Precalculus Examples

$3 x + y = 5$ , $4.5 y = c + 9 x$

Step 1

Move all terms containing variables to the left side of the equation.

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Step 1.1

Subtract $c$ from both sides of the equation.

$3 x + y = 5, 4.5 y - c = 9 x$

Step 1.2

Subtract $9 x$ from both sides of the equation.

$3 x + y = 5, 4.5 y - c - 9 x = 0$

Step 2

Reorder the polynomial.

$- c + 4.5 y - 9 x = 0$

$3 x + y = 5$

Step 3

Move $4.5 y$ .

$- c - 9 x + 4.5 y = 0$

$3 x + y = 5$

Step 4

Multiply each equation by the value that makes the coefficients of $x$ opposite.

$(\frac{1}{9}) \cdot (3 x + y) = (\frac{1}{9}) (5)$

$(\frac{1}{27}) \cdot (- c - 9 x + 4.5 y) = (\frac{1}{27}) (0)$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $(\frac{1}{9}) \cdot (3 x + y)$ .

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Step 5.1.1.1

Apply the distributive property.

$\frac{1}{9} \cdot (3 x) + \frac{1}{9} y = (\frac{1}{9}) (5)$

$(\frac{1}{27}) \cdot (- c - 9 x + 4.5 y) = (\frac{1}{27}) (0)$

Step 5.1.1.2

Cancel the common factor of $3$ .

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Step 5.1.1.2.1

Factor $3$ out of $9$ .

$\frac{1}{3 (3)} \cdot (3 x) + \frac{1}{9} y = (\frac{1}{9}) (5)$

$(\frac{1}{27}) \cdot (- c - 9 x + 4.5 y) = (\frac{1}{27}) (0)$

Step 5.1.1.2.2

Factor $3$ out of $3 x$ .

$\frac{1}{3 (3)} \cdot (3 (x)) + \frac{1}{9} y = (\frac{1}{9}) (5)$

$(\frac{1}{27}) \cdot (- c - 9 x + 4.5 y) = (\frac{1}{27}) (0)$

Step 5.1.1.2.3

Cancel the common factor.

$\frac{1}{3 \cdot 3} \cdot (3 x) + \frac{1}{9} y = (\frac{1}{9}) (5)$

$(\frac{1}{27}) \cdot (- c - 9 x + 4.5 y) = (\frac{1}{27}) (0)$

Step 5.1.1.2.4

Rewrite the expression.

$\frac{1}{3} x + \frac{1}{9} y = (\frac{1}{9}) (5)$

$(\frac{1}{27}) \cdot (- c - 9 x + 4.5 y) = (\frac{1}{27}) (0)$

$\frac{1}{3} x + \frac{1}{9} y = (\frac{1}{9}) (5)$

$(\frac{1}{27}) \cdot (- c - 9 x + 4.5 y) = (\frac{1}{27}) (0)$

Step 5.1.1.3

Combine $\frac{1}{3}$ and $x$ .

$\frac{x}{3} + \frac{1}{9} y = (\frac{1}{9}) (5)$

$(\frac{1}{27}) \cdot (- c - 9 x + 4.5 y) = (\frac{1}{27}) (0)$

Step 5.1.1.4

Combine $\frac{1}{9}$ and $y$ .

$\frac{x}{3} + \frac{y}{9} = (\frac{1}{9}) (5)$

$(\frac{1}{27}) \cdot (- c - 9 x + 4.5 y) = (\frac{1}{27}) (0)$

$\frac{x}{3} + \frac{y}{9} = (\frac{1}{9}) (5)$

$(\frac{1}{27}) \cdot (- c - 9 x + 4.5 y) = (\frac{1}{27}) (0)$

$\frac{x}{3} + \frac{y}{9} = (\frac{1}{9}) (5)$

$(\frac{1}{27}) \cdot (- c - 9 x + 4.5 y) = (\frac{1}{27}) (0)$

Step 5.2

Simplify the right side.

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Step 5.2.1

Combine $\frac{1}{9}$ and $5$ .

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$(\frac{1}{27}) \cdot (- c - 9 x + 4.5 y) = (\frac{1}{27}) (0)$

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$(\frac{1}{27}) \cdot (- c - 9 x + 4.5 y) = (\frac{1}{27}) (0)$

Step 5.3

Simplify the left side.

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Step 5.3.1

Simplify $(\frac{1}{27}) \cdot (- c - 9 x + 4.5 y)$ .

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Step 5.3.1.1

Apply the distributive property.

