Algebra Examples

$\frac{3 x - 3 y}{3} = \frac{x - 3}{6}$ , $(11, 12)$

Step 1

Solve $\frac{3 x - 3 y}{3} = \frac{x - 3}{6}$ .

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Step 1.1

Cancel the common factor of $3 x - 3 y$ and $3$ .

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Step 1.1.1

Factor $3$ out of $3 x$ .

$\frac{3 (x) - 3 y}{3} = \frac{x - 3}{6}$

Step 1.1.2

Factor $3$ out of $- 3 y$ .

$\frac{3 (x) + 3 (- y)}{3} = \frac{x - 3}{6}$

Step 1.1.3

Factor $3$ out of $3 (x) + 3 (- y)$ .

$\frac{3 (x - y)}{3} = \frac{x - 3}{6}$

Step 1.1.4

Cancel the common factors.

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Step 1.1.4.1

Factor $3$ out of $3$ .

$\frac{3 (x - y)}{3 (1)} = \frac{x - 3}{6}$

Step 1.1.4.2

Cancel the common factor.

$\frac{3 (x - y)}{3 \cdot 1} = \frac{x - 3}{6}$

Step 1.1.4.3

Rewrite the expression.

$\frac{x - y}{1} = \frac{x - 3}{6}$

Step 1.1.4.4

Divide $x - y$ by $1$ .

$x - y = \frac{x - 3}{6}$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.2.1.1

Subtract $x$ from both sides of the equation.

$- y = \frac{x - 3}{6} - x$

Step 1.2.1.2

Simplify each term.

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Step 1.2.1.2.1

Split the fraction $\frac{x - 3}{6}$ into two fractions.

$- y = \frac{x}{6} + \frac{- 3}{6} - x$

Step 1.2.1.2.2

Simplify each term.

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Step 1.2.1.2.2.1

Cancel the common factor of $- 3$ and $6$ .

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Step 1.2.1.2.2.1.1

Factor $3$ out of $- 3$ .

$- y = \frac{x}{6} + \frac{3 (- 1)}{6} - x$

Step 1.2.1.2.2.1.2

Cancel the common factors.

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Step 1.2.1.2.2.1.2.1

Factor $3$ out of $6$ .

$- y = \frac{x}{6} + \frac{3 \cdot - 1}{3 \cdot 2} - x$

Step 1.2.1.2.2.1.2.2

Cancel the common factor.

$- y = \frac{x}{6} + \frac{3 \cdot - 1}{3 \cdot 2} - x$

Step 1.2.1.2.2.1.2.3

Rewrite the expression.

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

Step 1.2.1.2.2.2

Move the negative in front of the fraction.

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

Step 1.2.2

Divide each term in $- y = \frac{x}{6} - \frac{1}{2} - x$ by $- 1$ and simplify.

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Step 1.2.2.1

Divide each term in $- y = \frac{x}{6} - \frac{1}{2} - x$ by $- 1$ .

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

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.2.2.2

Divide $y$ by $1$ .

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

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Combine the numerators over the common denominator.

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

Step 1.2.2.3.2

To write $- x$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Step 1.2.2.3.3

Simplify terms.

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Step 1.2.2.3.3.1

Combine $- x$ and $\frac{6}{6}$ .

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

Step 1.2.2.3.3.2

Combine the numerators over the common denominator.

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

Step 1.2.2.3.4

Simplify each term.

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Step 1.2.2.3.4.1

Simplify the numerator.

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Step 1.2.2.3.4.1.1

Factor $x$ out of $x - x \cdot 6$ .

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Step 1.2.2.3.4.1.1.1

Raise $x$ to the power of $1$ .

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

Step 1.2.2.3.4.1.1.2

Factor $x$ out of $x^{1}$ .

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

Step 1.2.2.3.4.1.1.3

Factor $x$ out of $- x \cdot 6$ .

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

Step 1.2.2.3.4.1.1.4

Factor $x$ out of $x \cdot 1 + x (- 1 \cdot 6)$ .

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

Step 1.2.2.3.4.1.2

Multiply $- 1$ by $6$ .

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

Step 1.2.2.3.4.1.3

Subtract $6$ from $1$ .

$y = \frac{\frac{x \cdot - 5}{6} - \frac{1}{2}}{- 1}$

Step 1.2.2.3.4.2

Move $- 5$ to the left of $x$ .

$y = \frac{\frac{- 5 \cdot x}{6} - \frac{1}{2}}{- 1}$

Step 1.2.2.3.4.3

Move the negative in front of the fraction.

$y = \frac{- \frac{5 x}{6} - \frac{1}{2}}{- 1}$

Step 1.2.2.3.5

Simplify by multiplying through.

