Algebra Examples

$\frac{x - 2}{4} - \frac{y - x}{2} = x - 7$ $\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1

Solve for $y$ in $\frac{x - 2}{4} - \frac{y - x}{2} = x - 7$ .

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Step 1.1

Simplify $\frac{x - 2}{4} - \frac{y - x}{2}$ .

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Step 1.1.1

To write $- \frac{y - x}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.1.2

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.2.1

Multiply $\frac{y - x}{2}$ by $\frac{2}{2}$ .

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

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.1.2.2

Multiply $2$ by $2$ .

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

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

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

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.1.3

Combine the numerators over the common denominator.

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

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.1.4

Simplify the numerator.

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Step 1.1.4.1

Apply the distributive property.

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

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.1.4.2

Multiply $- - x$ .

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Step 1.1.4.2.1

Multiply $- 1$ by $- 1$ .

$\frac{x - 2 + (- y + 1 x) \cdot 2}{4} = x - 7$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.1.4.2.2

Multiply $x$ by $1$ .

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

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

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

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.1.4.3

Apply the distributive property.

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

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.1.4.4

Multiply $2$ by $- 1$ .

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

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.1.4.5

Move $2$ to the left of $x$ .

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

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.1.4.6

Add $x$ and $2 x$ .

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

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

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

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

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

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.2

Multiply both sides by $4$ .

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

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.3

Simplify.

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Step 1.3.1

Simplify the left side.

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Step 1.3.1.1

Simplify $\frac{3 x - 2 - 2 y}{4} \cdot 4$ .

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Step 1.3.1.1.1

Cancel the common factor of $4$ .

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Step 1.3.1.1.1.1

Cancel the common factor.

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

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.3.1.1.1.2

Rewrite the expression.

$3 x - 2 - 2 y = (x - 7) \cdot 4$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

$3 x - 2 - 2 y = (x - 7) \cdot 4$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.3.1.1.2

Move $- 2$ .

$3 x - 2 y - 2 = (x - 7) \cdot 4$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

$3 x - 2 y - 2 = (x - 7) \cdot 4$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

$3 x - 2 y - 2 = (x - 7) \cdot 4$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.3.2

Simplify the right side.

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Step 1.3.2.1

Simplify $(x - 7) \cdot 4$ .

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Step 1.3.2.1.1

Apply the distributive property.

$3 x - 2 y - 2 = x \cdot 4 - 7 \cdot 4$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.3.2.1.2

Simplify the expression.

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Step 1.3.2.1.2.1

Move $4$ to the left of $x$ .

$3 x - 2 y - 2 = 4 \cdot x - 7 \cdot 4$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.3.2.1.2.2

Multiply $- 7$ by $4$ .

$3 x - 2 y - 2 = 4 x - 28$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

$3 x - 2 y - 2 = 4 x - 28$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

$3 x - 2 y - 2 = 4 x - 28$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

$3 x - 2 y - 2 = 4 x - 28$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

$3 x - 2 y - 2 = 4 x - 28$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.4

Solve for $y$ .

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Step 1.4.1

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

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Step 1.4.1.1

Subtract $3 x$ from both sides of the equation.

$- 2 y - 2 = 4 x - 28 - 3 x$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.4.1.2

Add $2$ to both sides of the equation.

$- 2 y = 4 x - 28 - 3 x + 2$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.4.1.3

Subtract $3 x$ from $4 x$ .

$- 2 y = x - 28 + 2$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.4.1.4

Add $- 28$ and $2$ .

$- 2 y = x - 26$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

$- 2 y = x - 26$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.4.2

Divide each term in $- 2 y = x - 26$ by $- 2$ and simplify.

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Step 1.4.2.1

Divide each term in $- 2 y = x - 26$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{x}{- 2} + \frac{- 26}{- 2}$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.4.2.2

Simplify the left side.

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Step 1.4.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.4.2.2.1.1

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{x}{- 2} + \frac{- 26}{- 2}$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{- 2} + \frac{- 26}{- 2}$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

$y = \frac{x}{- 2} + \frac{- 26}{- 2}$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

$y = \frac{x}{- 2} + \frac{- 26}{- 2}$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.4.2.3

Simplify the right side.

