Precalculus Examples

$15 x - 9 y = - 11$ , $10 x + a y = 17$

Step 1

Solve the equation for $x$ .

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Step 1.1

Add $9 y$ to both sides of the equation.

$15 x = - 11 + 9 y$

$10 x + a y = 17$

Step 1.2

Divide each term in $15 x = - 11 + 9 y$ by $15$ and simplify.

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Step 1.2.1

Divide each term in $15 x = - 11 + 9 y$ by $15$ .

$\frac{15 x}{15} = \frac{- 11}{15} + \frac{9 y}{15}$

$10 x + a y = 17$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $15$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{15 x}{15} = \frac{- 11}{15} + \frac{9 y}{15}$

$10 x + a y = 17$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 11}{15} + \frac{9 y}{15}$

$10 x + a y = 17$

$x = \frac{- 11}{15} + \frac{9 y}{15}$

$10 x + a y = 17$

$x = \frac{- 11}{15} + \frac{9 y}{15}$

$10 x + a y = 17$

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

Move the negative in front of the fraction.

$x = - \frac{11}{15} + \frac{9 y}{15}$

$10 x + a y = 17$

Step 1.2.3.1.2

Cancel the common factor of $9$ and $15$ .

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Step 1.2.3.1.2.1

Factor $3$ out of $9 y$ .

$x = - \frac{11}{15} + \frac{3 (3 y)}{15}$

$10 x + a y = 17$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $3$ out of $15$ .

$x = - \frac{11}{15} + \frac{3 (3 y)}{3 (5)}$

$10 x + a y = 17$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$x = - \frac{11}{15} + \frac{3 (3 y)}{3 \cdot 5}$

$10 x + a y = 17$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$x = - \frac{11}{15} + \frac{3 y}{5}$

$10 x + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

$10 x + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

$10 x + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

$10 x + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

$10 x + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

$10 x + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

$10 x + a y = 17$

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.

$10 (- \frac{11}{15}) + 10 (\frac{3 y}{5}) + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.1.2

Cancel the common factor of $5$ .

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Step 2.1.2.1

Move the leading negative in $- \frac{11}{15}$ into the numerator.

$10 (\frac{- 11}{15}) + 10 (\frac{3 y}{5}) + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.1.2.2

Factor $5$ out of $10$ .

$5 (2) (\frac{- 11}{15}) + 10 (\frac{3 y}{5}) + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.1.2.3

Factor $5$ out of $15$ .

$5 \cdot (2 (\frac{- 11}{5 \cdot 3})) + 10 (\frac{3 y}{5}) + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.1.2.4

Cancel the common factor.

$5 \cdot (2 (\frac{- 11}{5 \cdot 3})) + 10 (\frac{3 y}{5}) + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.1.2.5

Rewrite the expression.

$2 (\frac{- 11}{3}) + 10 (\frac{3 y}{5}) + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

$2 (\frac{- 11}{3}) + 10 (\frac{3 y}{5}) + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.1.3

Combine $2$ and $\frac{- 11}{3}$ .

$\frac{2 \cdot - 11}{3} + 10 (\frac{3 y}{5}) + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.1.4

Multiply $2$ by $- 11$ .

$\frac{- 22}{3} + 10 (\frac{3 y}{5}) + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.1.5

Cancel the common factor of $5$ .

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Step 2.1.5.1

Factor $5$ out of $10$ .

$\frac{- 22}{3} + 5 (2) (\frac{3 y}{5}) + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.1.5.2

Cancel the common factor.

$\frac{- 22}{3} + 5 \cdot (2 (\frac{3 y}{5})) + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.1.5.3

Rewrite the expression.

$\frac{- 22}{3} + 2 (3 y) + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

$\frac{- 22}{3} + 2 (3 y) + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.1.6

Multiply $3$ by $2$ .

$\frac{- 22}{3} + 6 y + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.1.7

Move the negative in front of the fraction.

$- \frac{22}{3} + 6 y + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

$- \frac{22}{3} + 6 y + a y = 17$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.2

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

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Step 2.2.1

Add $\frac{22}{3}$ to both sides of the equation.

$6 y + a y = 17 + \frac{22}{3}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.2.2

To write $17$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$6 y + a y = 17 \cdot \frac{3}{3} + \frac{22}{3}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.2.3

Combine $17$ and $\frac{3}{3}$ .

$6 y + a y = \frac{17 \cdot 3}{3} + \frac{22}{3}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.2.4

Combine the numerators over the common denominator.