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$\frac{1}{27} \cdot (- c) + \frac{1}{27} \cdot (- 9 x) + \frac{1}{27} \cdot (4.5 y) = (\frac{1}{27}) (0)$

Step 5.3.1.2

Simplify.

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Step 5.3.1.2.1

Combine $\frac{1}{27}$ and $c$ .

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$- \frac{c}{27} + \frac{1}{27} \cdot (- 9 x) + \frac{1}{27} \cdot (4.5 y) = (\frac{1}{27}) (0)$

Step 5.3.1.2.2

Cancel the common factor of $9$ .

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Step 5.3.1.2.2.1

Factor $9$ out of $27$ .

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$- \frac{c}{27} + \frac{1}{9 (3)} \cdot (- 9 x) + \frac{1}{27} \cdot (4.5 y) = (\frac{1}{27}) (0)$

Step 5.3.1.2.2.2

Factor $9$ out of $- 9 x$ .

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$- \frac{c}{27} + \frac{1}{9 (3)} \cdot (9 (- x)) + \frac{1}{27} \cdot (4.5 y) = (\frac{1}{27}) (0)$

Step 5.3.1.2.2.3

Cancel the common factor.

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$- \frac{c}{27} + \frac{1}{9 \cdot 3} \cdot (9 (- x)) + \frac{1}{27} \cdot (4.5 y) = (\frac{1}{27}) (0)$

Step 5.3.1.2.2.4

Rewrite the expression.

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$- \frac{c}{27} + \frac{1}{3} \cdot (- x) + \frac{1}{27} \cdot (4.5 y) = (\frac{1}{27}) (0)$

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$- \frac{c}{27} + \frac{1}{3} \cdot (- x) + \frac{1}{27} \cdot (4.5 y) = (\frac{1}{27}) (0)$

Step 5.3.1.2.3

Combine $\frac{1}{3}$ and $x$ .

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$- \frac{c}{27} - \frac{x}{3} + \frac{1}{27} \cdot (4.5 y) = (\frac{1}{27}) (0)$

Step 5.3.1.2.4

Cancel the common factor of $4.5$ .

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Step 5.3.1.2.4.1

Factor $4.5$ out of $27$ .

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$- \frac{c}{27} - \frac{x}{3} + \frac{1}{4.5 (6)} \cdot (4.5 y) = (\frac{1}{27}) (0)$

Step 5.3.1.2.4.2

Factor $4.5$ out of $4.5 y$ .

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$- \frac{c}{27} - \frac{x}{3} + \frac{1}{4.5 (6)} \cdot (4.5 (y)) = (\frac{1}{27}) (0)$

Step 5.3.1.2.4.3

Cancel the common factor.

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$- \frac{c}{27} - \frac{x}{3} + \frac{1}{4.5 \cdot 6} \cdot (4.5 y) = (\frac{1}{27}) (0)$

Step 5.3.1.2.4.4

Rewrite the expression.

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$- \frac{c}{27} - \frac{x}{3} + \frac{1}{6} y = (\frac{1}{27}) (0)$

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$- \frac{c}{27} - \frac{x}{3} + \frac{1}{6} y = (\frac{1}{27}) (0)$

Step 5.3.1.2.5

Combine $\frac{1}{6}$ and $y$ .

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$- \frac{c}{27} - \frac{x}{3} + \frac{y}{6} = (\frac{1}{27}) (0)$

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$- \frac{c}{27} - \frac{x}{3} + \frac{y}{6} = (\frac{1}{27}) (0)$

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$- \frac{c}{27} - \frac{x}{3} + \frac{y}{6} = (\frac{1}{27}) (0)$

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$- \frac{c}{27} - \frac{x}{3} + \frac{y}{6} = (\frac{1}{27}) (0)$

Step 5.4

Simplify the right side.

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Step 5.4.1

Multiply $\frac{1}{27}$ by $0$ .

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$- \frac{c}{27} - \frac{x}{3} + \frac{y}{6} = 0$

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$- \frac{c}{27} - \frac{x}{3} + \frac{y}{6} = 0$

$\frac{x}{3} + \frac{y}{9} = \frac{5}{9}$

$- \frac{c}{27} - \frac{x}{3} + \frac{y}{6} = 0$

Step 6

Add the two equations together to eliminate $x$ from the system.

$x$

$+$

$y$

$=$

$5$

$/$

$9$

$+$

$-$

$x$

$+$

$y$

$=$

$0$

$-$

$c$

$/$

$2$

$7$

$+$

$y$

$/$

$6$

$=$

$5$

$/$

$9$

Step 7

Simplify the equation and solve for $y$ .