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Step 1.4.2.3.1

Simplify each term.

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Step 1.4.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{x}{2} + \frac{- 26}{- 2}$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 1.4.2.3.1.2

Divide $- 26$ by $- 2$ .

$y = - \frac{x}{2} + 13$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

$y = - \frac{x}{2} + 13$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

$y = - \frac{x}{2} + 13$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

$y = - \frac{x}{2} + 13$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

$y = - \frac{x}{2} + 13$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

$y = - \frac{x}{2} + 13$

$\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$

Step 2

Replace all occurrences of $y$ with $- \frac{x}{2} + 13$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $\frac{3 x - y}{8} - \frac{3 y - x}{6} = y - 13$ with $- \frac{x}{2} + 13$ .

$\frac{3 x - (- \frac{x}{2} + 13)}{8} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2

Simplify $\frac{3 x - (- \frac{x}{2} + 13)}{8} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$ .

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Simplify $\frac{3 x - (- \frac{x}{2} + 13)}{8} - \frac{3 (- \frac{x}{2} + 13) - x}{6}$ .

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Step 2.2.1.1.1

Simplify each term.

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Step 2.2.1.1.1.1

Simplify the numerator.

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Step 2.2.1.1.1.1.1

Apply the distributive property.

$\frac{3 x + \frac{x}{2} - 1 \cdot 13}{8} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.1.2

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

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Step 2.2.1.1.1.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{3 x + 1 (\frac{x}{2}) - 1 \cdot 13}{8} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.1.2.2

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

$\frac{3 x + \frac{x}{2} - 1 \cdot 13}{8} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

$\frac{3 x + \frac{x}{2} - 1 \cdot 13}{8} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.1.3

Multiply $- 1$ by $13$ .

$\frac{3 x + \frac{x}{2} - 13}{8} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.1.4

To write $3 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 x \cdot \frac{2}{2} + \frac{x}{2} - 13}{8} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.1.5

Combine $3 x$ and $\frac{2}{2}$ .

$\frac{\frac{3 x \cdot 2}{2} + \frac{x}{2} - 13}{8} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.1.6

Combine the numerators over the common denominator.

$\frac{\frac{3 x \cdot 2 + x}{2} - 13}{8} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.1.7

To write $- 13$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{3 x \cdot 2 + x}{2} - 13 \cdot \frac{2}{2}}{8} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.1.8

Combine $- 13$ and $\frac{2}{2}$ .

$\frac{\frac{3 x \cdot 2 + x}{2} + \frac{- 13 \cdot 2}{2}}{8} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.1.9

Combine the numerators over the common denominator.

$\frac{\frac{3 x \cdot 2 + x - 13 \cdot 2}{2}}{8} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.1.10

Rewrite $\frac{3 x \cdot 2 + x - 13 \cdot 2}{2}$ in a factored form.

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Step 2.2.1.1.1.1.10.1

Multiply $2$ by $3$ .

$\frac{\frac{6 x + x - 13 \cdot 2}{2}}{8} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.1.10.2

Multiply $- 13$ by $2$ .

$\frac{\frac{6 x + x - 26}{2}}{8} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.1.10.3

Add $6 x$ and $x$ .

$\frac{\frac{7 x - 26}{2}}{8} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

$\frac{\frac{7 x - 26}{2}}{8} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

$\frac{\frac{7 x - 26}{2}}{8} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{7 x - 26}{2} \cdot \frac{1}{8} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.3

Multiply $\frac{7 x - 26}{2} \cdot \frac{1}{8}$ .

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Step 2.2.1.1.1.3.1

Multiply $\frac{7 x - 26}{2}$ by $\frac{1}{8}$ .

$\frac{7 x - 26}{2 \cdot 8} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.3.2

Multiply $2$ by $8$ .

$\frac{7 x - 26}{16} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

$\frac{7 x - 26}{16} - \frac{3 (- \frac{x}{2} + 13) - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.4

Simplify the numerator.

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Step 2.2.1.1.1.4.1

Apply the distributive property.

$\frac{7 x - 26}{16} - \frac{3 (- \frac{x}{2}) + 3 \cdot 13 - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.4.2

Multiply $3 (- \frac{x}{2})$ .

