Precalculus Examples

$- 8 x + 12 y = - 17$ , $12 x + a y = 4$

Step 1

Solve the equation for $a$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$12 (\frac{17}{8}) + 12 (\frac{3 y}{2}) + a y = 4$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.1.2

Cancel the common factor of $4$ .

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Step 1.1.2.1

Factor $4$ out of $12$ .

$4 (3) (\frac{17}{8}) + 12 (\frac{3 y}{2}) + a y = 4$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.1.2.2

Factor $4$ out of $8$ .

$4 \cdot (3 (\frac{17}{4 \cdot 2})) + 12 (\frac{3 y}{2}) + a y = 4$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.1.2.3

Cancel the common factor.

$4 \cdot (3 (\frac{17}{4 \cdot 2})) + 12 (\frac{3 y}{2}) + a y = 4$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.1.2.4

Rewrite the expression.

$3 (\frac{17}{2}) + 12 (\frac{3 y}{2}) + a y = 4$

$x = \frac{17}{8} + \frac{3 y}{2}$

$3 (\frac{17}{2}) + 12 (\frac{3 y}{2}) + a y = 4$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.1.3

Combine $3$ and $\frac{17}{2}$ .

$\frac{3 \cdot 17}{2} + 12 (\frac{3 y}{2}) + a y = 4$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.1.4

Multiply $3$ by $17$ .

$\frac{51}{2} + 12 (\frac{3 y}{2}) + a y = 4$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.1.5

Cancel the common factor of $2$ .

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Step 1.1.5.1

Factor $2$ out of $12$ .

$\frac{51}{2} + 2 (6) (\frac{3 y}{2}) + a y = 4$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.1.5.2

Cancel the common factor.

$\frac{51}{2} + 2 \cdot (6 (\frac{3 y}{2})) + a y = 4$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.1.5.3

Rewrite the expression.

$\frac{51}{2} + 6 (3 y) + a y = 4$

$x = \frac{17}{8} + \frac{3 y}{2}$

$\frac{51}{2} + 6 (3 y) + a y = 4$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.1.6

Multiply $3$ by $6$ .

$\frac{51}{2} + 18 y + a y = 4$

$x = \frac{17}{8} + \frac{3 y}{2}$

$\frac{51}{2} + 18 y + a y = 4$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.2

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

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Step 1.2.1

Subtract $\frac{51}{2}$ from both sides of the equation.

$18 y + a y = 4 - \frac{51}{2}$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.2.2

Subtract $18 y$ from both sides of the equation.

$a y = 4 - \frac{51}{2} - 18 y$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.2.3

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

$a y = 4 \cdot \frac{2}{2} - \frac{51}{2} - 18 y$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.2.4

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

$a y = \frac{4 \cdot 2}{2} - \frac{51}{2} - 18 y$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.2.5

Combine the numerators over the common denominator.

$a y = \frac{4 \cdot 2 - 51}{2} - 18 y$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $4$ by $2$ .

$a y = \frac{8 - 51}{2} - 18 y$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.2.6.2

Subtract $51$ from $8$ .

$a y = \frac{- 43}{2} - 18 y$

$x = \frac{17}{8} + \frac{3 y}{2}$

$a y = \frac{- 43}{2} - 18 y$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.2.7

Move the negative in front of the fraction.

$a y = - \frac{43}{2} - 18 y$

$x = \frac{17}{8} + \frac{3 y}{2}$

$a y = - \frac{43}{2} - 18 y$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.3

Divide each term in $a y = - \frac{43}{2} - 18 y$ by $y$ and simplify.

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Step 1.3.1

Divide each term in $a y = - \frac{43}{2} - 18 y$ by $y$ .

$\frac{a y}{y} = \frac{- \frac{43}{2}}{y} + \frac{- 18 y}{y}$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $y$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{a y}{y} = \frac{- \frac{43}{2}}{y} + \frac{- 18 y}{y}$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- \frac{43}{2}}{y} + \frac{- 18 y}{y}$

$x = \frac{17}{8} + \frac{3 y}{2}$

$a = \frac{- \frac{43}{2}}{y} + \frac{- 18 y}{y}$

$x = \frac{17}{8} + \frac{3 y}{2}$

$a = \frac{- \frac{43}{2}}{y} + \frac{- 18 y}{y}$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$a = - \frac{43}{2} \cdot \frac{1}{y} + \frac{- 18 y}{y}$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.3.3.1.2

Multiply $\frac{1}{y}$ by $\frac{43}{2}$ .

$a = - \frac{43}{y \cdot 2} + \frac{- 18 y}{y}$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.3.3.1.3

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

$a = - \frac{43}{2 \cdot y} + \frac{- 18 y}{y}$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.3.3.1.4

Cancel the common factor of $y$ .

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Step 1.3.3.1.4.1

Cancel the common factor.

$a = - \frac{43}{2 y} + \frac{- 18 y}{y}$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 1.3.3.1.4.2

Divide $- 18$ by $1$ .

$a = - \frac{43}{2 y} - 18$

$x = \frac{17}{8} + \frac{3 y}{2}$

$a = - \frac{43}{2 y} - 18$

$x = \frac{17}{8} + \frac{3 y}{2}$

$a = - \frac{43}{2 y} - 18$

$x = \frac{17}{8} + \frac{3 y}{2}$

$a = - \frac{43}{2 y} - 18$

$x = \frac{17}{8} + \frac{3 y}{2}$

$a = - \frac{43}{2 y} - 18$

$x = \frac{17}{8} + \frac{3 y}{2}$

$a = - \frac{43}{2 y} - 18$

$x = \frac{17}{8} + \frac{3 y}{2}$

Step 2

Simplify the right side.

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Step 2.1

Reorder $\frac{17}{8}$ and $\frac{3 y}{2}$ .

$x = \frac{3 y}{2} + \frac{17}{8}$

$a = - \frac{43}{2 y} - 18$

$x = \frac{3 y}{2} + \frac{17}{8}$

$a = - \frac{43}{2 y} - 18$