Finite Math Examples

$20 x + 8 y = 80$ , $5 x + 24 y = 60$ , $20 x + 40 y = 200$

Step 1

Solve for $x$ in $20 x + 8 y = 80$ .

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Step 1.1

Subtract $8 y$ from both sides of the equation.

$20 x = 80 - 8 y$

$5 x + 24 y = 60$

$20 x + 40 y = 200$

Step 1.2

Divide each term in $20 x = 80 - 8 y$ by $20$ and simplify.

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Step 1.2.1

Divide each term in $20 x = 80 - 8 y$ by $20$ .

$\frac{20 x}{20} = \frac{80}{20} + \frac{- 8 y}{20}$

$5 x + 24 y = 60$

$20 x + 40 y = 200$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $20$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{20 x}{20} = \frac{80}{20} + \frac{- 8 y}{20}$

$5 x + 24 y = 60$

$20 x + 40 y = 200$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{80}{20} + \frac{- 8 y}{20}$

$5 x + 24 y = 60$

$20 x + 40 y = 200$

$x = \frac{80}{20} + \frac{- 8 y}{20}$

$5 x + 24 y = 60$

$20 x + 40 y = 200$

$x = \frac{80}{20} + \frac{- 8 y}{20}$

$5 x + 24 y = 60$

$20 x + 40 y = 200$

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

Divide $80$ by $20$ .

$x = 4 + \frac{- 8 y}{20}$

$5 x + 24 y = 60$

$20 x + 40 y = 200$

Step 1.2.3.1.2

Cancel the common factor of $- 8$ and $20$ .

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Step 1.2.3.1.2.1

Factor $4$ out of $- 8 y$ .

$x = 4 + \frac{4 (- 2 y)}{20}$

$5 x + 24 y = 60$

$20 x + 40 y = 200$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $4$ out of $20$ .

$x = 4 + \frac{4 (- 2 y)}{4 (5)}$

$5 x + 24 y = 60$

$20 x + 40 y = 200$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$x = 4 + \frac{4 (- 2 y)}{4 \cdot 5}$

$5 x + 24 y = 60$

$20 x + 40 y = 200$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$x = 4 + \frac{- 2 y}{5}$

$5 x + 24 y = 60$

$20 x + 40 y = 200$

$x = 4 + \frac{- 2 y}{5}$

$5 x + 24 y = 60$

$20 x + 40 y = 200$

$x = 4 + \frac{- 2 y}{5}$

$5 x + 24 y = 60$

$20 x + 40 y = 200$

Step 1.2.3.1.3

Move the negative in front of the fraction.

$x = 4 - \frac{2 y}{5}$

$5 x + 24 y = 60$

$20 x + 40 y = 200$

$x = 4 - \frac{2 y}{5}$

$5 x + 24 y = 60$

$20 x + 40 y = 200$

$x = 4 - \frac{2 y}{5}$

$5 x + 24 y = 60$

$20 x + 40 y = 200$

$x = 4 - \frac{2 y}{5}$

$5 x + 24 y = 60$

$20 x + 40 y = 200$

$x = 4 - \frac{2 y}{5}$

$5 x + 24 y = 60$

$20 x + 40 y = 200$

Step 2

Replace all occurrences of $x$ with $4 - \frac{2 y}{5}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $5 x + 24 y = 60$ with $4 - \frac{2 y}{5}$ .

$5 (4 - \frac{2 y}{5}) + 24 y = 60$

$x = 4 - \frac{2 y}{5}$

$20 x + 40 y = 200$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $5 (4 - \frac{2 y}{5}) + 24 y$ .

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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.

$5 \cdot 4 + 5 (- \frac{2 y}{5}) + 24 y = 60$

$x = 4 - \frac{2 y}{5}$

$20 x + 40 y = 200$

Step 2.2.1.1.2

Multiply $5$ by $4$ .

$20 + 5 (- \frac{2 y}{5}) + 24 y = 60$

$x = 4 - \frac{2 y}{5}$

$20 x + 40 y = 200$

Step 2.2.1.1.3

Cancel the common factor of $5$ .

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Step 2.2.1.1.3.1

Move the leading negative in $- \frac{2 y}{5}$ into the numerator.

$20 + 5 (\frac{- 2 y}{5}) + 24 y = 60$

$x = 4 - \frac{2 y}{5}$

$20 x + 40 y = 200$

Step 2.2.1.1.3.2

Cancel the common factor.

