Precalculus Examples

$- 9 a = 4$ , $9 a + 9 c = 6$ , $2 b + 8 c = 15$

Step 1

Divide each term in $- 9 a = 4$ by $- 9$ and simplify.

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Step 1.1

Divide each term in $- 9 a = 4$ by $- 9$ .

$\frac{- 9 a}{- 9} = \frac{4}{- 9}$

$9 a + 9 c = 6$

$2 b + 8 c = 15$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $- 9$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{- 9 a}{- 9} = \frac{4}{- 9}$

$9 a + 9 c = 6$

$2 b + 8 c = 15$

Step 1.2.1.2

Divide $a$ by $1$ .

$a = \frac{4}{- 9}$

$9 a + 9 c = 6$

$2 b + 8 c = 15$

$a = \frac{4}{- 9}$

$9 a + 9 c = 6$

$2 b + 8 c = 15$

$a = \frac{4}{- 9}$

$9 a + 9 c = 6$

$2 b + 8 c = 15$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

$a = - \frac{4}{9}$

$9 a + 9 c = 6$

$2 b + 8 c = 15$

$a = - \frac{4}{9}$

$9 a + 9 c = 6$

$2 b + 8 c = 15$

$a = - \frac{4}{9}$

$9 a + 9 c = 6$

$2 b + 8 c = 15$

Step 2

Replace all occurrences of $a$ with $- \frac{4}{9}$ in each equation.

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Step 2.1

Replace all occurrences of $a$ in $9 a + 9 c = 6$ with $- \frac{4}{9}$ .

$9 (- \frac{4}{9}) + 9 c = 6$

$a = - \frac{4}{9}$

$2 b + 8 c = 15$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $9$ .

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Step 2.2.1.1

Move the leading negative in $- \frac{4}{9}$ into the numerator.

$9 (\frac{- 4}{9}) + 9 c = 6$

$a = - \frac{4}{9}$

$2 b + 8 c = 15$

Step 2.2.1.2

Cancel the common factor.

$9 (\frac{- 4}{9}) + 9 c = 6$

$a = - \frac{4}{9}$

$2 b + 8 c = 15$

Step 2.2.1.3

Rewrite the expression.

$- 4 + 9 c = 6$

$a = - \frac{4}{9}$

$2 b + 8 c = 15$

$- 4 + 9 c = 6$

$a = - \frac{4}{9}$

$2 b + 8 c = 15$

$- 4 + 9 c = 6$

$a = - \frac{4}{9}$

$2 b + 8 c = 15$

$- 4 + 9 c = 6$

$a = - \frac{4}{9}$

$2 b + 8 c = 15$

Step 3

Solve for $c$ in $- 4 + 9 c = 6$ .

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Step 3.1

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

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Step 3.1.1

Add $4$ to both sides of the equation.

$9 c = 6 + 4$

$a = - \frac{4}{9}$

$2 b + 8 c = 15$

Step 3.1.2

Add $6$ and $4$ .

$9 c = 10$

$a = - \frac{4}{9}$

$2 b + 8 c = 15$

$9 c = 10$

$a = - \frac{4}{9}$

$2 b + 8 c = 15$

Step 3.2

Divide each term in $9 c = 10$ by $9$ and simplify.

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Step 3.2.1

Divide each term in $9 c = 10$ by $9$ .

$\frac{9 c}{9} = \frac{10}{9}$

$a = - \frac{4}{9}$

$2 b + 8 c = 15$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $9$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{9 c}{9} = \frac{10}{9}$

$a = - \frac{4}{9}$

$2 b + 8 c = 15$

Step 3.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

$2 b + 8 c = 15$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

$2 b + 8 c = 15$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

$2 b + 8 c = 15$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

$2 b + 8 c = 15$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

$2 b + 8 c = 15$

Step 4

Replace all occurrences of $c$ with $\frac{10}{9}$ in each equation.

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Step 4.1

Replace all occurrences of $c$ in $2 b + 8 c = 15$ with $\frac{10}{9}$ .

$2 b + 8 (\frac{10}{9}) = 15$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Multiply $8 (\frac{10}{9})$ .

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Step 4.2.1.1

Combine $8$ and $\frac{10}{9}$ .

$2 b + \frac{8 \cdot 10}{9} = 15$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

Step 4.2.1.2

Multiply $8$ by $10$ .

$2 b + \frac{80}{9} = 15$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

$2 b + \frac{80}{9} = 15$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

$2 b + \frac{80}{9} = 15$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

$2 b + \frac{80}{9} = 15$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

Step 5

Solve for $b$ in $2 b + \frac{80}{9} = 15$ .

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Step 5.1

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

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Step 5.1.1

Subtract $\frac{80}{9}$ from both sides of the equation.

$2 b = 15 - \frac{80}{9}$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

Step 5.1.2

To write $15$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$2 b = 15 \cdot \frac{9}{9} - \frac{80}{9}$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

Step 5.1.3

Combine $15$ and $\frac{9}{9}$ .

$2 b = \frac{15 \cdot 9}{9} - \frac{80}{9}$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

Step 5.1.4

Combine the numerators over the common denominator.

$2 b = \frac{15 \cdot 9 - 80}{9}$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

Step 5.1.5

Simplify the numerator.

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Step 5.1.5.1

Multiply $15$ by $9$ .

$2 b = \frac{135 - 80}{9}$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

Step 5.1.5.2

Subtract $80$ from $135$ .

$2 b = \frac{55}{9}$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

$2 b = \frac{55}{9}$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

$2 b = \frac{55}{9}$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

Step 5.2

Divide each term in $2 b = \frac{55}{9}$ by $2$ and simplify.

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Step 5.2.1

Divide each term in $2 b = \frac{55}{9}$ by $2$ .

$\frac{2 b}{2} = \frac{\frac{55}{9}}{2}$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{2 b}{2} = \frac{\frac{55}{9}}{2}$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

Step 5.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{\frac{55}{9}}{2}$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

$b = \frac{\frac{55}{9}}{2}$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

$b = \frac{\frac{55}{9}}{2}$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$b = \frac{55}{9} \cdot \frac{1}{2}$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

Step 5.2.3.2

Multiply $\frac{55}{9} \cdot \frac{1}{2}$ .

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Step 5.2.3.2.1

Multiply $\frac{55}{9}$ by $\frac{1}{2}$ .

$b = \frac{55}{9 \cdot 2}$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

Step 5.2.3.2.2

Multiply $9$ by $2$ .

$b = \frac{55}{18}$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

$b = \frac{55}{18}$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

$b = \frac{55}{18}$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

$b = \frac{55}{18}$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

$b = \frac{55}{18}$

$c = \frac{10}{9}$

$a = - \frac{4}{9}$

Step 6

Solve the system of equations.

$b = \frac{55}{18} c = \frac{10}{9} a = - \frac{4}{9}$

Step 7

List all of the solutions.

$b = \frac{55}{18}, c = \frac{10}{9}, a = - \frac{4}{9}$