Precalculus Examples

$- 6 x + b y = - 14$ , $15 x + 20 y = 13$

Step 1

Solve the equation for $x$ .

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Step 1.1

Subtract $b y$ from both sides of the equation.

$- 6 x = - 14 - b y$

$15 x + 20 y = 13$

Step 1.2

Divide each term in $- 6 x = - 14 - b y$ by $- 6$ and simplify.

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Step 1.2.1

Divide each term in $- 6 x = - 14 - b y$ by $- 6$ .

$\frac{- 6 x}{- 6} = \frac{- 14}{- 6} + \frac{- b y}{- 6}$

$15 x + 20 y = 13$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 6$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 6 x}{- 6} = \frac{- 14}{- 6} + \frac{- b y}{- 6}$

$15 x + 20 y = 13$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 14}{- 6} + \frac{- b y}{- 6}$

$15 x + 20 y = 13$

$x = \frac{- 14}{- 6} + \frac{- b y}{- 6}$

$15 x + 20 y = 13$

$x = \frac{- 14}{- 6} + \frac{- b y}{- 6}$

$15 x + 20 y = 13$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Cancel the common factor of $- 14$ and $- 6$ .

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Step 1.2.3.1.1.1

Factor $- 2$ out of $- 14$ .

$x = \frac{- 2 \cdot 7}{- 6} + \frac{- b y}{- 6}$

$15 x + 20 y = 13$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $- 2$ out of $- 6$ .

$x = \frac{- 2 \cdot 7}{- 2 \cdot 3} + \frac{- b y}{- 6}$

$15 x + 20 y = 13$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{- 2 \cdot 7}{- 2 \cdot 3} + \frac{- b y}{- 6}$

$15 x + 20 y = 13$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{7}{3} + \frac{- b y}{- 6}$

$15 x + 20 y = 13$

$x = \frac{7}{3} + \frac{- b y}{- 6}$

$15 x + 20 y = 13$

$x = \frac{7}{3} + \frac{- b y}{- 6}$

$15 x + 20 y = 13$

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

$x = \frac{7}{3} + \frac{b y}{6}$

$15 x + 20 y = 13$

$x = \frac{7}{3} + \frac{b y}{6}$

$15 x + 20 y = 13$

$x = \frac{7}{3} + \frac{b y}{6}$

$15 x + 20 y = 13$

$x = \frac{7}{3} + \frac{b y}{6}$

$15 x + 20 y = 13$

$x = \frac{7}{3} + \frac{b y}{6}$

$15 x + 20 y = 13$

Step 2

Solve the equation for $y$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$15 (\frac{7}{3}) + 15 (\frac{b y}{6}) + 20 y = 13$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.1.2

Cancel the common factor of $3$ .

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Step 2.1.2.1

Factor $3$ out of $15$ .

$3 (5) (\frac{7}{3}) + 15 (\frac{b y}{6}) + 20 y = 13$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.1.2.2

Cancel the common factor.

$3 \cdot (5 (\frac{7}{3})) + 15 (\frac{b y}{6}) + 20 y = 13$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.1.2.3

Rewrite the expression.

$5 \cdot 7 + 15 (\frac{b y}{6}) + 20 y = 13$

$x = \frac{7}{3} + \frac{b y}{6}$

$5 \cdot 7 + 15 (\frac{b y}{6}) + 20 y = 13$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.1.3

Multiply $5$ by $7$ .

$35 + 15 (\frac{b y}{6}) + 20 y = 13$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.1.4

Cancel the common factor of $3$ .

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Step 2.1.4.1

Factor $3$ out of $15$ .

$35 + 3 (5) (\frac{b y}{6}) + 20 y = 13$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.1.4.2

Factor $3$ out of $6$ .

$35 + 3 \cdot (5 (\frac{b y}{3 \cdot 2})) + 20 y = 13$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.1.4.3

Cancel the common factor.

$35 + 3 \cdot (5 (\frac{b y}{3 \cdot 2})) + 20 y = 13$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.1.4.4

Rewrite the expression.

$35 + 5 (\frac{b y}{2}) + 20 y = 13$

$x = \frac{7}{3} + \frac{b y}{6}$

$35 + 5 (\frac{b y}{2}) + 20 y = 13$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.1.5

Combine $5$ and $\frac{b y}{2}$ .

