Finite Math Examples

$4 x + 3 y = 21$ , $4 x + 8 y = 32$

Step 1

Solve for $x$ in $4 x + 3 y = 21$ .

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Step 1.1

Subtract $3 y$ from both sides of the equation.

$4 x = 21 - 3 y$

$4 x + 8 y = 32$

Step 1.2

Divide each term in $4 x = 21 - 3 y$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x = 21 - 3 y$ by $4$ .

$\frac{4 x}{4} = \frac{21}{4} + \frac{- 3 y}{4}$

$4 x + 8 y = 32$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{21}{4} + \frac{- 3 y}{4}$

$4 x + 8 y = 32$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{21}{4} + \frac{- 3 y}{4}$

$4 x + 8 y = 32$

$x = \frac{21}{4} + \frac{- 3 y}{4}$

$4 x + 8 y = 32$

$x = \frac{21}{4} + \frac{- 3 y}{4}$

$4 x + 8 y = 32$

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.

$x = \frac{21}{4} - \frac{3 y}{4}$

$4 x + 8 y = 32$

$x = \frac{21}{4} - \frac{3 y}{4}$

$4 x + 8 y = 32$

$x = \frac{21}{4} - \frac{3 y}{4}$

$4 x + 8 y = 32$

$x = \frac{21}{4} - \frac{3 y}{4}$

$4 x + 8 y = 32$

Step 2

Replace all occurrences of $x$ with $\frac{21}{4} - \frac{3 y}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $4 x + 8 y = 32$ with $\frac{21}{4} - \frac{3 y}{4}$ .

$4 (\frac{21}{4} - \frac{3 y}{4}) + 8 y = 32$

$x = \frac{21}{4} - \frac{3 y}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $4 (\frac{21}{4} - \frac{3 y}{4}) + 8 y$ .

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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.

$4 (\frac{21}{4}) + 4 (- \frac{3 y}{4}) + 8 y = 32$

$x = \frac{21}{4} - \frac{3 y}{4}$

Step 2.2.1.1.2

Cancel the common factor of $4$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

$4 (\frac{21}{4}) + 4 (- \frac{3 y}{4}) + 8 y = 32$

$x = \frac{21}{4} - \frac{3 y}{4}$

Step 2.2.1.1.2.2

Rewrite the expression.

$21 + 4 (- \frac{3 y}{4}) + 8 y = 32$

$x = \frac{21}{4} - \frac{3 y}{4}$

$21 + 4 (- \frac{3 y}{4}) + 8 y = 32$

$x = \frac{21}{4} - \frac{3 y}{4}$

Step 2.2.1.1.3

Cancel the common factor of $4$ .

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Step 2.2.1.1.3.1

Move the leading negative in $- \frac{3 y}{4}$ into the numerator.

$21 + 4 (\frac{- 3 y}{4}) + 8 y = 32$

$x = \frac{21}{4} - \frac{3 y}{4}$

Step 2.2.1.1.3.2

Cancel the common factor.

$21 + 4 (\frac{- 3 y}{4}) + 8 y = 32$

$x = \frac{21}{4} - \frac{3 y}{4}$

Step 2.2.1.1.3.3

Rewrite the expression.

$21 - 3 y + 8 y = 32$

$x = \frac{21}{4} - \frac{3 y}{4}$

$21 - 3 y + 8 y = 32$

$x = \frac{21}{4} - \frac{3 y}{4}$

$21 - 3 y + 8 y = 32$

$x = \frac{21}{4} - \frac{3 y}{4}$

Step 2.2.1.2

Add $- 3 y$ and $8 y$ .

$21 + 5 y = 32$

$x = \frac{21}{4} - \frac{3 y}{4}$

$21 + 5 y = 32$

$x = \frac{21}{4} - \frac{3 y}{4}$

$21 + 5 y = 32$

$x = \frac{21}{4} - \frac{3 y}{4}$

$21 + 5 y = 32$

$x = \frac{21}{4} - \frac{3 y}{4}$

Step 3

Solve for $y$ in $21 + 5 y = 32$ .