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Step 7.1

Simplify $\frac{y}{9} - \frac{c}{27} + \frac{y}{6}$ .

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Step 7.1.1

To write $\frac{y}{9}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{c}{27} + \frac{y}{9} \cdot \frac{2}{2} + \frac{y}{6} = \frac{5}{9}$

Step 7.1.2

To write $\frac{y}{6}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{c}{27} + \frac{y}{9} \cdot \frac{2}{2} + \frac{y}{6} \cdot \frac{3}{3} = \frac{5}{9}$

Step 7.1.3

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1.3.1

Multiply $\frac{y}{9}$ by $\frac{2}{2}$ .

$- \frac{c}{27} + \frac{y \cdot 2}{9 \cdot 2} + \frac{y}{6} \cdot \frac{3}{3} = \frac{5}{9}$

Step 7.1.3.2

Multiply $9$ by $2$ .

$- \frac{c}{27} + \frac{y \cdot 2}{18} + \frac{y}{6} \cdot \frac{3}{3} = \frac{5}{9}$

Step 7.1.3.3

Multiply $\frac{y}{6}$ by $\frac{3}{3}$ .

$- \frac{c}{27} + \frac{y \cdot 2}{18} + \frac{y \cdot 3}{6 \cdot 3} = \frac{5}{9}$

Step 7.1.3.4

Multiply $6$ by $3$ .

$- \frac{c}{27} + \frac{y \cdot 2}{18} + \frac{y \cdot 3}{18} = \frac{5}{9}$

Step 7.1.4

Combine the numerators over the common denominator.

$- \frac{c}{27} + \frac{y \cdot 2 + y \cdot 3}{18} = \frac{5}{9}$

Step 7.1.5

Reorder $y$ and $3$ .

$- \frac{c}{27} + \frac{2 \cdot y + 3 \cdot y}{18} = \frac{5}{9}$

Step 7.1.6

Add $2 \cdot y$ and $3 \cdot y$ .

$- \frac{c}{27} + \frac{5 y}{18} = \frac{5}{9}$

Step 7.2

Add $\frac{c}{27}$ to both sides of the equation.

$\frac{5 y}{18} = \frac{5}{9} + \frac{c}{27}$

Step 7.3

Multiply both sides of the equation by $\frac{18}{5}$ .

$\frac{18}{5} \cdot \frac{5 y}{18} = \frac{18}{5} (\frac{5}{9} + \frac{c}{27})$

Step 7.4

Simplify both sides of the equation.

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Step 7.4.1

Simplify the left side.

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Step 7.4.1.1

Simplify $\frac{18}{5} \cdot \frac{5 y}{18}$ .

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Step 7.4.1.1.1

Cancel the common factor of $18$ .

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Step 7.4.1.1.1.1

Cancel the common factor.

$\frac{18}{5} \cdot \frac{5 y}{18} = \frac{18}{5} (\frac{5}{9} + \frac{c}{27})$

Step 7.4.1.1.1.2

Rewrite the expression.

$\frac{1}{5} (5 y) = \frac{18}{5} (\frac{5}{9} + \frac{c}{27})$

Step 7.4.1.1.2

Cancel the common factor of $5$ .

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Step 7.4.1.1.2.1

Factor $5$ out of $5 y$ .

$\frac{1}{5} (5 (y)) = \frac{18}{5} (\frac{5}{9} + \frac{c}{27})$

Step 7.4.1.1.2.2

Cancel the common factor.

$\frac{1}{5} (5 y) = \frac{18}{5} (\frac{5}{9} + \frac{c}{27})$

Step 7.4.1.1.2.3

Rewrite the expression.

$y = \frac{18}{5} (\frac{5}{9} + \frac{c}{27})$

Step 7.4.2

Simplify the right side.

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Step 7.4.2.1

Simplify $\frac{18}{5} (\frac{5}{9} + \frac{c}{27})$ .

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Step 7.4.2.1.1

Apply the distributive property.

$y = \frac{18}{5} \cdot \frac{5}{9} + \frac{18}{5} \cdot \frac{c}{27}$

Step 7.4.2.1.2

Cancel the common factor of $9$ .

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Step 7.4.2.1.2.1

Factor $9$ out of $18$ .

$y = \frac{9 (2)}{5} \cdot \frac{5}{9} + \frac{18}{5} \cdot \frac{c}{27}$

Step 7.4.2.1.2.2

Cancel the common factor.