$20 + 5 (\frac{- 2 y}{5}) + 24 y = 60$

$x = 4 - \frac{2 y}{5}$

$20 x + 40 y = 200$

Step 2.2.1.1.3.3

Rewrite the expression.

$20 - 2 y + 24 y = 60$

$x = 4 - \frac{2 y}{5}$

$20 x + 40 y = 200$

$20 - 2 y + 24 y = 60$

$x = 4 - \frac{2 y}{5}$

$20 x + 40 y = 200$

$20 - 2 y + 24 y = 60$

$x = 4 - \frac{2 y}{5}$

$20 x + 40 y = 200$

Step 2.2.1.2

Add $- 2 y$ and $24 y$ .

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

$20 x + 40 y = 200$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

$20 x + 40 y = 200$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

$20 x + 40 y = 200$

Step 2.3

Replace all occurrences of $x$ in $20 x + 40 y = 200$ with $4 - \frac{2 y}{5}$ .

$20 (4 - \frac{2 y}{5}) + 40 y = 200$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $20 (4 - \frac{2 y}{5}) + 40 y$ .

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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.

$20 \cdot 4 + 20 (- \frac{2 y}{5}) + 40 y = 200$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

Step 2.4.1.1.2

Multiply $20$ by $4$ .

$80 + 20 (- \frac{2 y}{5}) + 40 y = 200$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

Step 2.4.1.1.3

Cancel the common factor of $5$ .

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Step 2.4.1.1.3.1

Move the leading negative in $- \frac{2 y}{5}$ into the numerator.

$80 + 20 (\frac{- 2 y}{5}) + 40 y = 200$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

Step 2.4.1.1.3.2

Factor $5$ out of $20$ .

$80 + 5 (4) (\frac{- 2 y}{5}) + 40 y = 200$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

Step 2.4.1.1.3.3

Cancel the common factor.

$80 + 5 \cdot (4 (\frac{- 2 y}{5})) + 40 y = 200$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

Step 2.4.1.1.3.4

Rewrite the expression.

$80 + 4 (- 2 y) + 40 y = 200$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

$80 + 4 (- 2 y) + 40 y = 200$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

Step 2.4.1.1.4

Multiply $- 2$ by $4$ .

$80 - 8 y + 40 y = 200$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

$80 - 8 y + 40 y = 200$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

Step 2.4.1.2

Add $- 8 y$ and $40 y$ .

$80 + 32 y = 200$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

$80 + 32 y = 200$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

$80 + 32 y = 200$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

$80 + 32 y = 200$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

Step 3

Solve for $y$ in $80 + 32 y = 200$ .

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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 $80$ from both sides of the equation.

$32 y = 200 - 80$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

Step 3.1.2

Subtract $80$ from $200$ .

$32 y = 120$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

$32 y = 120$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

Step 3.2

Divide each term in $32 y = 120$ by $32$ and simplify.

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Step 3.2.1

Divide each term in $32 y = 120$ by $32$ .

$\frac{32 y}{32} = \frac{120}{32}$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $32$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{32 y}{32} = \frac{120}{32}$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{120}{32}$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

$y = \frac{120}{32}$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

$y = \frac{120}{32}$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $120$ and $32$ .

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Step 3.2.3.1.1

Factor $8$ out of $120$ .

$y = \frac{8 (15)}{32}$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

Step 3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.1.2.1

Factor $8$ out of $32$ .

$y = \frac{8 \cdot 15}{8 \cdot 4}$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

Step 3.2.3.1.2.2

Cancel the common factor.

$y = \frac{8 \cdot 15}{8 \cdot 4}$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

Step 3.2.3.1.2.3

Rewrite the expression.

$y = \frac{15}{4}$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

$y = \frac{15}{4}$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

$y = \frac{15}{4}$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

$y = \frac{15}{4}$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

$y = \frac{15}{4}$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

$y = \frac{15}{4}$

$20 + 22 y = 60$

$x = 4 - \frac{2 y}{5}$

Step 4

Replace all occurrences of $y$ with $\frac{15}{4}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $20 + 22 y = 60$ with $\frac{15}{4}$ .

$20 + 22 (\frac{15}{4}) = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{2 y}{5}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $20 + 22 (\frac{15}{4})$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Cancel the common factor of $2$ .

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Step 4.2.1.1.1.1

Factor $2$ out of $22$ .

$20 + 2 (11) (\frac{15}{4}) = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{2 y}{5}$

Step 4.2.1.1.1.2

Factor $2$ out of $4$ .

$20 + 2 \cdot (11 (\frac{15}{2 \cdot 2})) = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{2 y}{5}$

Step 4.2.1.1.1.3

Cancel the common factor.