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Step 2.2.1.1.1.4.2.1

Multiply $- 1$ by $3$ .

$\frac{7 x - 26}{16} - \frac{- 3 \frac{x}{2} + 3 \cdot 13 - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.4.2.2

Combine $- 3$ and $\frac{x}{2}$ .

$\frac{7 x - 26}{16} - \frac{\frac{- 3 x}{2} + 3 \cdot 13 - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

$\frac{7 x - 26}{16} - \frac{\frac{- 3 x}{2} + 3 \cdot 13 - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.4.3

Multiply $3$ by $13$ .

$\frac{7 x - 26}{16} - \frac{\frac{- 3 x}{2} + 39 - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.4.4

Move the negative in front of the fraction.

$\frac{7 x - 26}{16} - \frac{- \frac{3 x}{2} + 39 - x}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.4.5

To write $- x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{7 x - 26}{16} - \frac{- \frac{3 x}{2} - x \cdot \frac{2}{2} + 39}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.4.6

Combine $- x$ and $\frac{2}{2}$ .

$\frac{7 x - 26}{16} - \frac{- \frac{3 x}{2} + \frac{- x \cdot 2}{2} + 39}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.4.7

Combine the numerators over the common denominator.

$\frac{7 x - 26}{16} - \frac{\frac{- 3 x - x \cdot 2}{2} + 39}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.4.8

To write $39$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{7 x - 26}{16} - \frac{\frac{- 3 x - x \cdot 2}{2} + 39 \cdot \frac{2}{2}}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.4.9

Combine $39$ and $\frac{2}{2}$ .

$\frac{7 x - 26}{16} - \frac{\frac{- 3 x - x \cdot 2}{2} + \frac{39 \cdot 2}{2}}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.4.10

Combine the numerators over the common denominator.

$\frac{7 x - 26}{16} - \frac{\frac{- 3 x - x \cdot 2 + 39 \cdot 2}{2}}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.4.11

Reorder terms.

$\frac{7 x - 26}{16} - \frac{\frac{- 1 \cdot (2 x) - 3 x + 2 \cdot 39}{2}}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.4.12

Rewrite $\frac{- 1 \cdot 2 x - 3 x + 2 \cdot 39}{2}$ in a factored form.

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Step 2.2.1.1.1.4.12.1

Multiply $- 1$ by $2$ .

$\frac{7 x - 26}{16} - \frac{\frac{- 2 x - 3 x + 2 \cdot 39}{2}}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.4.12.2

Multiply $2$ by $39$ .

$\frac{7 x - 26}{16} - \frac{\frac{- 2 x - 3 x + 78}{2}}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.4.12.3

Subtract $3 x$ from $- 2 x$ .

$\frac{7 x - 26}{16} - \frac{\frac{- 5 x + 78}{2}}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

$\frac{7 x - 26}{16} - \frac{\frac{- 5 x + 78}{2}}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

$\frac{7 x - 26}{16} - \frac{\frac{- 5 x + 78}{2}}{6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{7 x - 26}{16} - (\frac{- 5 x + 78}{2} \cdot \frac{1}{6}) = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.6

Multiply $\frac{- 5 x + 78}{2} \cdot \frac{1}{6}$ .

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Step 2.2.1.1.1.6.1

Multiply $\frac{- 5 x + 78}{2}$ by $\frac{1}{6}$ .

$\frac{7 x - 26}{16} - \frac{- 5 x + 78}{2 \cdot 6} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.1.6.2

Multiply $2$ by $6$ .

$\frac{7 x - 26}{16} - \frac{- 5 x + 78}{12} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

$\frac{7 x - 26}{16} - \frac{- 5 x + 78}{12} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

$\frac{7 x - 26}{16} - \frac{- 5 x + 78}{12} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.2

To write $\frac{7 x - 26}{16}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{7 x - 26}{16} \cdot \frac{3}{3} - \frac{- 5 x + 78}{12} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.3

To write $- \frac{- 5 x + 78}{12}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{7 x - 26}{16} \cdot \frac{3}{3} - \frac{- 5 x + 78}{12} \cdot \frac{4}{4} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.4

Write each expression with a common denominator of $48$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.1.4.1

Multiply $\frac{7 x - 26}{16}$ by $\frac{3}{3}$ .