$y = \frac{9 \cdot 2}{5} \cdot \frac{5}{9} + \frac{18}{5} \cdot \frac{c}{27}$

Step 7.4.2.1.2.3

Rewrite the expression.

$y = \frac{2}{5} \cdot 5 + \frac{18}{5} \cdot \frac{c}{27}$

Step 7.4.2.1.3

Cancel the common factor of $5$ .

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Step 7.4.2.1.3.1

Cancel the common factor.

$y = \frac{2}{5} \cdot 5 + \frac{18}{5} \cdot \frac{c}{27}$

Step 7.4.2.1.3.2

Rewrite the expression.

$y = 2 + \frac{18}{5} \cdot \frac{c}{27}$

Step 7.4.2.1.4

Cancel the common factor of $9$ .

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Step 7.4.2.1.4.1

Factor $9$ out of $18$ .

$y = 2 + \frac{9 (2)}{5} \cdot \frac{c}{27}$

Step 7.4.2.1.4.2

Factor $9$ out of $27$ .

$y = 2 + \frac{9 \cdot 2}{5} \cdot \frac{c}{9 \cdot 3}$

Step 7.4.2.1.4.3

Cancel the common factor.

$y = 2 + \frac{9 \cdot 2}{5} \cdot \frac{c}{9 \cdot 3}$

Step 7.4.2.1.4.4

Rewrite the expression.

$y = 2 + \frac{2}{5} \cdot \frac{c}{3}$

Step 7.4.2.1.5

Multiply $\frac{2}{5}$ by $\frac{c}{3}$ .

$y = 2 + \frac{2 c}{5 \cdot 3}$

Step 7.4.2.1.6

Multiply $5$ by $3$ .

$y = 2 + \frac{2 c}{15}$

Step 7.5

Reorder $2$ and $\frac{2 c}{15}$ .

$y = \frac{2 c}{15} + 2$

Step 8

Substitute the value found for $y$ into one of the original equations, then solve for $x$ .

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Step 8.1

Substitute the value found for $y$ into one of the original equations to solve for $x$ .

$\frac{x}{3} + \frac{\frac{2 c}{15} + 2}{9} = \frac{5}{9}$

Step 8.2

Simplify $\frac{x}{3} + \frac{\frac{2 c}{15} + 2}{9}$ .

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Simplify the numerator.

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Step 8.2.1.1.1

Factor $2$ out of $\frac{2 c}{15} + 2$ .

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Step 8.2.1.1.1.1

Factor $2$ out of $\frac{2 c}{15}$ .

$\frac{x}{3} + \frac{2 \frac{c}{15} + 2}{9} = \frac{5}{9}$

Step 8.2.1.1.1.2

Factor $2$ out of $2$ .

$\frac{x}{3} + \frac{2 \frac{c}{15} + 2 (1)}{9} = \frac{5}{9}$

Step 8.2.1.1.1.3

Factor $2$ out of $2 \frac{c}{15} + 2 (1)$ .

$\frac{x}{3} + \frac{2 (\frac{c}{15} + 1)}{9} = \frac{5}{9}$

Step 8.2.1.1.2

Write $1$ as a fraction with a common denominator.

$\frac{x}{3} + \frac{2 (\frac{c}{15} + \frac{15}{15})}{9} = \frac{5}{9}$

Step 8.2.1.1.3

Combine the numerators over the common denominator.

$\frac{x}{3} + \frac{2 \frac{c + 15}{15}}{9} = \frac{5}{9}$

Step 8.2.1.2

Combine $2$ and $\frac{c + 15}{15}$ .

$\frac{x}{3} + \frac{\frac{2 (c + 15)}{15}}{9} = \frac{5}{9}$

Step 8.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{x}{3} + \frac{2 (c + 15)}{15} \cdot \frac{1}{9} = \frac{5}{9}$

Step 8.2.1.4

Multiply $\frac{2 (c + 15)}{15} \cdot \frac{1}{9}$ .

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Step 8.2.1.4.1

Multiply $\frac{2 (c + 15)}{15}$ by $\frac{1}{9}$ .

$\frac{x}{3} + \frac{2 (c + 15)}{15 \cdot 9} = \frac{5}{9}$

Step 8.2.1.4.2

Multiply $15$ by $9$ .

$\frac{x}{3} + \frac{2 (c + 15)}{135} = \frac{5}{9}$

Step 8.2.2

To write $\frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{45}{45}$ .

