Finite Math Examples

$3 x - y = 12$ , $3 x + 3 y = 5$ , $5 x + 2 y = 17$

Step 1

Solve for $x$ in $3 x - y = 12$ .

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Step 1.1

Add $y$ to both sides of the equation.

$3 x = 12 + y$

$3 x + 3 y = 5$

$5 x + 2 y = 17$

Step 1.2

Divide each term in $3 x = 12 + y$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = 12 + y$ by $3$ .

$\frac{3 x}{3} = \frac{12}{3} + \frac{y}{3}$

$3 x + 3 y = 5$

$5 x + 2 y = 17$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{12}{3} + \frac{y}{3}$

$3 x + 3 y = 5$

$5 x + 2 y = 17$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{12}{3} + \frac{y}{3}$

$3 x + 3 y = 5$

$5 x + 2 y = 17$

$x = \frac{12}{3} + \frac{y}{3}$

$3 x + 3 y = 5$

$5 x + 2 y = 17$

$x = \frac{12}{3} + \frac{y}{3}$

$3 x + 3 y = 5$

$5 x + 2 y = 17$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $12$ by $3$ .

$x = 4 + \frac{y}{3}$

$3 x + 3 y = 5$

$5 x + 2 y = 17$

$x = 4 + \frac{y}{3}$

$3 x + 3 y = 5$

$5 x + 2 y = 17$

$x = 4 + \frac{y}{3}$

$3 x + 3 y = 5$

$5 x + 2 y = 17$

$x = 4 + \frac{y}{3}$

$3 x + 3 y = 5$

$5 x + 2 y = 17$

Step 2

Replace all occurrences of $x$ with $4 + \frac{y}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 x + 3 y = 5$ with $4 + \frac{y}{3}$ .

$3 (4 + \frac{y}{3}) + 3 y = 5$

$x = 4 + \frac{y}{3}$

$5 x + 2 y = 17$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (4 + \frac{y}{3}) + 3 y$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$3 \cdot 4 + 3 (\frac{y}{3}) + 3 y = 5$

$x = 4 + \frac{y}{3}$

$5 x + 2 y = 17$

Step 2.2.1.1.2

Multiply $3$ by $4$ .

$12 + 3 (\frac{y}{3}) + 3 y = 5$

$x = 4 + \frac{y}{3}$

$5 x + 2 y = 17$

Step 2.2.1.1.3

Cancel the common factor of $3$ .

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Step 2.2.1.1.3.1

Cancel the common factor.

$12 + 3 (\frac{y}{3}) + 3 y = 5$

$x = 4 + \frac{y}{3}$

$5 x + 2 y = 17$

Step 2.2.1.1.3.2

Rewrite the expression.

$12 + y + 3 y = 5$

$x = 4 + \frac{y}{3}$

$5 x + 2 y = 17$

$12 + y + 3 y = 5$

$x = 4 + \frac{y}{3}$

$5 x + 2 y = 17$

$12 + y + 3 y = 5$

$x = 4 + \frac{y}{3}$

$5 x + 2 y = 17$

Step 2.2.1.2

Add $y$ and $3 y$ .

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$5 x + 2 y = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$5 x + 2 y = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$5 x + 2 y = 17$

Step 2.3

Replace all occurrences of $x$ in $5 x + 2 y = 17$ with $4 + \frac{y}{3}$ .

$5 (4 + \frac{y}{3}) + 2 y = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $5 (4 + \frac{y}{3}) + 2 y$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Apply the distributive property.

$5 \cdot 4 + 5 (\frac{y}{3}) + 2 y = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 2.4.1.1.2

Multiply $5$ by $4$ .

$20 + 5 (\frac{y}{3}) + 2 y = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 2.4.1.1.3

Combine $5$ and $\frac{y}{3}$ .

$20 + \frac{5 y}{3} + 2 y = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$20 + \frac{5 y}{3} + 2 y = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 2.4.1.2

To write $2 y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$20 + \frac{5 y}{3} + 2 y \cdot \frac{3}{3} = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 2.4.1.3

Simplify terms.

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Step 2.4.1.3.1

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

$20 + \frac{5 y}{3} + \frac{2 y \cdot 3}{3} = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 2.4.1.3.2

Combine the numerators over the common denominator.

