Finite Math Examples

$24 x - 5 y = - 29$ , $6 x + 7 y = 67$

Step 1

Solve for $x$ in $24 x - 5 y = - 29$ .

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Step 1.1

Add $5 y$ to both sides of the equation.

$24 x = - 29 + 5 y$

$6 x + 7 y = 67$

Step 1.2

Divide each term in $24 x = - 29 + 5 y$ by $24$ and simplify.

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Step 1.2.1

Divide each term in $24 x = - 29 + 5 y$ by $24$ .

$\frac{24 x}{24} = \frac{- 29}{24} + \frac{5 y}{24}$

$6 x + 7 y = 67$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $24$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{24 x}{24} = \frac{- 29}{24} + \frac{5 y}{24}$

$6 x + 7 y = 67$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 29}{24} + \frac{5 y}{24}$

$6 x + 7 y = 67$

$x = \frac{- 29}{24} + \frac{5 y}{24}$

$6 x + 7 y = 67$

$x = \frac{- 29}{24} + \frac{5 y}{24}$

$6 x + 7 y = 67$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$x = - \frac{29}{24} + \frac{5 y}{24}$

$6 x + 7 y = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$6 x + 7 y = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$6 x + 7 y = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$6 x + 7 y = 67$

Step 2

Replace all occurrences of $x$ with $- \frac{29}{24} + \frac{5 y}{24}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $6 x + 7 y = 67$ with $- \frac{29}{24} + \frac{5 y}{24}$ .

$6 (- \frac{29}{24} + \frac{5 y}{24}) + 7 y = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $6 (- \frac{29}{24} + \frac{5 y}{24}) + 7 y$ .

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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.

$6 (- \frac{29}{24}) + 6 (\frac{5 y}{24}) + 7 y = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 2.2.1.1.2

Cancel the common factor of $6$ .

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Step 2.2.1.1.2.1

Move the leading negative in $- \frac{29}{24}$ into the numerator.

$6 (\frac{- 29}{24}) + 6 (\frac{5 y}{24}) + 7 y = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 2.2.1.1.2.2

Factor $6$ out of $24$ .

$6 (\frac{- 29}{6 (4)}) + 6 (\frac{5 y}{24}) + 7 y = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 2.2.1.1.2.3

Cancel the common factor.

$6 (\frac{- 29}{6 \cdot 4}) + 6 (\frac{5 y}{24}) + 7 y = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 2.2.1.1.2.4

Rewrite the expression.

$\frac{- 29}{4} + 6 (\frac{5 y}{24}) + 7 y = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$\frac{- 29}{4} + 6 (\frac{5 y}{24}) + 7 y = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 2.2.1.1.3

Cancel the common factor of $6$ .

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Step 2.2.1.1.3.1

Factor $6$ out of $24$ .

$\frac{- 29}{4} + 6 (\frac{5 y}{6 (4)}) + 7 y = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 2.2.1.1.3.2

Cancel the common factor.

$\frac{- 29}{4} + 6 (\frac{5 y}{6 \cdot 4}) + 7 y = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 2.2.1.1.3.3

Rewrite the expression.

$\frac{- 29}{4} + \frac{5 y}{4} + 7 y = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$\frac{- 29}{4} + \frac{5 y}{4} + 7 y = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 2.2.1.1.4

Move the negative in front of the fraction.

$- \frac{29}{4} + \frac{5 y}{4} + 7 y = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$- \frac{29}{4} + \frac{5 y}{4} + 7 y = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 2.2.1.2

To write $7 y$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{29}{4} + \frac{5 y}{4} + 7 y \cdot \frac{4}{4} = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 2.2.1.3

Combine $7 y$ and $\frac{4}{4}$ .

$- \frac{29}{4} + \frac{5 y}{4} + \frac{7 y \cdot 4}{4} = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$- \frac{29}{4} + \frac{5 y + 7 y \cdot 4}{4} = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{- 29 + 5 y + 7 y \cdot 4}{4} = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 2.2.1.6

Multiply $4$ by $7$ .

$\frac{- 29 + 5 y + 28 y}{4} = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 2.2.1.7

Add $5 y$ and $28 y$ .

$\frac{- 29 + 33 y}{4} = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 2.2.1.8

Rewrite $- 29$ as $- 1 (29)$ .

$\frac{- 1 \cdot 29 + 33 y}{4} = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 2.2.1.9

Factor $- 1$ out of $33 y$ .

