Algebra Examples

Through

(3, 5)

$(3, 5)$ ; perpendicular to

x - 2 y = 2

$x - 2 y = 2$

Step 1

Solve

x - 2 y = 2

$x - 2 y = 2$ .

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Step 1.1

Subtract

x

$x$ from both sides of the equation.

- 2 y = 2 - x

$- 2 y = 2 - x$

Step 1.2

Divide each term in

- 2 y = 2 - x

$- 2 y = 2 - x$ by

- 2

$- 2$ and simplify.

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Step 1.2.1

Divide each term in

- 2 y = 2 - x

$- 2 y = 2 - x$ by

- 2

$- 2$ .

\frac{- 2 y}{- 2} = \frac{2}{- 2} + \frac{- x}{- 2}

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of

- 2

$- 2$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.1.2

Divide

$y$ by

$1$ .

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

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

$2$ by

$- 2$ .

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

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

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

Step 2

Find the slope when

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

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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

$\frac{x}{2}$ .

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

Step 2.1.3

Reorder terms.

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

Step 2.2

Using the slope-intercept form, the slope is

$\frac{1}{2}$ .

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

Step 3

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

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

Step 4

Simplify

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

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 \cdot 2)$

Step 4.2

Multiply

$- (1 \cdot 2)$ .

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Step 4.2.1

Multiply

$2$ by

$1$ .

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

Step 4.2.2

Multiply

$- 1$ by

$2$ .

$m_{perpendicular} = - 2$

Step 5

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

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Step 5.1

Use the slope

$- 2$ and a given point

$(3, 5)$ 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

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = - 2 \cdot (x - (3))$

Step 5.2

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

$y - 5 = - 2 \cdot (x - 3)$

Step 6

Solve for

$y$ .

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Step 6.1

Simplify

$- 2 \cdot (x - 3)$ .

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Step 6.1.1

Rewrite.

$y - 5 = 0 + 0 - 2 \cdot (x - 3)$

Step 6.1.2

Simplify by adding zeros.

$y - 5 = - 2 \cdot (x - 3)$

Step 6.1.3

Apply the distributive property.

$y - 5 = - 2 x - 2 \cdot - 3$

Step 6.1.4

Multiply

$- 2$ by

$- 3$ .

$y - 5 = - 2 x + 6$

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

$5$ to both sides of the equation.

$y = - 2 x + 6 + 5$

Step 6.2.2

Add

$6$ and

$5$ .

$y = - 2 x + 11$

Step 7