Algebra Examples

(- \frac{2}{3}, \frac{7}{8})

$(- \frac{2}{3}, \frac{7}{8})$ ,

3 x + 4 y = 7

$3 x + 4 y = 7$

Step 1

Solve

3 x + 4 y = 7

$3 x + 4 y = 7$ .

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Step 1.1

Subtract

3 x

$3 x$ from both sides of the equation.

4 y = 7 - 3 x

$4 y = 7 - 3 x$

Step 1.2

Divide each term in

4 y = 7 - 3 x

$4 y = 7 - 3 x$ by

4

$4$ and simplify.

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Step 1.2.1

Divide each term in

4 y = 7 - 3 x

$4 y = 7 - 3 x$ by

4

$4$ .

\frac{4 y}{4} = \frac{7}{4} + \frac{- 3 x}{4}

$\frac{4 y}{4} = \frac{7}{4} + \frac{- 3 x}{4}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of

4

$4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{7}{4} + \frac{- 3 x}{4}$

Step 1.2.2.1.2

Divide

$y$ by

$1$ .

$y = \frac{7}{4} + \frac{- 3 x}{4}$

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.

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

Step 2

Find the slope when

$y = \frac{7}{4} - \frac{3 x}{4}$ .

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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

$\frac{7}{4}$ and

$- \frac{3 x}{4}$ .

$y = - \frac{3 x}{4} + \frac{7}{4}$

Step 2.1.3

Write in

$y = m x + b$ form.

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Step 2.1.3.1

Reorder terms.

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

Step 2.1.3.2

Remove parentheses.

$y = - \frac{3}{4} x + \frac{7}{4}$

Step 2.2

Using the slope-intercept form, the slope is

$- \frac{3}{4}$ .

$m = - \frac{3}{4}$

Step 3

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

$m_{perpendicular} = - \frac{1}{- \frac{3}{4}}$

Step 4

Simplify

$- \frac{1}{- \frac{3}{4}}$ 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

$- 1 (- 1)$ .

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{3}{4}}$

Step 4.1.2

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{3}{4}}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{4}{3})$

Step 4.3

Multiply

$\frac{4}{3}$ by

$1$ .

$m_{perpendicular} = \frac{4}{3}$

Step 4.4

Multiply

$- - \frac{4}{3}$ .

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Step 4.4.1

Multiply

$- 1$ by

$- 1$ .

$m_{perpendicular} = 1 (\frac{4}{3})$

Step 4.4.2

Multiply

$\frac{4}{3}$ by

$1$ .

$m_{perpendicular} = \frac{4}{3}$

Step 5

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

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Step 5.1

Use the slope

$\frac{4}{3}$ and a given point

$(- \frac{2}{3}, \frac{7}{8})$ 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

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

$y - (\frac{7}{8}) = \frac{4}{3} \cdot (x - (- \frac{2}{3}))$

Step 5.2

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

$y - \frac{7}{8} = \frac{4}{3} \cdot (x + \frac{2}{3})$

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

$\frac{4}{3} \cdot (x + \frac{2}{3})$ .

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Step 6.1.1.1

Rewrite.

$y - \frac{7}{8} = 0 + 0 + \frac{4}{3} \cdot (x + \frac{2}{3})$

Step 6.1.1.2

Simplify by adding zeros.

$y - \frac{7}{8} = \frac{4}{3} \cdot (x + \frac{2}{3})$

Step 6.1.1.3

Apply the distributive property.

$y - \frac{7}{8} = \frac{4}{3} x + \frac{4}{3} \cdot \frac{2}{3}$

Step 6.1.1.4

Combine

$\frac{4}{3}$ and

$x$ .

$y - \frac{7}{8} = \frac{4 x}{3} + \frac{4}{3} \cdot \frac{2}{3}$

Step 6.1.1.5

Multiply

$\frac{4}{3} \cdot \frac{2}{3}$ .

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Step 6.1.1.5.1

Multiply

$\frac{4}{3}$ by

$\frac{2}{3}$ .

$y - \frac{7}{8} = \frac{4 x}{3} + \frac{4 \cdot 2}{3 \cdot 3}$

Step 6.1.1.5.2

Multiply

$4$ by

$2$ .

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

Step 6.1.1.5.3

Multiply

$3$ by

$3$ .

$y - \frac{7}{8} = \frac{4 x}{3} + \frac{8}{9}$

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

$\frac{7}{8}$ to both sides of the equation.

$y = \frac{4 x}{3} + \frac{8}{9} + \frac{7}{8}$

Step 6.1.2.2

To write

$\frac{8}{9}$ as a fraction with a common denominator, multiply by

$\frac{8}{8}$ .

$y = \frac{4 x}{3} + \frac{8}{9} \cdot \frac{8}{8} + \frac{7}{8}$

Step 6.1.2.3

To write

$\frac{7}{8}$ as a fraction with a common denominator, multiply by

$\frac{9}{9}$ .

$y = \frac{4 x}{3} + \frac{8}{9} \cdot \frac{8}{8} + \frac{7}{8} \cdot \frac{9}{9}$

Step 6.1.2.4

Write each expression with a common denominator of

$72$ , by multiplying each by an appropriate factor of

$1$ .

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Step 6.1.2.4.1

Multiply

$\frac{8}{9}$ by

$\frac{8}{8}$ .

$y = \frac{4 x}{3} + \frac{8 \cdot 8}{9 \cdot 8} + \frac{7}{8} \cdot \frac{9}{9}$

Step 6.1.2.4.2

Multiply

$9$ by

$8$ .

$y = \frac{4 x}{3} + \frac{8 \cdot 8}{72} + \frac{7}{8} \cdot \frac{9}{9}$

Step 6.1.2.4.3

Multiply

$\frac{7}{8}$ by

$\frac{9}{9}$ .

$y = \frac{4 x}{3} + \frac{8 \cdot 8}{72} + \frac{7 \cdot 9}{8 \cdot 9}$

Step 6.1.2.4.4

Multiply

$8$ by

$9$ .

$y = \frac{4 x}{3} + \frac{8 \cdot 8}{72} + \frac{7 \cdot 9}{72}$

Step 6.1.2.5

Combine the numerators over the common denominator.

$y = \frac{4 x}{3} + \frac{8 \cdot 8 + 7 \cdot 9}{72}$

Step 6.1.2.6

Simplify the numerator.

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Step 6.1.2.6.1

Multiply

$8$ by

$8$ .

$y = \frac{4 x}{3} + \frac{64 + 7 \cdot 9}{72}$

Step 6.1.2.6.2

Multiply

$7$ by

$9$ .

$y = \frac{4 x}{3} + \frac{64 + 63}{72}$

Step 6.1.2.6.3

Add

$64$ and

$63$ .

$y = \frac{4 x}{3} + \frac{127}{72}$

Step 6.2

Reorder terms.

$y = \frac{4}{3} x + \frac{127}{72}$

Step 7