Algebra Examples

(5, - 4)

$(5, - 4)$ that is parallel to the line

5 x + 6 y = 7

$5 x + 6 y = 7$

Step 1

Solve

5 x + 6 y = 7

$5 x + 6 y = 7$ .

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Step 1.1

Subtract

5 x

$5 x$ from both sides of the equation.

6 y = 7 - 5 x

$6 y = 7 - 5 x$

Step 1.2

Divide each term in

6 y = 7 - 5 x

$6 y = 7 - 5 x$ by

6

$6$ and simplify.

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Step 1.2.1

Divide each term in

6 y = 7 - 5 x

$6 y = 7 - 5 x$ by

6

$6$ .

\frac{6 y}{6} = \frac{7}{6} + \frac{- 5 x}{6}

$\frac{6 y}{6} = \frac{7}{6} + \frac{- 5 x}{6}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of

6

$6$ .

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Step 1.2.2.1.1

Cancel the common factor.

\frac{6 y}{6} = \frac{7}{6} + \frac{- 5 x}{6}

$\frac{6 y}{6} = \frac{7}{6} + \frac{- 5 x}{6}$

Step 1.2.2.1.2

Divide

y

$y$ by

1

$1$ .

y = \frac{7}{6} + \frac{- 5 x}{6}

$y = \frac{7}{6} + \frac{- 5 x}{6}$

y = \frac{7}{6} + \frac{- 5 x}{6}

$y = \frac{7}{6} + \frac{- 5 x}{6}$

y = \frac{7}{6} + \frac{- 5 x}{6}

$y = \frac{7}{6} + \frac{- 5 x}{6}$

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}{6} - \frac{5 x}{6}

$y = \frac{7}{6} - \frac{5 x}{6}$

y = \frac{7}{6} - \frac{5 x}{6}

$y = \frac{7}{6} - \frac{5 x}{6}$

y = \frac{7}{6} - \frac{5 x}{6}

$y = \frac{7}{6} - \frac{5 x}{6}$

y = \frac{7}{6} - \frac{5 x}{6}

$y = \frac{7}{6} - \frac{5 x}{6}$

Step 2

Find the slope when

y = \frac{7}{6} - \frac{5 x}{6}

$y = \frac{7}{6} - \frac{5 x}{6}$ .

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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

$y = m x + b$ , where

m

$m$ is the slope and

b

$b$ is the y-intercept.

y = m x + b

$y = m x + b$

Step 2.1.2

Reorder

\frac{7}{6}

$\frac{7}{6}$ and

- \frac{5 x}{6}

$- \frac{5 x}{6}$ .

y = - \frac{5 x}{6} + \frac{7}{6}

$y = - \frac{5 x}{6} + \frac{7}{6}$

Step 2.1.3

Write in

y = m x + b

$y = m x + b$ form.

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Step 2.1.3.1

Reorder terms.

y = - (\frac{5}{6} x) + \frac{7}{6}

$y = - (\frac{5}{6} x) + \frac{7}{6}$

Step 2.1.3.2

Remove parentheses.

y = - \frac{5}{6} x + \frac{7}{6}

$y = - \frac{5}{6} x + \frac{7}{6}$

y = - \frac{5}{6} x + \frac{7}{6}

$y = - \frac{5}{6} x + \frac{7}{6}$

y = - \frac{5}{6} x + \frac{7}{6}

$y = - \frac{5}{6} x + \frac{7}{6}$

Step 2.2

Using the slope-intercept form, the slope is

- \frac{5}{6}

$- \frac{5}{6}$ .

m = - \frac{5}{6}

$m = - \frac{5}{6}$

m = - \frac{5}{6}

$m = - \frac{5}{6}$

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}{6}}

$m_{perpendicular} = - \frac{1}{- \frac{5}{6}}$

Step 4

Simplify

- \frac{1}{- \frac{5}{6}}

$- \frac{1}{- \frac{5}{6}}$ to find the slope of the perpendicular line.

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Step 4.1

Cancel the common factor of

1

$1$ and

- 1

$- 1$ .

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Step 4.1.1

Rewrite

1

$1$ as

- 1 (- 1)

$- 1 (- 1)$ .

m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{5}{6}}

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{5}{6}}$

Step 4.1.2

Move the negative in front of the fraction.

m_{perpendicular} = \frac{1}{\frac{5}{6}}

$m_{perpendicular} = \frac{1}{\frac{5}{6}}$

m_{perpendicular} = \frac{1}{\frac{5}{6}}

$m_{perpendicular} = \frac{1}{\frac{5}{6}}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

m_{perpendicular} = 1 (\frac{6}{5})

$m_{perpendicular} = 1 (\frac{6}{5})$

Step 4.3

Multiply

\frac{6}{5}

$\frac{6}{5}$ by

1

$1$ .

m_{perpendicular} = \frac{6}{5}

$m_{perpendicular} = \frac{6}{5}$

Step 4.4

Multiply

- - \frac{6}{5}

$- - \frac{6}{5}$ .

