Finite Math Examples

$[\begin{array}{c} 3 & - 2 \\ 4 & 3 \end{array}] [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} - 6 \\ 11 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 3 & - 2 \\ 4 & 3 \end{array}] [\begin{array}{c} x \\ y \end{array}]$ .

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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.

$[\begin{array}{c} 3 x - 2 y \\ 4 x + 3 y \end{array}] = [\begin{array}{c} - 6 \\ 11 \end{array}]$

Step 2

Write as a linear system of equations.

$3 x - 2 y = - 6$

$4 x + 3 y = 11$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $x$ in $3 x - 2 y = - 6$ .

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Step 3.1.1

Add $2 y$ to both sides of the equation.

$3 x = - 6 + 2 y$

$4 x + 3 y = 11$

Step 3.1.2

Divide each term in $3 x = - 6 + 2 y$ by $3$ and simplify.

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Step 3.1.2.1

Divide each term in $3 x = - 6 + 2 y$ by $3$ .

$\frac{3 x}{3} = \frac{- 6}{3} + \frac{2 y}{3}$

$4 x + 3 y = 11$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $3$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 6}{3} + \frac{2 y}{3}$

$4 x + 3 y = 11$

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{3} + \frac{2 y}{3}$

$4 x + 3 y = 11$

$x = \frac{- 6}{3} + \frac{2 y}{3}$

$4 x + 3 y = 11$

$x = \frac{- 6}{3} + \frac{2 y}{3}$

$4 x + 3 y = 11$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $- 6$ by $3$ .

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

$4 x + 3 y = 11$

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

$4 x + 3 y = 11$

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

$4 x + 3 y = 11$

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

$4 x + 3 y = 11$

Step 3.2

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

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Step 3.2.1

Replace all occurrences of $x$ in $4 x + 3 y = 11$ with $- 2 + \frac{2 y}{3}$ .

$4 (- 2 + \frac{2 y}{3}) + 3 y = 11$

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

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

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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.

$4 \cdot - 2 + 4 (\frac{2 y}{3}) + 3 y = 11$

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

Step 3.2.2.1.1.2

Multiply $4$ by $- 2$ .

$- 8 + 4 (\frac{2 y}{3}) + 3 y = 11$

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

Step 3.2.2.1.1.3

Multiply $4 \frac{2 y}{3}$ .

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Step 3.2.2.1.1.3.1

Combine $4$ and $\frac{2 y}{3}$ .

$- 8 + \frac{4 (2 y)}{3} + 3 y = 11$

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

Step 3.2.2.1.1.3.2

Multiply $2$ by $4$ .

$- 8 + \frac{8 y}{3} + 3 y = 11$

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

$- 8 + \frac{8 y}{3} + 3 y = 11$

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

$- 8 + \frac{8 y}{3} + 3 y = 11$

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

Step 3.2.2.1.2

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

$- 8 + \frac{8 y}{3} + 3 y \cdot \frac{3}{3} = 11$

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

Step 3.2.2.1.3

Simplify terms.

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Step 3.2.2.1.3.1

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

$- 8 + \frac{8 y}{3} + \frac{3 y \cdot 3}{3} = 11$

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

Step 3.2.2.1.3.2

Combine the numerators over the common denominator.

$- 8 + \frac{8 y + 3 y \cdot 3}{3} = 11$

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

$- 8 + \frac{8 y + 3 y \cdot 3}{3} = 11$

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

Step 3.2.2.1.4

Simplify each term.

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Step 3.2.2.1.4.1

Simplify the numerator.

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Step 3.2.2.1.4.1.1

Factor $y$ out of $8 y + 3 y \cdot 3$ .

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Step 3.2.2.1.4.1.1.1

Factor $y$ out of $8 y$ .

$- 8 + \frac{y \cdot 8 + 3 y \cdot 3}{3} = 11$

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

Step 3.2.2.1.4.1.1.2

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

$- 8 + \frac{y \cdot 8 + y (3 \cdot 3)}{3} = 11$

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

Step 3.2.2.1.4.1.1.3

Factor $y$ out of $y \cdot 8 + y (3 \cdot 3)$ .

$- 8 + \frac{y (8 + 3 \cdot 3)}{3} = 11$

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

$- 8 + \frac{y (8 + 3 \cdot 3)}{3} = 11$

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

Step 3.2.2.1.4.1.2

Multiply $3$ by $3$ .

$- 8 + \frac{y (8 + 9)}{3} = 11$

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

Step 3.2.2.1.4.1.3

Add $8$ and $9$ .

$- 8 + \frac{y \cdot 17}{3} = 11$

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

$- 8 + \frac{y \cdot 17}{3} = 11$

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

Step 3.2.2.1.4.2

Move $17$ to the left of $y$ .

$- 8 + \frac{17 y}{3} = 11$

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

$- 8 + \frac{17 y}{3} = 11$

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

$- 8 + \frac{17 y}{3} = 11$

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

$- 8 + \frac{17 y}{3} = 11$

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

$- 8 + \frac{17 y}{3} = 11$

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

Step 3.3

Solve for $y$ in $- 8 + \frac{17 y}{3} = 11$ .

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Step 3.3.1

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

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Step 3.3.1.1

Add $8$ to both sides of the equation.

$\frac{17 y}{3} = 11 + 8$

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

Step 3.3.1.2

Add $11$ and $8$ .

