Linear Algebra Examples

$[\begin{array}{c} 5 & 3 \\ 2 & - 6 \end{array}] \cdot [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 3 \\ 30 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 5 & 3 \\ 2 & - 6 \end{array}] [\begin{array}{c} x \\ y \end{array}]$ .

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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.

$[\begin{array}{c} 5 x + 3 y \\ 2 x - 6 y \end{array}] = [\begin{array}{c} 3 \\ 30 \end{array}]$

Step 2

Write as a linear system of equations.

$5 x + 3 y = 3$

$2 x - 6 y = 30$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $x$ in $5 x + 3 y = 3$ .

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Step 3.1.1

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

$5 x = 3 - 3 y$

$2 x - 6 y = 30$

Step 3.1.2

Divide each term in $5 x = 3 - 3 y$ by $5$ and simplify.

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Step 3.1.2.1

Divide each term in $5 x = 3 - 3 y$ by $5$ .

$\frac{5 x}{5} = \frac{3}{5} + \frac{- 3 y}{5}$

$2 x - 6 y = 30$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $5$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{3}{5} + \frac{- 3 y}{5}$

$2 x - 6 y = 30$

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{5} + \frac{- 3 y}{5}$

$2 x - 6 y = 30$

$x = \frac{3}{5} + \frac{- 3 y}{5}$

$2 x - 6 y = 30$

$x = \frac{3}{5} + \frac{- 3 y}{5}$

$2 x - 6 y = 30$

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.

$x = \frac{3}{5} - \frac{3 y}{5}$

$2 x - 6 y = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

$2 x - 6 y = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

$2 x - 6 y = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

$2 x - 6 y = 30$

Step 3.2

Replace all occurrences of $x$ with $\frac{3}{5} - \frac{3 y}{5}$ in each equation.

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Step 3.2.1

Replace all occurrences of $x$ in $2 x - 6 y = 30$ with $\frac{3}{5} - \frac{3 y}{5}$ .

$2 (\frac{3}{5} - \frac{3 y}{5}) - 6 y = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify $2 (\frac{3}{5} - \frac{3 y}{5}) - 6 y$ .

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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.

$2 (\frac{3}{5}) + 2 (- \frac{3 y}{5}) - 6 y = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.2.2.1.1.2

Multiply $2 (\frac{3}{5})$ .

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Step 3.2.2.1.1.2.1

Combine $2$ and $\frac{3}{5}$ .

$\frac{2 \cdot 3}{5} + 2 (- \frac{3 y}{5}) - 6 y = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.2.2.1.1.2.2

Multiply $2$ by $3$ .

$\frac{6}{5} + 2 (- \frac{3 y}{5}) - 6 y = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

$\frac{6}{5} + 2 (- \frac{3 y}{5}) - 6 y = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.2.2.1.1.3

Multiply $2 (- \frac{3 y}{5})$ .

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Step 3.2.2.1.1.3.1

Multiply $- 1$ by $2$ .

$\frac{6}{5} - 2 \frac{3 y}{5} - 6 y = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.2.2.1.1.3.2

Combine $- 2$ and $\frac{3 y}{5}$ .

$\frac{6}{5} + \frac{- 2 (3 y)}{5} - 6 y = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.2.2.1.1.3.3

Multiply $3$ by $- 2$ .

$\frac{6}{5} + \frac{- 6 y}{5} - 6 y = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

$\frac{6}{5} + \frac{- 6 y}{5} - 6 y = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.2.2.1.1.4

Move the negative in front of the fraction.

$\frac{6}{5} - \frac{6 y}{5} - 6 y = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

$\frac{6}{5} - \frac{6 y}{5} - 6 y = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.2.2.1.2

To write $- 6 y$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{6}{5} - \frac{6 y}{5} - 6 y \cdot \frac{5}{5} = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.2.2.1.3

Combine $- 6 y$ and $\frac{5}{5}$ .

$\frac{6}{5} - \frac{6 y}{5} + \frac{- 6 y \cdot 5}{5} = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.2.2.1.4

Combine the numerators over the common denominator.

$\frac{6}{5} + \frac{- 6 y - 6 y \cdot 5}{5} = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.2.2.1.5

Combine the numerators over the common denominator.

$\frac{6 - 6 y - 6 y \cdot 5}{5} = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.2.2.1.6

Multiply $5$ by $- 6$ .

$\frac{6 - 6 y - 30 y}{5} = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.2.2.1.7

Subtract $30 y$ from $- 6 y$ .

$\frac{6 - 36 y}{5} = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.2.2.1.8

Factor $6$ out of $6 - 36 y$ .

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Step 3.2.2.1.8.1

Factor $6$ out of $6$ .

$\frac{6 (1) - 36 y}{5} = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.2.2.1.8.2

Factor $6$ out of $- 36 y$ .

$\frac{6 (1) + 6 (- 6 y)}{5} = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.2.2.1.8.3

Factor $6$ out of $6 (1) + 6 (- 6 y)$ .