$20 + \frac{5 y + 2 y \cdot 3}{3} = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$20 + \frac{5 y + 2 y \cdot 3}{3} = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 2.4.1.4

Simplify each term.

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Step 2.4.1.4.1

Simplify the numerator.

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Step 2.4.1.4.1.1

Factor $y$ out of $5 y + 2 y \cdot 3$ .

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Step 2.4.1.4.1.1.1

Factor $y$ out of $5 y$ .

$20 + \frac{y \cdot 5 + 2 y \cdot 3}{3} = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 2.4.1.4.1.1.2

Factor $y$ out of $2 y \cdot 3$ .

$20 + \frac{y \cdot 5 + y (2 \cdot 3)}{3} = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 2.4.1.4.1.1.3

Factor $y$ out of $y \cdot 5 + y (2 \cdot 3)$ .

$20 + \frac{y (5 + 2 \cdot 3)}{3} = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$20 + \frac{y (5 + 2 \cdot 3)}{3} = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 2.4.1.4.1.2

Multiply $2$ by $3$ .

$20 + \frac{y (5 + 6)}{3} = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 2.4.1.4.1.3

Add $5$ and $6$ .

$20 + \frac{y \cdot 11}{3} = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$20 + \frac{y \cdot 11}{3} = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 2.4.1.4.2

Move $11$ to the left of $y$ .

$20 + \frac{11 y}{3} = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$20 + \frac{11 y}{3} = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$20 + \frac{11 y}{3} = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$20 + \frac{11 y}{3} = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$20 + \frac{11 y}{3} = 17$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 3

Solve for $y$ in $20 + \frac{11 y}{3} = 17$ .

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Step 3.1

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

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Step 3.1.1

Subtract $20$ from both sides of the equation.

$\frac{11 y}{3} = 17 - 20$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 3.1.2

Subtract $20$ from $17$ .

$\frac{11 y}{3} = - 3$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$\frac{11 y}{3} = - 3$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 3.2

Multiply both sides of the equation by $\frac{3}{11}$ .

$\frac{3}{11} \cdot \frac{11 y}{3} = \frac{3}{11} \cdot - 3$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $\frac{3}{11} \cdot \frac{11 y}{3}$ .

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Step 3.3.1.1.1

Cancel the common factor of $3$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{3}{11} \cdot \frac{11 y}{3} = \frac{3}{11} \cdot - 3$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{11} \cdot (11 y) = \frac{3}{11} \cdot - 3$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$\frac{1}{11} \cdot (11 y) = \frac{3}{11} \cdot - 3$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 3.3.1.1.2

Cancel the common factor of $11$ .

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Step 3.3.1.1.2.1

Factor $11$ out of $11 y$ .

$\frac{1}{11} \cdot (11 (y)) = \frac{3}{11} \cdot - 3$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{1}{11} \cdot (11 y) = \frac{3}{11} \cdot - 3$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = \frac{3}{11} \cdot - 3$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$y = \frac{3}{11} \cdot - 3$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$y = \frac{3}{11} \cdot - 3$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$y = \frac{3}{11} \cdot - 3$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $\frac{3}{11} \cdot - 3$ .

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Step 3.3.2.1.1

Multiply $\frac{3}{11} \cdot - 3$ .

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Step 3.3.2.1.1.1

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

$y = \frac{3 \cdot - 3}{11}$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 3.3.2.1.1.2

Multiply $3$ by $- 3$ .

$y = \frac{- 9}{11}$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$y = \frac{- 9}{11}$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 3.3.2.1.2

Move the negative in front of the fraction.

$y = - \frac{9}{11}$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$y = - \frac{9}{11}$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$y = - \frac{9}{11}$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$y = - \frac{9}{11}$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

$y = - \frac{9}{11}$

$12 + 4 y = 5$

$x = 4 + \frac{y}{3}$

Step 4

Replace all occurrences of $y$ with $- \frac{9}{11}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $12 + 4 y = 5$ with $- \frac{9}{11}$ .

$12 + 4 (- \frac{9}{11}) = 5$

$y = - \frac{9}{11}$

$x = 4 + \frac{y}{3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $12 + 4 (- \frac{9}{11})$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Multiply $4 (- \frac{9}{11})$ .

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Step 4.2.1.1.1.1

Multiply $- 1$ by $4$ .

