor of $2$ .

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Step 6.1.1.5.1

Factor $2$ out of $- 8$ .

$y - 1 = \frac{x}{2} + \frac{1}{2} \cdot (2 (- 4))$

Step 6.1.1.5.2

Cancel the common factor.

$y - 1 = \frac{x}{2} + \frac{1}{2} \cdot (2 \cdot - 4)$

Step 6.1.1.5.3

Rewrite the expression.

$y - 1 = \frac{x}{2} - 4$

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 $1$ to both sides of the equation.

$y = \frac{x}{2} - 4 + 1$

Step 6.1.2.2

Add $- 4$ and $1$ .

$y = \frac{x}{2} - 3$

Step 6.2

Reorder terms.

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

Step 7