$35 + \frac{5 b y}{2} + 20 y = 13$

$x = \frac{7}{3} + \frac{b y}{6}$

$35 + \frac{5 b y}{2} + 20 y = 13$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.2

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

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Step 2.2.1

Subtract $35$ from both sides of the equation.

$\frac{5 b y}{2} + 20 y = 13 - 35$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.2.2

Subtract $35$ from $13$ .

$\frac{5 b y}{2} + 20 y = - 22$

$x = \frac{7}{3} + \frac{b y}{6}$

$\frac{5 b y}{2} + 20 y = - 22$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.3

Factor $5 y$ out of $\frac{5 b y}{2} + 20 y$ .

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Step 2.3.1

Factor $5 y$ out of $\frac{5 b y}{2}$ .

$5 y (\frac{b}{2}) + 20 y = - 22$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.3.2

Factor $5 y$ out of $20 y$ .

$5 y (\frac{b}{2}) + 5 y (4) = - 22$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.3.3

Factor $5 y$ out of $5 y \frac{b}{2} + 5 y (4)$ .

$5 y (\frac{b}{2} + 4) = - 22$

$x = \frac{7}{3} + \frac{b y}{6}$

$5 y (\frac{b}{2} + 4) = - 22$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.4

Divide each term in $5 y (\frac{b}{2} + 4) = - 22$ by $5 (\frac{b}{2} + 4)$ and simplify.

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Step 2.4.1

Divide each term in $5 y (\frac{b}{2} + 4) = - 22$ by $5 (\frac{b}{2} + 4)$ .

$\frac{5 y (\frac{b}{2} + 4)}{5 (\frac{b}{2} + 4)} = \frac{- 22}{5 (\frac{b}{2} + 4)}$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $5$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{5 y (\frac{b}{2} + 4)}{5 (\frac{b}{2} + 4)} = \frac{- 22}{5 (\frac{b}{2} + 4)}$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.4.2.1.2

Rewrite the expression.

$\frac{y (\frac{b}{2} + 4)}{\frac{b}{2} + 4} = \frac{- 22}{5 (\frac{b}{2} + 4)}$

$x = \frac{7}{3} + \frac{b y}{6}$

$\frac{y (\frac{b}{2} + 4)}{\frac{b}{2} + 4} = \frac{- 22}{5 (\frac{b}{2} + 4)}$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.4.2.2

Cancel the common factor of $\frac{b}{2} + 4$ .

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Step 2.4.2.2.1

Cancel the common factor.

$\frac{y (\frac{b}{2} + 4)}{\frac{b}{2} + 4} = \frac{- 22}{5 (\frac{b}{2} + 4)}$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.4.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 22}{5 (\frac{b}{2} + 4)}$

$x = \frac{7}{3} + \frac{b y}{6}$

$y = \frac{- 22}{5 (\frac{b}{2} + 4)}$

$x = \frac{7}{3} + \frac{b y}{6}$

$y = \frac{- 22}{5 (\frac{b}{2} + 4)}$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Simplify the denominator.

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Step 2.4.3.1.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{- 22}{5 (\frac{b}{2} + 4 \cdot \frac{2}{2})}$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.4.3.1.2

Combine $4$ and $\frac{2}{2}$ .

$y = \frac{- 22}{5 (\frac{b}{2} + \frac{4 \cdot 2}{2})}$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.4.3.1.3

Combine the numerators over the common denominator.

$y = \frac{- 22}{5 (\frac{b + 4 \cdot 2}{2})}$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.4.3.1.4

Multiply $4$ by $2$ .

$y = \frac{- 22}{5 (\frac{b + 8}{2})}$

$x = \frac{7}{3} + \frac{b y}{6}$

$y = \frac{- 22}{5 (\frac{b + 8}{2})}$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.4.3.2

Combine $5$ and $\frac{b + 8}{2}$ .

$y = \frac{- 22}{\frac{5 (b + 8)}{2}}$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.4.3.3

Multiply the numerator by the reciprocal of the denominator.

$y = - 22 \frac{2}{5 (b + 8)}$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.4.3.4

Multiply $- 22 \frac{2}{5 (b + 8)}$ .

