Algebra Examples

5 x + 3 y = 0

$5 x + 3 y = 0$ ,

(\frac{7}{8}, \frac{3}{4})

$(\frac{7}{8}, \frac{3}{4})$

Step 1

Solve

5 x + 3 y = 0

$5 x + 3 y = 0$ .

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Step 1.1

Subtract

5 x

$5 x$ from both sides of the equation.

3 y = - 5 x

$3 y = - 5 x$

Step 1.2

Divide each term in

3 y = - 5 x

$3 y = - 5 x$ by

3

$3$ and simplify.

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Step 1.2.1

Divide each term in

3 y = - 5 x

$3 y = - 5 x$ by

3

$3$ .

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

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of

3

$3$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.1.2

Divide

$y$ by

$1$ .

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

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

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

Step 2

Find the slope when

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

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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

Write in

$y = m x + b$ form.

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Step 2.1.2.1

Reorder terms.

$y = - (\frac{5}{3} x)$

Step 2.1.2.2

Remove parentheses.

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

Step 2.2

Using the slope-intercept form, the slope is

$- \frac{5}{3}$ .

$m = - \frac{5}{3}$

Step 3

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

$m_{perpendicular} = - \frac{1}{- \frac{5}{3}}$

Step 4

Simplify

$- \frac{1}{- \frac{5}{3}}$ to find the slope of the perpendicular line.

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Step 4.1

Cancel the common factor of

$1$ and

$- 1$ .

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Step 4.1.1

Rewrite

$1$ as

$- 1 (- 1)$ .

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{5}{3}}$

Step 4.1.2

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{5}{3}}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{3}{5})$

Step 4.3

Multiply

$\frac{3}{5}$ by

$1$ .

$m_{perpendicular} = \frac{3}{5}$

Step 4.4

Multiply

$- - \frac{3}{5}$ .

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Step 4.4.1

Multiply

$- 1$ by

$- 1$ .

$m_{perpendicular} = 1 (\frac{3}{5})$

Step 4.4.2

Multiply

$\frac{3}{5}$ by

$1$ .

$m_{perpendicular} = \frac{3}{5}$

Step 5

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

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Step 5.1

Use the slope

$\frac{3}{5}$ and a given point

$(\frac{7}{8}, \frac{3}{4})$ 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

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

$y - (\frac{3}{4}) = \frac{3}{5} \cdot (x - (\frac{7}{8}))$

Step 5.2

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

$y - \frac{3}{4} = \frac{3}{5} \cdot (x - \frac{7}{8})$

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

$\frac{3}{5} \cdot (x - \frac{7}{8})$ .

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Step 6.1.1.1

Rewrite.

$y - \frac{3}{4} = 0 + 0 + \frac{3}{5} \cdot (x - \frac{7}{8})$

Step 6.1.1.2

Simplify by adding zeros.

$y - \frac{3}{4} = \frac{3}{5} \cdot (x - \frac{7}{8})$

Step 6.1.1.3

Apply the distributive property.

$y - \frac{3}{4} = \frac{3}{5} x + \frac{3}{5} (- \frac{7}{8})$

Step 6.1.1.4

Combine

$\frac{3}{5}$ and

$x$ .

$y - \frac{3}{4} = \frac{3 x}{5} + \frac{3}{5} (- \frac{7}{8})$

Step 6.1.1.5

Multiply

$\frac{3}{5} (- \frac{7}{8})$ .

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Step 6.1.1.5.1

Multiply

$\frac{3}{5}$ by

$\frac{7}{8}$ .

$y - \frac{3}{4} = \frac{3 x}{5} - \frac{3 \cdot 7}{5 \cdot 8}$

Step 6.1.1.5.2

Multiply

$3$ by

$7$ .

$y - \frac{3}{4} = \frac{3 x}{5} - \frac{21}{5 \cdot 8}$

Step 6.1.1.5.3

Multiply

$5$ by

$8$ .

$y - \frac{3}{4} = \frac{3 x}{5} - \frac{21}{40}$

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

$\frac{3}{4}$ to both sides of the equation.

$y = \frac{3 x}{5} - \frac{21}{40} + \frac{3}{4}$

Step 6.1.2.2

To write

$\frac{3}{4}$ as a fraction with a common denominator, multiply by

$\frac{10}{10}$ .

$y = \frac{3 x}{5} - \frac{21}{40} + \frac{3}{4} \cdot \frac{10}{10}$

Step 6.1.2.3

Write each expression with a common denominator of

$40$ , by multiplying each by an appropriate factor of

$1$ .

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Step 6.1.2.3.1

Multiply

$\frac{3}{4}$ by

$\frac{10}{10}$ .

$y = \frac{3 x}{5} - \frac{21}{40} + \frac{3 \cdot 10}{4 \cdot 10}$

Step 6.1.2.3.2

Multiply

$4$ by

$10$ .

$y = \frac{3 x}{5} - \frac{21}{40} + \frac{3 \cdot 10}{40}$

Step 6.1.2.4

Combine the numerators over the common denominator.

$y = \frac{3 x}{5} + \frac{- 21 + 3 \cdot 10}{40}$

Step 6.1.2.5

Simplify the numerator.

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Step 6.1.2.5.1

Multiply

$3$ by

$10$ .

$y = \frac{3 x}{5} + \frac{- 21 + 30}{40}$

Step 6.1.2.5.2

Add

$- 21$ and

$30$ .

$y = \frac{3 x}{5} + \frac{9}{40}$

Step 6.2

Reorder terms.

$y = \frac{3}{5} x + \frac{9}{40}$

Step 7