$\frac{x}{3} \cdot \frac{45}{45} + \frac{2 (c + 15)}{135} = \frac{5}{9}$

Step 8.2.3

Write each expression with a common denominator of $135$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.2.3.1

Multiply $\frac{x}{3}$ by $\frac{45}{45}$ .

$\frac{x \cdot 45}{3 \cdot 45} + \frac{2 (c + 15)}{135} = \frac{5}{9}$

Step 8.2.3.2

Multiply $3$ by $45$ .

$\frac{x \cdot 45}{135} + \frac{2 (c + 15)}{135} = \frac{5}{9}$

Step 8.2.4

Combine the numerators over the common denominator.

$\frac{x \cdot 45 + 2 (c + 15)}{135} = \frac{5}{9}$

Step 8.2.5

Simplify the numerator.

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Step 8.2.5.1

Move $45$ to the left of $x$ .

$\frac{45 \cdot x + 2 (c + 15)}{135} = \frac{5}{9}$

Step 8.2.5.2

Apply the distributive property.

$\frac{45 x + 2 c + 2 \cdot 15}{135} = \frac{5}{9}$

Step 8.2.5.3

Multiply $2$ by $15$ .

$\frac{45 x + 2 c + 30}{135} = \frac{5}{9}$

Step 8.3

Multiply both sides by $135$ .

$\frac{45 x + 2 c + 30}{135} \cdot 135 = \frac{5}{9} \cdot 135$

Step 8.4

Simplify.

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Step 8.4.1

Simplify the left side.

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Step 8.4.1.1

Simplify $\frac{45 x + 2 c + 30}{135} \cdot 135$ .

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Step 8.4.1.1.1

Cancel the common factor of $135$ .

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Step 8.4.1.1.1.1

Cancel the common factor.

$\frac{45 x + 2 c + 30}{135} \cdot 135 = \frac{5}{9} \cdot 135$

Step 8.4.1.1.1.2

Rewrite the expression.

$45 x + 2 c + 30 = \frac{5}{9} \cdot 135$

Step 8.4.1.1.2

Reorder $45 x$ and $2 c$ .

$2 c + 45 x + 30 = \frac{5}{9} \cdot 135$

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

Simplify $\frac{5}{9} \cdot 135$ .

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Step 8.4.2.1.1

Cancel the common factor of $9$ .

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Step 8.4.2.1.1.1

Factor $9$ out of $135$ .

$2 c + 45 x + 30 = \frac{5}{9} \cdot (9 (15))$

Step 8.4.2.1.1.2

Cancel the common factor.

$2 c + 45 x + 30 = \frac{5}{9} \cdot (9 \cdot 15)$

Step 8.4.2.1.1.3

Rewrite the expression.

$2 c + 45 x + 30 = 5 \cdot 15$

Step 8.4.2.1.2

Multiply $5$ by $15$ .

$2 c + 45 x + 30 = 75$

Step 8.5

Solve for $x$ .

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Step 8.5.1

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

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Step 8.5.1.1

Subtract $2 c$ from both sides of the equation.

$45 x + 30 = 75 - 2 c$

Step 8.5.1.2

Subtract $30$ from both sides of the equation.

$45 x = 75 - 2 c - 30$

Step 8.5.1.3

Subtract $30$ from $75$ .

$45 x = - 2 c + 45$

Step 8.5.2

Divide each term in $45 x = - 2 c + 45$ by $45$ and simplify.

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Step 8.5.2.1

Divide each term in $45 x = - 2 c + 45$ by $45$ .

$\frac{45 x}{45} = \frac{- 2 c}{45} + \frac{45}{45}$

Step 8.5.2.2

Simplify the left side.

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Step 8.5.2.2.1

Cancel the common factor of $45$ .

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Step 8.5.2.2.1.1

Cancel the common factor.

$\frac{45 x}{45} = \frac{- 2 c}{45} + \frac{45}{45}$

Step 8.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2 c}{45} + \frac{45}{45}$

Step 8.5.2.3

Simplify the right side.

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Step 8.5.2.3.1

Simplify each term.

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Step 8.5.2.3.1.1

Move the negative in front of the fraction.

$x = - \frac{2 c}{45} + \frac{45}{45}$

Step 8.5.2.3.1.2

Divide $45$ by $45$ .

$x = - \frac{2 c}{45} + 1$

Step 9

This is the final solution to the independent system of equations.

$y = \frac{2 c}{15} + 2$

$x = - \frac{2 c}{45} + 1$