$12 - 4 (\frac{9}{11}) = 5$

$y = - \frac{9}{11}$

$x = 4 + \frac{y}{3}$

Step 4.2.1.1.1.2

Combine $- 4$ and $\frac{9}{11}$ .

$12 + \frac{- 4 \cdot 9}{11} = 5$

$y = - \frac{9}{11}$

$x = 4 + \frac{y}{3}$

Step 4.2.1.1.1.3

Multiply $- 4$ by $9$ .

$12 + \frac{- 36}{11} = 5$

$y = - \frac{9}{11}$

$x = 4 + \frac{y}{3}$

$12 + \frac{- 36}{11} = 5$

$y = - \frac{9}{11}$

$x = 4 + \frac{y}{3}$

Step 4.2.1.1.2

Move the negative in front of the fraction.

$12 - \frac{36}{11} = 5$

$y = - \frac{9}{11}$

$x = 4 + \frac{y}{3}$

$12 - \frac{36}{11} = 5$

$y = - \frac{9}{11}$

$x = 4 + \frac{y}{3}$

Step 4.2.1.2

To write $12$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$12 \cdot \frac{11}{11} - \frac{36}{11} = 5$

$y = - \frac{9}{11}$

$x = 4 + \frac{y}{3}$

Step 4.2.1.3

Combine $12$ and $\frac{11}{11}$ .

$\frac{12 \cdot 11}{11} - \frac{36}{11} = 5$

$y = - \frac{9}{11}$

$x = 4 + \frac{y}{3}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$\frac{12 \cdot 11 - 36}{11} = 5$

$y = - \frac{9}{11}$

$x = 4 + \frac{y}{3}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $12$ by $11$ .

$\frac{132 - 36}{11} = 5$

$y = - \frac{9}{11}$

$x = 4 + \frac{y}{3}$

Step 4.2.1.5.2

Subtract $36$ from $132$ .

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

$x = 4 + \frac{y}{3}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

$x = 4 + \frac{y}{3}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

$x = 4 + \frac{y}{3}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

$x = 4 + \frac{y}{3}$

Step 4.3

Replace all occurrences of $y$ in $x = 4 + \frac{y}{3}$ with $- \frac{9}{11}$ .

$x = 4 + \frac{- \frac{9}{11}}{3}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $4 + \frac{- \frac{9}{11}}{3}$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = 4 - \frac{9}{11} \cdot \frac{1}{3}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

Step 4.4.1.1.2

Cancel the common factor of $3$ .

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Step 4.4.1.1.2.1

Move the leading negative in $- \frac{9}{11}$ into the numerator.

$x = 4 + \frac{- 9}{11} \cdot \frac{1}{3}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

Step 4.4.1.1.2.2

Factor $3$ out of $- 9$ .

$x = 4 + \frac{3 (- 3)}{11} \cdot \frac{1}{3}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

Step 4.4.1.1.2.3

Cancel the common factor.

$x = 4 + \frac{3 \cdot - 3}{11} \cdot \frac{1}{3}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

Step 4.4.1.1.2.4

Rewrite the expression.

$x = 4 + \frac{- 3}{11}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

$x = 4 + \frac{- 3}{11}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

Step 4.4.1.1.3

Move the negative in front of the fraction.

$x = 4 - \frac{3}{11}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

$x = 4 - \frac{3}{11}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

Step 4.4.1.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$x = 4 \cdot \frac{11}{11} - \frac{3}{11}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

Step 4.4.1.3

Combine $4$ and $\frac{11}{11}$ .

$x = \frac{4 \cdot 11}{11} - \frac{3}{11}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

Step 4.4.1.4

Combine the numerators over the common denominator.

$x = \frac{4 \cdot 11 - 3}{11}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

Step 4.4.1.5

Simplify the numerator.

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Step 4.4.1.5.1

Multiply $4$ by $11$ .

$x = \frac{44 - 3}{11}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

Step 4.4.1.5.2

Subtract $3$ from $44$ .

$x = \frac{41}{11}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

$x = \frac{41}{11}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

$x = \frac{41}{11}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

$x = \frac{41}{11}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

$x = \frac{41}{11}$

$\frac{96}{11} = 5$

$y = - \frac{9}{11}$

Step 5

Since $\frac{96}{11} = 5$ is not true, there is no solution.

No solution

Step 6