$20 + 2 \cdot (11 (\frac{15}{2 \cdot 2})) = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{2 y}{5}$

Step 4.2.1.1.1.4

Rewrite the expression.

$20 + 11 (\frac{15}{2}) = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{2 y}{5}$

$20 + 11 (\frac{15}{2}) = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{2 y}{5}$

Step 4.2.1.1.2

Combine $11$ and $\frac{15}{2}$ .

$20 + \frac{11 \cdot 15}{2} = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{2 y}{5}$

Step 4.2.1.1.3

Multiply $11$ by $15$ .

$20 + \frac{165}{2} = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{2 y}{5}$

$20 + \frac{165}{2} = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{2 y}{5}$

Step 4.2.1.2

To write $20$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$20 \cdot \frac{2}{2} + \frac{165}{2} = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{2 y}{5}$

Step 4.2.1.3

Combine $20$ and $\frac{2}{2}$ .

$\frac{20 \cdot 2}{2} + \frac{165}{2} = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{2 y}{5}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$\frac{20 \cdot 2 + 165}{2} = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{2 y}{5}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $20$ by $2$ .

$\frac{40 + 165}{2} = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{2 y}{5}$

Step 4.2.1.5.2

Add $40$ and $165$ .

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{2 y}{5}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{2 y}{5}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{2 y}{5}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{2 y}{5}$

Step 4.3

Replace all occurrences of $y$ in $x = 4 - \frac{2 y}{5}$ with $\frac{15}{4}$ .

$x = 4 - \frac{2 (\frac{15}{4})}{5}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $4 - \frac{2 (\frac{15}{4})}{5}$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Combine $2$ and $\frac{15}{4}$ .

$x = 4 - \frac{\frac{2 \cdot 15}{4}}{5}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

Step 4.4.1.1.2

Multiply $2$ by $15$ .

$x = 4 - \frac{\frac{30}{4}}{5}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

Step 4.4.1.1.3

Cancel the common factor of $30$ and $4$ .

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Step 4.4.1.1.3.1

Factor $2$ out of $30$ .

$x = 4 - \frac{\frac{2 (15)}{4}}{5}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

Step 4.4.1.1.3.2

Cancel the common factors.

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Step 4.4.1.1.3.2.1

Factor $2$ out of $4$ .

$x = 4 - \frac{\frac{2 \cdot 15}{2 \cdot 2}}{5}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

Step 4.4.1.1.3.2.2

Cancel the common factor.

$x = 4 - \frac{\frac{2 \cdot 15}{2 \cdot 2}}{5}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

Step 4.4.1.1.3.2.3

Rewrite the expression.

$x = 4 - \frac{\frac{15}{2}}{5}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{\frac{15}{2}}{5}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{\frac{15}{2}}{5}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

Step 4.4.1.1.4

Multiply the numerator by the reciprocal of the denominator.

$x = 4 - (\frac{15}{2} \cdot \frac{1}{5})$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

Step 4.4.1.1.5

Cancel the common factor of $5$ .

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Step 4.4.1.1.5.1

Factor $5$ out of $15$ .

$x = 4 - (\frac{5 (3)}{2} \cdot \frac{1}{5})$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

Step 4.4.1.1.5.2

Cancel the common factor.

$x = 4 - (\frac{5 \cdot 3}{2} \cdot \frac{1}{5})$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

Step 4.4.1.1.5.3

Rewrite the expression.

$x = 4 - \frac{3}{2}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{3}{2}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

$x = 4 - \frac{3}{2}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

Step 4.4.1.2

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

$x = 4 \cdot \frac{2}{2} - \frac{3}{2}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

Step 4.4.1.3

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

$x = \frac{4 \cdot 2}{2} - \frac{3}{2}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

Step 4.4.1.4

Combine the numerators over the common denominator.

$x = \frac{4 \cdot 2 - 3}{2}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

Step 4.4.1.5

Simplify the numerator.

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Step 4.4.1.5.1

Multiply $4$ by $2$ .

$x = \frac{8 - 3}{2}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

Step 4.4.1.5.2

Subtract $3$ from $8$ .

$x = \frac{5}{2}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

$x = \frac{5}{2}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

$x = \frac{5}{2}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

$x = \frac{5}{2}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

$x = \frac{5}{2}$

$\frac{205}{2} = 60$

$y = \frac{15}{4}$

Step 5

Since $\frac{205}{2} = 60$ is not true, there is no solution.

No solution

Step 6