Finite Math Examples

$[\begin{array}{c} 2 & 3 \\ 3 & 5 \end{array}] [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 13 \\ 21 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 2 & 3 \\ 3 & 5 \end{array}] [\begin{array}{c} x \\ y \end{array}]$ .

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Step 1.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $2 \times 2$ and the second matrix is $2 \times 1$ .

Step 1.2

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} 2 x + 3 y \\ 3 x + 5 y \end{array}] = [\begin{array}{c} 13 \\ 21 \end{array}]$

Step 2

Write as a linear system of equations.

$2 x + 3 y = 13$

$3 x + 5 y = 21$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $x$ in $2 x + 3 y = 13$ .

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Step 3.1.1

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

$2 x = 13 - 3 y$

$3 x + 5 y = 21$

Step 3.1.2

Divide each term in $2 x = 13 - 3 y$ by $2$ and simplify.

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Step 3.1.2.1

Divide each term in $2 x = 13 - 3 y$ by $2$ .

$\frac{2 x}{2} = \frac{13}{2} + \frac{- 3 y}{2}$

$3 x + 5 y = 21$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $2$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{13}{2} + \frac{- 3 y}{2}$

$3 x + 5 y = 21$

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{13}{2} + \frac{- 3 y}{2}$

$3 x + 5 y = 21$

$x = \frac{13}{2} + \frac{- 3 y}{2}$

$3 x + 5 y = 21$

$x = \frac{13}{2} + \frac{- 3 y}{2}$

$3 x + 5 y = 21$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Move the negative in front of the fraction.

$x = \frac{13}{2} - \frac{3 y}{2}$

$3 x + 5 y = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

$3 x + 5 y = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

$3 x + 5 y = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

$3 x + 5 y = 21$

Step 3.2

Replace all occurrences of $x$ with $\frac{13}{2} - \frac{3 y}{2}$ in each equation.

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Step 3.2.1

Replace all occurrences of $x$ in $3 x + 5 y = 21$ with $\frac{13}{2} - \frac{3 y}{2}$ .

$3 (\frac{13}{2} - \frac{3 y}{2}) + 5 y = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify $3 (\frac{13}{2} - \frac{3 y}{2}) + 5 y$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Apply the distributive property.

$3 (\frac{13}{2}) + 3 (- \frac{3 y}{2}) + 5 y = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

Step 3.2.2.1.1.2

Multiply $3 (\frac{13}{2})$ .

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Step 3.2.2.1.1.2.1

Combine $3$ and $\frac{13}{2}$ .

$\frac{3 \cdot 13}{2} + 3 (- \frac{3 y}{2}) + 5 y = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

Step 3.2.2.1.1.2.2

Multiply $3$ by $13$ .

$\frac{39}{2} + 3 (- \frac{3 y}{2}) + 5 y = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

$\frac{39}{2} + 3 (- \frac{3 y}{2}) + 5 y = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

Step 3.2.2.1.1.3

Multiply $3 (- \frac{3 y}{2})$ .

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Step 3.2.2.1.1.3.1

Multiply $- 1$ by $3$ .

$\frac{39}{2} - 3 \frac{3 y}{2} + 5 y = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

Step 3.2.2.1.1.3.2

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

$\frac{39}{2} + \frac{- 3 (3 y)}{2} + 5 y = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

Step 3.2.2.1.1.3.3

Multiply $3$ by $- 3$ .

$\frac{39}{2} + \frac{- 9 y}{2} + 5 y = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

$\frac{39}{2} + \frac{- 9 y}{2} + 5 y = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

Step 3.2.2.1.1.4

Move the negative in front of the fraction.

$\frac{39}{2} - \frac{9 y}{2} + 5 y = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

$\frac{39}{2} - \frac{9 y}{2} + 5 y = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

Step 3.2.2.1.2

To write $5 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{39}{2} - \frac{9 y}{2} + 5 y \cdot \frac{2}{2} = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

Step 3.2.2.1.3

Combine $5 y$ and $\frac{2}{2}$ .

$\frac{39}{2} - \frac{9 y}{2} + \frac{5 y \cdot 2}{2} = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

Step 3.2.2.1.4

Combine the numerators over the common denominator.

$\frac{39}{2} + \frac{- 9 y + 5 y \cdot 2}{2} = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

Step 3.2.2.1.5

Combine the numerators over the common denominator.

$\frac{39 - 9 y + 5 y \cdot 2}{2} = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

Step 3.2.2.1.6

Multiply $2$ by $5$ .

$\frac{39 - 9 y + 10 y}{2} = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

Step 3.2.2.1.7

Add $- 9 y$ and $10 y$ .

$\frac{39 + y}{2} = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

$\frac{39 + y}{2} = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

$\frac{39 + y}{2} = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

$\frac{39 + y}{2} = 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

Step 3.3

Solve for $y$ in $\frac{39 + y}{2} = 21$ .

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Step 3.3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{39 + y}{2}) = 2 \cdot 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

Step 3.3.2

Simplify both sides of the equation.

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Step 3.3.2.1

Simplify the left side.

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Step 3.3.2.1.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1.1

Cancel the common factor.

$2 (\frac{39 + y}{2}) = 2 \cdot 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

Step 3.3.2.1.1.2

Rewrite the expression.

$39 + y = 2 \cdot 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

$39 + y = 2 \cdot 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

$39 + y = 2 \cdot 21$

$x = \frac{13}{2} - \frac{3 y}{2}$

Step 3.3.2.2

Simplify the right side.

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Step 3.3.2.2.1

Multiply $2$ by $21$ .

$39 + y = 42$

$x = \frac{13}{2} - \frac{3 y}{2}$

$39 + y = 42$

$x = \frac{13}{2} - \frac{3 y}{2}$

$39 + y = 42$

$x = \frac{13}{2} - \frac{3 y}{2}$

Step 3.3.3

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

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Step 3.3.3.1

Subtract $39$ from both sides of the equation.

$y = 42 - 39$

$x = \frac{13}{2} - \frac{3 y}{2}$

Step 3.3.3.2

Subtract $39$ from $42$ .

$y = 3$

$x = \frac{13}{2} - \frac{3 y}{2}$

$y = 3$

$x = \frac{13}{2} - \frac{3 y}{2}$

$y = 3$

$x = \frac{13}{2} - \frac{3 y}{2}$

Step 3.4

Replace all occurrences of $y$ with $3$ in each equation.

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Step 3.4.1

Replace all occurrences of $y$ in $x = \frac{13}{2} - \frac{3 y}{2}$ with $3$ .

$x = \frac{13}{2} - \frac{3 (3)}{2}$

$y = 3$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $\frac{13}{2} - \frac{3 (3)}{2}$ .

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Step 3.4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{13 - 3 \cdot 3}{2}$

$y = 3$

Step 3.4.2.1.2

Simplify the expression.

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Step 3.4.2.1.2.1

Multiply $- 3$ by $3$ .

$x = \frac{13 - 9}{2}$

$y = 3$

Step 3.4.2.1.2.2

Subtract $9$ from $13$ .

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

$y = 3$

Step 3.4.2.1.2.3

Divide $4$ by $2$ .

$x = 2$

$y = 3$

$x = 2$

$y = 3$

$x = 2$

$y = 3$

$x = 2$

$y = 3$

$x = 2$

$y = 3$

Step 3.5

List all of the solutions.

$x = 2, y = 3$