$\frac{- 1 \cdot 29 - (- 33 y)}{4} = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 2.2.1.10

Factor $- 1$ out of $- 1 (29) - (- 33 y)$ .

$\frac{- 1 (29 - 33 y)}{4} = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 2.2.1.11

Move the negative in front of the fraction.

$- \frac{29 - 33 y}{4} = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$- \frac{29 - 33 y}{4} = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$- \frac{29 - 33 y}{4} = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$- \frac{29 - 33 y}{4} = 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 3

Solve for $y$ in $- \frac{29 - 33 y}{4} = 67$ .

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Step 3.1

Multiply both sides of the equation by $- 4$ .

$- 4 (- \frac{29 - 33 y}{4}) = - 4 \cdot 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $- 4 (- \frac{29 - 33 y}{4})$ .

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Step 3.2.1.1.1

Cancel the common factor of $4$ .

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Step 3.2.1.1.1.1

Move the leading negative in $- \frac{29 - 33 y}{4}$ into the numerator.

$- 4 \frac{- (29 - 33 y)}{4} = - 4 \cdot 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 3.2.1.1.1.2

Factor $4$ out of $- 4$ .

$4 (- 1) (\frac{- (29 - 33 y)}{4}) = - 4 \cdot 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 3.2.1.1.1.3

Cancel the common factor.

$4 \cdot (- 1 \frac{- (29 - 33 y)}{4}) = - 4 \cdot 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 3.2.1.1.1.4

Rewrite the expression.

$29 - 33 y = - 4 \cdot 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$29 - 33 y = - 4 \cdot 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 3.2.1.1.2

Multiply.

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Step 3.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (29 - 33 y) = - 4 \cdot 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 3.2.1.1.2.2

Multiply $29 - 33 y$ by $1$ .

$29 - 33 y = - 4 \cdot 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$29 - 33 y = - 4 \cdot 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$29 - 33 y = - 4 \cdot 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$29 - 33 y = - 4 \cdot 67$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 4$ by $67$ .

$29 - 33 y = - 268$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$29 - 33 y = - 268$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$29 - 33 y = - 268$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 3.3

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

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Step 3.3.1

Subtract $29$ from both sides of the equation.

$- 33 y = - 268 - 29$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 3.3.2

Subtract $29$ from $- 268$ .

$- 33 y = - 297$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$- 33 y = - 297$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 3.4

Divide each term in $- 33 y = - 297$ by $- 33$ and simplify.

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Step 3.4.1

Divide each term in $- 33 y = - 297$ by $- 33$ .

$\frac{- 33 y}{- 33} = \frac{- 297}{- 33}$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $- 33$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{- 33 y}{- 33} = \frac{- 297}{- 33}$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 297}{- 33}$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$y = \frac{- 297}{- 33}$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$y = \frac{- 297}{- 33}$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $- 297$ by $- 33$ .

$y = 9$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$y = 9$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$y = 9$

$x = - \frac{29}{24} + \frac{5 y}{24}$

$y = 9$

$x = - \frac{29}{24} + \frac{5 y}{24}$

Step 4

Replace all occurrences of $y$ with $9$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{29}{24} + \frac{5 y}{24}$ with $9$ .

$x = - \frac{29}{24} + \frac{5 (9)}{24}$

$y = 9$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{29}{24} + \frac{5 (9)}{24}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{- 29 + 5 (9)}{24}$

$y = 9$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $5$ by $9$ .

$x = \frac{- 29 + 45}{24}$

$y = 9$

Step 4.2.1.2.2

Add $- 29$ and $45$ .

$x = \frac{16}{24}$

$y = 9$

$x = \frac{16}{24}$

$y = 9$

Step 4.2.1.3

Cancel the common factor of $16$ and $24$ .

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Step 4.2.1.3.1

Factor $8$ out of $16$ .

$x = \frac{8 (2)}{24}$

$y = 9$

Step 4.2.1.3.2

Cancel the common factors.

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Step 4.2.1.3.2.1

Factor $8$ out of $24$ .

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

$y = 9$

Step 4.2.1.3.2.2

Cancel the common factor.

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

$y = 9$

Step 4.2.1.3.2.3

Rewrite the expression.

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

$y = 9$

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

$y = 9$

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

$y = 9$

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

$y = 9$

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

$y = 9$

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

$y = 9$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(\frac{2}{3}, 9)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{2}{3}, 9)$

Equation Form:

$x = \frac{2}{3}, y = 9$

Step 7