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Step 2.4.3.4.1

Combine $- 22$ and $\frac{2}{5 (b + 8)}$ .

$y = \frac{- 22 \cdot 2}{5 (b + 8)}$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.4.3.4.2

Multiply $- 22$ by $2$ .

$y = \frac{- 44}{5 (b + 8)}$

$x = \frac{7}{3} + \frac{b y}{6}$

$y = \frac{- 44}{5 (b + 8)}$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 2.4.3.5

Move the negative in front of the fraction.

$y = - \frac{44}{5 (b + 8)}$

$x = \frac{7}{3} + \frac{b y}{6}$

$y = - \frac{44}{5 (b + 8)}$

$x = \frac{7}{3} + \frac{b y}{6}$

$y = - \frac{44}{5 (b + 8)}$

$x = \frac{7}{3} + \frac{b y}{6}$

$y = - \frac{44}{5 (b + 8)}$

$x = \frac{7}{3} + \frac{b y}{6}$

Step 3

Simplify the right side.

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Step 3.1

Simplify each term.

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Step 3.1.1

Combine $b$ and $\frac{44}{5 (b + 8)}$ .

$x = \frac{7}{3} + \frac{- \frac{b \cdot 44}{5 (b + 8)}}{6}$

$y = - \frac{44}{5 (b + 8)}$

Step 3.1.2

Move $44$ to the left of $b$ .

$x = \frac{7}{3} + \frac{- \frac{44 b}{5 (b + 8)}}{6}$

$y = - \frac{44}{5 (b + 8)}$

Step 3.1.3

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{7}{3} - \frac{44 b}{5 (b + 8)} \cdot \frac{1}{6}$

$y = - \frac{44}{5 (b + 8)}$

Step 3.1.4

Cancel the common factor of $2$ .

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Step 3.1.4.1

Move the leading negative in $- \frac{44 b}{5 (b + 8)}$ into the numerator.

$x = \frac{7}{3} + \frac{- 44 b}{5 (b + 8)} \cdot \frac{1}{6}$

$y = - \frac{44}{5 (b + 8)}$

Step 3.1.4.2

Factor $2$ out of $- 44 b$ .

$x = \frac{7}{3} + \frac{2 (- 22 b)}{5 (b + 8)} \cdot \frac{1}{6}$

$y = - \frac{44}{5 (b + 8)}$

Step 3.1.4.3

Factor $2$ out of $6$ .

$x = \frac{7}{3} + \frac{2 (- 22 b)}{5 (b + 8)} \cdot \frac{1}{2 (3)}$

$y = - \frac{44}{5 (b + 8)}$

Step 3.1.4.4

Cancel the common factor.

$x = \frac{7}{3} + \frac{2 (- 22 b)}{5 (b + 8)} \cdot \frac{1}{2 \cdot 3}$

$y = - \frac{44}{5 (b + 8)}$

Step 3.1.4.5

Rewrite the expression.

$x = \frac{7}{3} + \frac{- 22 b}{5 (b + 8)} \cdot \frac{1}{3}$

$y = - \frac{44}{5 (b + 8)}$

$x = \frac{7}{3} + \frac{- 22 b}{5 (b + 8)} \cdot \frac{1}{3}$

$y = - \frac{44}{5 (b + 8)}$

Step 3.1.5

Multiply $\frac{- 22 b}{5 (b + 8)}$ by $\frac{1}{3}$ .

$x = \frac{7}{3} + \frac{- 22 b}{5 (b + 8) \cdot 3}$

$y = - \frac{44}{5 (b + 8)}$

Step 3.1.6

Multiply $3$ by $5$ .

$x = \frac{7}{3} + \frac{- 22 b}{15 (b + 8)}$

$y = - \frac{44}{5 (b + 8)}$

Step 3.1.7

Move the negative in front of the fraction.

$x = \frac{7}{3} - \frac{22 b}{15 (b + 8)}$

$y = - \frac{44}{5 (b + 8)}$

$x = \frac{7}{3} - \frac{22 b}{15 (b + 8)}$

$y = - \frac{44}{5 (b + 8)}$

$x = \frac{7}{3} - \frac{22 b}{15 (b + 8)}$

$y = - \frac{44}{5 (b + 8)}$