$\frac{(7 x - 26) \cdot 3}{16 \cdot 3} - \frac{- 5 x + 78}{12} \cdot \frac{4}{4} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.4.2

Multiply $16$ by $3$ .

$\frac{(7 x - 26) \cdot 3}{48} - \frac{- 5 x + 78}{12} \cdot \frac{4}{4} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.4.3

Multiply $\frac{- 5 x + 78}{12}$ by $\frac{4}{4}$ .

$\frac{(7 x - 26) \cdot 3}{48} - \frac{(- 5 x + 78) \cdot 4}{12 \cdot 4} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.4.4

Multiply $12$ by $4$ .

$\frac{(7 x - 26) \cdot 3}{48} - \frac{(- 5 x + 78) \cdot 4}{48} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

$\frac{(7 x - 26) \cdot 3}{48} - \frac{(- 5 x + 78) \cdot 4}{48} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.5

Combine the numerators over the common denominator.

$\frac{(7 x - 26) \cdot 3 - (- 5 x + 78) \cdot 4}{48} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.6

Simplify the numerator.

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Step 2.2.1.1.6.1

Apply the distributive property.

$\frac{7 x \cdot 3 - 26 \cdot 3 - (- 5 x + 78) \cdot 4}{48} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.6.2

Multiply $3$ by $7$ .

$\frac{21 x - 26 \cdot 3 - (- 5 x + 78) \cdot 4}{48} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.6.3

Multiply $- 26$ by $3$ .

$\frac{21 x - 78 - (- 5 x + 78) \cdot 4}{48} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.6.4

Apply the distributive property.

$\frac{21 x - 78 + (- (- 5 x) - 1 \cdot 78) \cdot 4}{48} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.6.5

Multiply $- 5$ by $- 1$ .

$\frac{21 x - 78 + (5 x - 1 \cdot 78) \cdot 4}{48} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.6.6

Multiply $- 1$ by $78$ .

$\frac{21 x - 78 + (5 x - 78) \cdot 4}{48} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.6.7

Apply the distributive property.

$\frac{21 x - 78 + 5 x \cdot 4 - 78 \cdot 4}{48} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.6.8

Multiply $4$ by $5$ .

$\frac{21 x - 78 + 20 x - 78 \cdot 4}{48} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.6.9

Multiply $- 78$ by $4$ .

$\frac{21 x - 78 + 20 x - 312}{48} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.6.10

Add $21 x$ and $20 x$ .

$\frac{41 x - 78 - 312}{48} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.1.1.6.11

Subtract $312$ from $- 78$ .

$\frac{41 x - 390}{48} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

$\frac{41 x - 390}{48} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

$\frac{41 x - 390}{48} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

$\frac{41 x - 390}{48} = (- \frac{x}{2} + 13) - 13$

$y = - \frac{x}{2} + 13$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Combine the opposite terms in $(- \frac{x}{2} + 13) - 13$ .

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Step 2.2.2.1.1

Subtract $13$ from $13$ .

$\frac{41 x - 390}{48} = - \frac{x}{2} + 0$

$y = - \frac{x}{2} + 13$

Step 2.2.2.1.2

Add $- \frac{x}{2}$ and $0$ .

$\frac{41 x - 390}{48} = - \frac{x}{2}$

$y = - \frac{x}{2} + 13$

$\frac{41 x - 390}{48} = - \frac{x}{2}$

$y = - \frac{x}{2} + 13$

$\frac{41 x - 390}{48} = - \frac{x}{2}$

$y = - \frac{x}{2} + 13$

$\frac{41 x - 390}{48} = - \frac{x}{2}$

$y = - \frac{x}{2} + 13$

$\frac{41 x - 390}{48} = - \frac{x}{2}$

$y = - \frac{x}{2} + 13$

Step 3

Solve for $x$ in $\frac{41 x - 390}{48} = - \frac{x}{2}$ .

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Step 3.1

Multiply both sides by $48$ .

$\frac{41 x - 390}{48} \cdot 48 = - \frac{x}{2} \cdot 48$

$y = - \frac{x}{2} + 13$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of $48$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{41 x - 390}{48} \cdot 48 = - \frac{x}{2} \cdot 48$

$y = - \frac{x}{2} + 13$

Step 3.2.1.1.2

Rewrite the expression.