$\frac{6 (1 - 6 y)}{5} = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

$\frac{6 (1 - 6 y)}{5} = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

$\frac{6 (1 - 6 y)}{5} = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

$\frac{6 (1 - 6 y)}{5} = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

$\frac{6 (1 - 6 y)}{5} = 30$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.3

Solve for $y$ in $\frac{6 (1 - 6 y)}{5} = 30$ .

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Step 3.3.1

Multiply both sides by $5$ .

$\frac{6 (1 - 6 y)}{5} \cdot 5 = 30 \cdot 5$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Simplify the left side.

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Step 3.3.2.1.1

Simplify $\frac{6 (1 - 6 y)}{5} \cdot 5$ .

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Step 3.3.2.1.1.1

Cancel the common factor of $5$ .

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Step 3.3.2.1.1.1.1

Cancel the common factor.

$\frac{6 (1 - 6 y)}{5} \cdot 5 = 30 \cdot 5$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.3.2.1.1.1.2

Rewrite the expression.

$6 (1 - 6 y) = 30 \cdot 5$

$x = \frac{3}{5} - \frac{3 y}{5}$

$6 (1 - 6 y) = 30 \cdot 5$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.3.2.1.1.2

Apply the distributive property.

$6 \cdot 1 + 6 (- 6 y) = 30 \cdot 5$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.3.2.1.1.3

Simplify the expression.

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Step 3.3.2.1.1.3.1

Multiply $6$ by $1$ .

$6 + 6 (- 6 y) = 30 \cdot 5$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.3.2.1.1.3.2

Multiply $- 6$ by $6$ .

$6 - 36 y = 30 \cdot 5$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.3.2.1.1.3.3

Reorder $6$ and $- 36 y$ .

$- 36 y + 6 = 30 \cdot 5$

$x = \frac{3}{5} - \frac{3 y}{5}$

$- 36 y + 6 = 30 \cdot 5$

$x = \frac{3}{5} - \frac{3 y}{5}$

$- 36 y + 6 = 30 \cdot 5$

$x = \frac{3}{5} - \frac{3 y}{5}$

$- 36 y + 6 = 30 \cdot 5$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.3.2.2

Simplify the right side.

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Step 3.3.2.2.1

Multiply $30$ by $5$ .

$- 36 y + 6 = 150$

$x = \frac{3}{5} - \frac{3 y}{5}$

$- 36 y + 6 = 150$

$x = \frac{3}{5} - \frac{3 y}{5}$

$- 36 y + 6 = 150$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.3.3

Solve for $y$ .

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Step 3.3.3.1

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

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Step 3.3.3.1.1

Subtract $6$ from both sides of the equation.

$- 36 y = 150 - 6$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.3.3.1.2

Subtract $6$ from $150$ .

$- 36 y = 144$

$x = \frac{3}{5} - \frac{3 y}{5}$

$- 36 y = 144$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.3.3.2

Divide each term in $- 36 y = 144$ by $- 36$ and simplify.

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Step 3.3.3.2.1

Divide each term in $- 36 y = 144$ by $- 36$ .

$\frac{- 36 y}{- 36} = \frac{144}{- 36}$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.3.3.2.2

Simplify the left side.

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Step 3.3.3.2.2.1

Cancel the common factor of $- 36$ .

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Step 3.3.3.2.2.1.1

Cancel the common factor.

$\frac{- 36 y}{- 36} = \frac{144}{- 36}$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.3.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{144}{- 36}$

$x = \frac{3}{5} - \frac{3 y}{5}$

$y = \frac{144}{- 36}$

$x = \frac{3}{5} - \frac{3 y}{5}$

$y = \frac{144}{- 36}$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.3.3.2.3

Simplify the right side.

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Step 3.3.3.2.3.1

Divide $144$ by $- 36$ .

$y = - 4$

$x = \frac{3}{5} - \frac{3 y}{5}$

$y = - 4$

$x = \frac{3}{5} - \frac{3 y}{5}$

$y = - 4$

$x = \frac{3}{5} - \frac{3 y}{5}$

$y = - 4$

$x = \frac{3}{5} - \frac{3 y}{5}$

$y = - 4$

$x = \frac{3}{5} - \frac{3 y}{5}$

Step 3.4

Replace all occurrences of $y$ with $- 4$ in each equation.

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Step 3.4.1

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

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

$y = - 4$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

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

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Step 3.4.2.1.1

Combine the numerators over the common denominator.

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

$y = - 4$

Step 3.4.2.1.2

Simplify the expression.

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Step 3.4.2.1.2.1

Multiply $- 3$ by $- 4$ .

$x = \frac{3 + 12}{5}$

$y = - 4$

Step 3.4.2.1.2.2

Add $3$ and $12$ .

$x = \frac{15}{5}$

$y = - 4$

Step 3.4.2.1.2.3

Divide $15$ by $5$ .

$x = 3$

$y = - 4$

$x = 3$

$y = - 4$

$x = 3$

$y = - 4$

$x = 3$

$y = - 4$

$x = 3$

$y = - 4$

Step 3.5

List all of the solutions.

$x = 3, y = - 4$