$6 y + a y = \frac{17 \cdot 3 + 22}{3}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply $17$ by $3$ .

$6 y + a y = \frac{51 + 22}{3}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.2.5.2

Add $51$ and $22$ .

$6 y + a y = \frac{73}{3}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

$6 y + a y = \frac{73}{3}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

$6 y + a y = \frac{73}{3}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.3

Factor $y$ out of $6 y + a y$ .

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Step 2.3.1

Factor $y$ out of $6 y$ .

$y \cdot 6 + a y = \frac{73}{3}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.3.2

Factor $y$ out of $a y$ .

$y \cdot 6 + y a = \frac{73}{3}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.3.3

Factor $y$ out of $y \cdot 6 + y a$ .

$y \cdot (6 + a) = \frac{73}{3}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.3.4

Multiply $y$ by $6 + a$ .

$y (6 + a) = \frac{73}{3}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

$y (6 + a) = \frac{73}{3}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.4

Divide each term in $y (6 + a) = \frac{73}{3}$ by $6 + a$ and simplify.

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Step 2.4.1

Divide each term in $y (6 + a) = \frac{73}{3}$ by $6 + a$ .

$\frac{y (6 + a)}{6 + a} = \frac{\frac{73}{3}}{6 + a}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $6 + a$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{y (6 + a)}{6 + a} = \frac{\frac{73}{3}}{6 + a}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{73}{3}}{6 + a}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

$y = \frac{\frac{73}{3}}{6 + a}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

$y = \frac{\frac{73}{3}}{6 + a}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{73}{3} \cdot \frac{1}{6 + a}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

Step 2.4.3.2

Multiply $\frac{73}{3}$ by $\frac{1}{6 + a}$ .

$y = \frac{73}{3 (6 + a)}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

$y = \frac{73}{3 (6 + a)}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

$y = \frac{73}{3 (6 + a)}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

$y = \frac{73}{3 (6 + a)}$

$x = - \frac{11}{15} + \frac{3 y}{5}$

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 $3$ and $\frac{73}{3 (6 + a)}$ .

$x = - \frac{11}{15} + \frac{\frac{3 \cdot 73}{3 (6 + a)}}{5}$

$y = \frac{73}{3 (6 + a)}$

Step 3.1.2

Multiply $3$ by $73$ .

$x = - \frac{11}{15} + \frac{\frac{219}{3 (6 + a)}}{5}$

$y = \frac{73}{3 (6 + a)}$

Step 3.1.3

Reduce the expression $\frac{219}{3 (6 + a)}$ by cancelling the common factors.

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Step 3.1.3.1

Factor $3$ out of $219$ .

$x = - \frac{11}{15} + \frac{\frac{3 \cdot 73}{3 (6 + a)}}{5}$

$y = \frac{73}{3 (6 + a)}$

Step 3.1.3.2

Cancel the common factor.

$x = - \frac{11}{15} + \frac{\frac{3 \cdot 73}{3 (6 + a)}}{5}$

$y = \frac{73}{3 (6 + a)}$

Step 3.1.3.3

Rewrite the expression.

$x = - \frac{11}{15} + \frac{\frac{73}{6 + a}}{5}$

$y = \frac{73}{3 (6 + a)}$

$x = - \frac{11}{15} + \frac{\frac{73}{6 + a}}{5}$

$y = \frac{73}{3 (6 + a)}$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{11}{15} + \frac{73}{6 + a} \cdot \frac{1}{5}$

$y = \frac{73}{3 (6 + a)}$

Step 3.1.5

Multiply $\frac{73}{6 + a}$ by $\frac{1}{5}$ .

$x = - \frac{11}{15} + \frac{73}{(6 + a) \cdot 5}$

$y = \frac{73}{3 (6 + a)}$

Step 3.1.6

Move $5$ to the left of $6 + a$ .

$x = - \frac{11}{15} + \frac{73}{5 (6 + a)}$

$y = \frac{73}{3 (6 + a)}$

$x = - \frac{11}{15} + \frac{73}{5 (6 + a)}$

$y = \frac{73}{3 (6 + a)}$

$x = - \frac{11}{15} + \frac{73}{5 (6 + a)}$

$y = \frac{73}{3 (6 + a)}$

Step 4

Simplify the right side.

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Step 4.1

Reorder $6$ and $a$ .

$x = - \frac{11}{15} + \frac{73}{5 (a + 6)}$

$y = \frac{73}{3 (6 + a)}$

$x = - \frac{11}{15} + \frac{73}{5 (a + 6)}$

$y = \frac{73}{3 (6 + a)}$