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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 $21$ from both sides of the equation.

$5 y = 32 - 21$

$x = \frac{21}{4} - \frac{3 y}{4}$

Step 3.1.2

Subtract $21$ from $32$ .

$5 y = 11$

$x = \frac{21}{4} - \frac{3 y}{4}$

$5 y = 11$

$x = \frac{21}{4} - \frac{3 y}{4}$

Step 3.2

Divide each term in $5 y = 11$ by $5$ and simplify.

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Step 3.2.1

Divide each term in $5 y = 11$ by $5$ .

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

$x = \frac{21}{4} - \frac{3 y}{4}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1

Cancel the common factor.

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

$x = \frac{21}{4} - \frac{3 y}{4}$

Step 3.2.2.1.2

Divide $y$ by $1$ .

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

$x = \frac{21}{4} - \frac{3 y}{4}$

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

$x = \frac{21}{4} - \frac{3 y}{4}$

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

$x = \frac{21}{4} - \frac{3 y}{4}$

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

$x = \frac{21}{4} - \frac{3 y}{4}$

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

$x = \frac{21}{4} - \frac{3 y}{4}$

Step 4

Replace all occurrences of $y$ with $\frac{11}{5}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{21}{4} - \frac{3 y}{4}$ with $\frac{11}{5}$ .

$x = \frac{21}{4} - \frac{3 (\frac{11}{5})}{4}$

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

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{21}{4} - \frac{3 (\frac{11}{5})}{4}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{21 - 3 (\frac{11}{5})}{4}$

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

Step 4.2.1.2

Simplify each term.

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Step 4.2.1.2.1

Multiply $- 3 (\frac{11}{5})$ .

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Step 4.2.1.2.1.1

Combine $- 3$ and $\frac{11}{5}$ .

$x = \frac{21 + \frac{- 3 \cdot 11}{5}}{4}$

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

Step 4.2.1.2.1.2

Multiply $- 3$ by $11$ .

$x = \frac{21 + \frac{- 33}{5}}{4}$

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

$x = \frac{21 + \frac{- 33}{5}}{4}$

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

Step 4.2.1.2.2

Move the negative in front of the fraction.

$x = \frac{21 - \frac{33}{5}}{4}$

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

$x = \frac{21 - \frac{33}{5}}{4}$

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

Step 4.2.1.3

To write $21$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x = \frac{21 \cdot \frac{5}{5} - \frac{33}{5}}{4}$

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

Step 4.2.1.4

Combine $21$ and $\frac{5}{5}$ .

$x = \frac{\frac{21 \cdot 5}{5} - \frac{33}{5}}{4}$

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

Step 4.2.1.5

Combine the numerators over the common denominator.

$x = \frac{\frac{21 \cdot 5 - 33}{5}}{4}$

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

Step 4.2.1.6

Simplify the numerator.

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Step 4.2.1.6.1

Multiply $21$ by $5$ .

$x = \frac{\frac{105 - 33}{5}}{4}$

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

Step 4.2.1.6.2

Subtract $33$ from $105$ .

$x = \frac{\frac{72}{5}}{4}$

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

$x = \frac{\frac{72}{5}}{4}$

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

Step 4.2.1.7

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{72}{5} \cdot \frac{1}{4}$

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

Step 4.2.1.8

Cancel the common factor of $4$ .

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Step 4.2.1.8.1

Factor $4$ out of $72$ .

$x = \frac{4 (18)}{5} \cdot \frac{1}{4}$

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

Step 4.2.1.8.2

Cancel the common factor.

$x = \frac{4 \cdot 18}{5} \cdot \frac{1}{4}$

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

Step 4.2.1.8.3

Rewrite the expression.

$x = \frac{18}{5}$

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

$x = \frac{18}{5}$

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

$x = \frac{18}{5}$

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

$x = \frac{18}{5}$

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

$x = \frac{18}{5}$

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

Step 5

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

$(\frac{18}{5}, \frac{11}{5})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{18}{5}, \frac{11}{5})$

Equation Form:

$x = \frac{18}{5}, y = \frac{11}{5}$

Step 7