$\frac{17 y}{3} = 19$

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

$\frac{17 y}{3} = 19$

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

Step 3.3.2

Multiply both sides of the equation by $\frac{3}{17}$ .

$\frac{3}{17} \cdot \frac{17 y}{3} = \frac{3}{17} \cdot 19$

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

Step 3.3.3

Simplify both sides of the equation.

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Step 3.3.3.1

Simplify the left side.

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Step 3.3.3.1.1

Simplify $\frac{3}{17} \cdot \frac{17 y}{3}$ .

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Step 3.3.3.1.1.1

Cancel the common factor of $3$ .

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Step 3.3.3.1.1.1.1

Cancel the common factor.

$\frac{3}{17} \cdot \frac{17 y}{3} = \frac{3}{17} \cdot 19$

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

Step 3.3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{17} \cdot (17 y) = \frac{3}{17} \cdot 19$

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

$\frac{1}{17} \cdot (17 y) = \frac{3}{17} \cdot 19$

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

Step 3.3.3.1.1.2

Cancel the common factor of $17$ .

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Step 3.3.3.1.1.2.1

Factor $17$ out of $17 y$ .

$\frac{1}{17} \cdot (17 (y)) = \frac{3}{17} \cdot 19$

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

Step 3.3.3.1.1.2.2

Cancel the common factor.

$\frac{1}{17} \cdot (17 y) = \frac{3}{17} \cdot 19$

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

Step 3.3.3.1.1.2.3

Rewrite the expression.

$y = \frac{3}{17} \cdot 19$

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

$y = \frac{3}{17} \cdot 19$

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

$y = \frac{3}{17} \cdot 19$

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

$y = \frac{3}{17} \cdot 19$

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

Step 3.3.3.2

Simplify the right side.

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Step 3.3.3.2.1

Multiply $\frac{3}{17} \cdot 19$ .

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Step 3.3.3.2.1.1

Combine $\frac{3}{17}$ and $19$ .

$y = \frac{3 \cdot 19}{17}$

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

Step 3.3.3.2.1.2

Multiply $3$ by $19$ .

$y = \frac{57}{17}$

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

$y = \frac{57}{17}$

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

$y = \frac{57}{17}$

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

$y = \frac{57}{17}$

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

$y = \frac{57}{17}$

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

Step 3.4

Replace all occurrences of $y$ with $\frac{57}{17}$ in each equation.

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Step 3.4.1

Replace all occurrences of $y$ in $x = - 2 + \frac{2 y}{3}$ with $\frac{57}{17}$ .

$x = - 2 + \frac{2 (\frac{57}{17})}{3}$

$y = \frac{57}{17}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- 2 + \frac{2 (\frac{57}{17})}{3}$ .

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Step 3.4.2.1.1

Simplify each term.

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Step 3.4.2.1.1.1

Combine $2$ and $\frac{57}{17}$ .

$x = - 2 + \frac{\frac{2 \cdot 57}{17}}{3}$

$y = \frac{57}{17}$

Step 3.4.2.1.1.2

Multiply $2$ by $57$ .

$x = - 2 + \frac{\frac{114}{17}}{3}$

$y = \frac{57}{17}$

Step 3.4.2.1.1.3

Multiply the numerator by the reciprocal of the denominator.

$x = - 2 + \frac{114}{17} \cdot \frac{1}{3}$

$y = \frac{57}{17}$

Step 3.4.2.1.1.4

Cancel the common factor of $3$ .

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Step 3.4.2.1.1.4.1

Factor $3$ out of $114$ .

$x = - 2 + \frac{3 (38)}{17} \cdot \frac{1}{3}$

$y = \frac{57}{17}$

Step 3.4.2.1.1.4.2

Cancel the common factor.

$x = - 2 + \frac{3 \cdot 38}{17} \cdot \frac{1}{3}$

$y = \frac{57}{17}$

Step 3.4.2.1.1.4.3

Rewrite the expression.

$x = - 2 + \frac{38}{17}$

$y = \frac{57}{17}$

$x = - 2 + \frac{38}{17}$

$y = \frac{57}{17}$

$x = - 2 + \frac{38}{17}$

$y = \frac{57}{17}$

Step 3.4.2.1.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{17}{17}$ .

$x = - 2 \cdot \frac{17}{17} + \frac{38}{17}$

$y = \frac{57}{17}$

Step 3.4.2.1.3

Combine $- 2$ and $\frac{17}{17}$ .

$x = \frac{- 2 \cdot 17}{17} + \frac{38}{17}$

$y = \frac{57}{17}$

Step 3.4.2.1.4

Combine the numerators over the common denominator.

$x = \frac{- 2 \cdot 17 + 38}{17}$

$y = \frac{57}{17}$

Step 3.4.2.1.5

Simplify the numerator.

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Step 3.4.2.1.5.1

Multiply $- 2$ by $17$ .

$x = \frac{- 34 + 38}{17}$

$y = \frac{57}{17}$

Step 3.4.2.1.5.2

Add $- 34$ and $38$ .

$x = \frac{4}{17}$

$y = \frac{57}{17}$

$x = \frac{4}{17}$

$y = \frac{57}{17}$

$x = \frac{4}{17}$

$y = \frac{57}{17}$

$x = \frac{4}{17}$

$y = \frac{57}{17}$

$x = \frac{4}{17}$

$y = \frac{57}{17}$

Step 3.5

List all of the solutions.

$x = \frac{4}{17}, y = \frac{57}{17}$