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Step 4.4.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

m_{perpendicular} = 1 (\frac{6}{5})

$m_{perpendicular} = 1 (\frac{6}{5})$

Step 4.4.2

Multiply

\frac{6}{5}

$\frac{6}{5}$ by

1

$1$ .

m_{perpendicular} = \frac{6}{5}

$m_{perpendicular} = \frac{6}{5}$

m_{perpendicular} = \frac{6}{5}

$m_{perpendicular} = \frac{6}{5}$

m_{perpendicular} = \frac{6}{5}

$m_{perpendicular} = \frac{6}{5}$

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{6}{5}

$\frac{6}{5}$ and a given point

(5, - 4)

$(5, - 4)$ to substitute for

x_{1}

$x_{1}$ and

y_{1}

$y_{1}$ in the point-slope form

y - y_{1} = m (x - x_{1})

$y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation

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

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

y - (- 4) = \frac{6}{5} \cdot (x - (5))

$y - (- 4) = \frac{6}{5} \cdot (x - (5))$

Step 5.2

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

y + 4 = \frac{6}{5} \cdot (x - 5)

$y + 4 = \frac{6}{5} \cdot (x - 5)$

y + 4 = \frac{6}{5} \cdot (x - 5)

$y + 4 = \frac{6}{5} \cdot (x - 5)$

Step 6

Write in

y = m x + b

$y = m x + b$ form.

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Step 6.1

Solve for

y

$y$ .

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Step 6.1.1

Simplify

\frac{6}{5} \cdot (x - 5)

$\frac{6}{5} \cdot (x - 5)$ .

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Step 6.1.1.1

Rewrite.

y + 4 = 0 + 0 + \frac{6}{5} \cdot (x - 5)

$y + 4 = 0 + 0 + \frac{6}{5} \cdot (x - 5)$

Step 6.1.1.2

Simplify by adding zeros.

y + 4 = \frac{6}{5} \cdot (x - 5)

$y + 4 = \frac{6}{5} \cdot (x - 5)$

Step 6.1.1.3

Apply the distributive property.

y + 4 = \frac{6}{5} x + \frac{6}{5} \cdot - 5

$y + 4 = \frac{6}{5} x + \frac{6}{5} \cdot - 5$

Step 6.1.1.4

Combine

\frac{6}{5}

$\frac{6}{5}$ and

x

$x$ .

y + 4 = \frac{6 x}{5} + \frac{6}{5} \cdot - 5

$y + 4 = \frac{6 x}{5} + \frac{6}{5} \cdot - 5$

Step 6.1.1.5

Cancel the common factor of

5

$5$ .

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Step 6.1.1.5.1

Factor

5

$5$ out of

- 5

$- 5$ .

y + 4 = \frac{6 x}{5} + \frac{6}{5} \cdot (5 (- 1))

$y + 4 = \frac{6 x}{5} + \frac{6}{5} \cdot (5 (- 1))$

Step 6.1.1.5.2

Cancel the common factor.

y + 4 = \frac{6 x}{5} + \frac{6}{5} \cdot (5 \cdot - 1)

$y + 4 = \frac{6 x}{5} + \frac{6}{5} \cdot (5 \cdot - 1)$

Step 6.1.1.5.3

Rewrite the expression.

y + 4 = \frac{6 x}{5} + 6 \cdot - 1

$y + 4 = \frac{6 x}{5} + 6 \cdot - 1$

y + 4 = \frac{6 x}{5} + 6 \cdot - 1

$y + 4 = \frac{6 x}{5} + 6 \cdot - 1$

Step 6.1.1.6

Multiply

6

$6$ by

- 1

$- 1$ .

y + 4 = \frac{6 x}{5} - 6

$y + 4 = \frac{6 x}{5} - 6$

y + 4 = \frac{6 x}{5} - 6

$y + 4 = \frac{6 x}{5} - 6$

Step 6.1.2

Move all terms not containing

y

$y$ to the right side of the equation.

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Step 6.1.2.1

Subtract

4

$4$ from both sides of the equation.

y = \frac{6 x}{5} - 6 - 4

$y = \frac{6 x}{5} - 6 - 4$

Step 6.1.2.2

Subtract

4

$4$ from

- 6

$- 6$ .

y = \frac{6 x}{5} - 10

$y = \frac{6 x}{5} - 10$

y = \frac{6 x}{5} - 10

$y = \frac{6 x}{5} - 10$

y = \frac{6 x}{5} - 10

$y = \frac{6 x}{5} - 10$

Step 6.2

Reorder terms.

y = \frac{6}{5} x - 10

$y = \frac{6}{5} x - 10$

y = \frac{6}{5} x - 10

$y = \frac{6}{5} x - 10$

Step 7