$41 x - 390 = - \frac{x}{2} \cdot 48$

$y = - \frac{x}{2} + 13$

$41 x - 390 = - \frac{x}{2} \cdot 48$

$y = - \frac{x}{2} + 13$

$41 x - 390 = - \frac{x}{2} \cdot 48$

$y = - \frac{x}{2} + 13$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- \frac{x}{2} \cdot 48$ .

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Step 3.2.2.1.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1.1

Move the leading negative in $- \frac{x}{2}$ into the numerator.

$41 x - 390 = \frac{- x}{2} \cdot 48$

$y = - \frac{x}{2} + 13$

Step 3.2.2.1.1.2

Factor $2$ out of $48$ .

$41 x - 390 = \frac{- x}{2} \cdot (2 (24))$

$y = - \frac{x}{2} + 13$

Step 3.2.2.1.1.3

Cancel the common factor.

$41 x - 390 = \frac{- x}{2} \cdot (2 \cdot 24)$

$y = - \frac{x}{2} + 13$

Step 3.2.2.1.1.4

Rewrite the expression.

$41 x - 390 = - x \cdot 24$

$y = - \frac{x}{2} + 13$

$41 x - 390 = - x \cdot 24$

$y = - \frac{x}{2} + 13$

Step 3.2.2.1.2

Multiply $24$ by $- 1$ .

$41 x - 390 = - 24 x$

$y = - \frac{x}{2} + 13$

$41 x - 390 = - 24 x$

$y = - \frac{x}{2} + 13$

$41 x - 390 = - 24 x$

$y = - \frac{x}{2} + 13$

$41 x - 390 = - 24 x$

$y = - \frac{x}{2} + 13$

Step 3.3

Solve for $x$ .

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Step 3.3.1

Move all terms containing $x$ to the left side of the equation.

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Step 3.3.1.1

Add $24 x$ to both sides of the equation.

$41 x - 390 + 24 x = 0$

$y = - \frac{x}{2} + 13$

Step 3.3.1.2

Add $41 x$ and $24 x$ .

$65 x - 390 = 0$

$y = - \frac{x}{2} + 13$

$65 x - 390 = 0$

$y = - \frac{x}{2} + 13$

Step 3.3.2

Add $390$ to both sides of the equation.

$65 x = 390$

$y = - \frac{x}{2} + 13$

Step 3.3.3

Divide each term in $65 x = 390$ by $65$ and simplify.

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Step 3.3.3.1

Divide each term in $65 x = 390$ by $65$ .

$\frac{65 x}{65} = \frac{390}{65}$

$y = - \frac{x}{2} + 13$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $65$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{65 x}{65} = \frac{390}{65}$

$y = - \frac{x}{2} + 13$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{390}{65}$

$y = - \frac{x}{2} + 13$

$x = \frac{390}{65}$

$y = - \frac{x}{2} + 13$

$x = \frac{390}{65}$

$y = - \frac{x}{2} + 13$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $390$ by $65$ .

$x = 6$

$y = - \frac{x}{2} + 13$

$x = 6$

$y = - \frac{x}{2} + 13$

$x = 6$

$y = - \frac{x}{2} + 13$

$x = 6$

$y = - \frac{x}{2} + 13$

$x = 6$

$y = - \frac{x}{2} + 13$

Step 4

Replace all occurrences of $x$ with $6$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = - \frac{x}{2} + 13$ with $6$ .

$y = - \frac{6}{2} + 13$

$x = 6$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{6}{2} + 13$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Divide $6$ by $2$ .

$y = - 1 \cdot 3 + 13$

$x = 6$

Step 4.2.1.1.2

Multiply $- 1$ by $3$ .

$y = - 3 + 13$

$x = 6$

$y = - 3 + 13$

$x = 6$

Step 4.2.1.2

Add $- 3$ and $13$ .

$y = 10$

$x = 6$

$y = 10$

$x = 6$

$y = 10$

$x = 6$

$y = 10$

$x = 6$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(6, 10)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(6, 10)$

Equation Form:

$x = 6, y = 10$

Step 7