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Step 1.2.2.3.5.1

Move the negative one from the denominator of $\frac{- \frac{5 x}{6} - \frac{1}{2}}{- 1}$ .

$y = - 1 \cdot (- \frac{5 x}{6} - \frac{1}{2})$

Step 1.2.2.3.5.2

Rewrite $- 1 \cdot (- \frac{5 x}{6} - \frac{1}{2})$ as $- (- \frac{5 x}{6} - \frac{1}{2})$ .

$y = - (- \frac{5 x}{6} - \frac{1}{2})$

Step 1.2.2.3.5.3

Apply the distributive property.

$y = - - \frac{5 x}{6} - - \frac{1}{2}$

Step 1.2.2.3.6

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

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Step 1.2.2.3.6.1

Multiply $- 1$ by $- 1$ .

$y = 1 \frac{5 x}{6} - - \frac{1}{2}$

Step 1.2.2.3.6.2

Multiply $\frac{5 x}{6}$ by $1$ .

$y = \frac{5 x}{6} - - \frac{1}{2}$

Step 1.2.2.3.7

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

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Step 1.2.2.3.7.1

Multiply $- 1$ by $- 1$ .

$y = \frac{5 x}{6} + 1 (\frac{1}{2})$

Step 1.2.2.3.7.2

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

$y = \frac{5 x}{6} + \frac{1}{2}$

Step 2

Find the slope when $y = \frac{5 x}{6} + \frac{1}{2}$ .

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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 terms.

$y = \frac{5}{6} x + \frac{1}{2}$

Step 2.2

Using the slope-intercept form, the slope is $\frac{5}{6}$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

Multiply $\frac{6}{5}$ by $1$ .

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

Step 5

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

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Step 5.1

Use the slope $- \frac{6}{5}$ and a given point $(11, 12)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (12) = - \frac{6}{5} \cdot (x - (11))$

Step 5.2

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

$y - 12 = - \frac{6}{5} \cdot (x - 11)$

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 $- \frac{6}{5} \cdot (x - 11)$ .

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Step 6.1.1.1

Rewrite.

$y - 12 = 0 + 0 - \frac{6}{5} \cdot (x - 11)$

Step 6.1.1.2

Simplify by adding zeros.

$y - 12 = - \frac{6}{5} \cdot (x - 11)$

Step 6.1.1.3

Apply the distributive property.

$y - 12 = - \frac{6}{5} x - \frac{6}{5} \cdot - 11$

Step 6.1.1.4

Combine $x$ and $\frac{6}{5}$ .

$y - 12 = - \frac{x \cdot 6}{5} - \frac{6}{5} \cdot - 11$

Step 6.1.1.5

Multiply $- \frac{6}{5} \cdot - 11$ .

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Step 6.1.1.5.1

Multiply $- 11$ by $- 1$ .

$y - 12 = - \frac{x \cdot 6}{5} + 11 (\frac{6}{5})$

Step 6.1.1.5.2

Combine $11$ and $\frac{6}{5}$ .

$y - 12 = - \frac{x \cdot 6}{5} + \frac{11 \cdot 6}{5}$

Step 6.1.1.5.3

Multiply $11$ by $6$ .

$y - 12 = - \frac{x \cdot 6}{5} + \frac{66}{5}$

Step 6.1.1.6

Move $6$ to the left of $x$ .

$y - 12 = - \frac{6 x}{5} + \frac{66}{5}$

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 $12$ to both sides of the equation.

$y = - \frac{6 x}{5} + \frac{66}{5} + 12$

Step 6.1.2.2

To write $12$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = - \frac{6 x}{5} + \frac{66}{5} + 12 \cdot \frac{5}{5}$

Step 6.1.2.3

Combine $12$ and $\frac{5}{5}$ .

$y = - \frac{6 x}{5} + \frac{66}{5} + \frac{12 \cdot 5}{5}$

Step 6.1.2.4

Combine the numerators over the common denominator.

$y = - \frac{6 x}{5} + \frac{66 + 12 \cdot 5}{5}$

Step 6.1.2.5

Simplify the numerator.

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Step 6.1.2.5.1

Multiply $12$ by $5$ .

$y = - \frac{6 x}{5} + \frac{66 + 60}{5}$

Step 6.1.2.5.2

Add $66$ and $60$ .

$y = - \frac{6 x}{5} + \frac{126}{5}$

Step 6.2

Reorder terms.

$y = - (\frac{6}{5} x) + \frac{126}{5}$

Step 6.3

Remove parentheses.

$y = - \frac{6}{5} x + \frac{126}{5}$

Step 7