Linear Algebra Examples

$[\begin{array}{c} 2 x + 3 y \\ 3 x - y \end{array}] = [\begin{array}{c} 7 \\ 16 \end{array}]$

Step 1

Write as a linear system of equations.

$2 x + 3 y = 7$

$3 x - y = 16$

Step 2

Solve the system of equations.

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Step 2.1

Solve for $x$ in $2 x + 3 y = 7$ .

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Step 2.1.1

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

$2 x = 7 - 3 y$

$3 x - y = 16$

Step 2.1.2

Divide each term in $2 x = 7 - 3 y$ by $2$ and simplify.

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Step 2.1.2.1

Divide each term in $2 x = 7 - 3 y$ by $2$ .

$\frac{2 x}{2} = \frac{7}{2} + \frac{- 3 y}{2}$

$3 x - y = 16$

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $2$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{7}{2} + \frac{- 3 y}{2}$

$3 x - y = 16$

Step 2.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{2} + \frac{- 3 y}{2}$

$3 x - y = 16$

$x = \frac{7}{2} + \frac{- 3 y}{2}$

$3 x - y = 16$

$x = \frac{7}{2} + \frac{- 3 y}{2}$

$3 x - y = 16$

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Move the negative in front of the fraction.

$x = \frac{7}{2} - \frac{3 y}{2}$

$3 x - y = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

$3 x - y = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

$3 x - y = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

$3 x - y = 16$

Step 2.2

Replace all occurrences of $x$ with $\frac{7}{2} - \frac{3 y}{2}$ in each equation.

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Step 2.2.1

Replace all occurrences of $x$ in $3 x - y = 16$ with $\frac{7}{2} - \frac{3 y}{2}$ .

$3 (\frac{7}{2} - \frac{3 y}{2}) - y = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify $3 (\frac{7}{2} - \frac{3 y}{2}) - y$ .

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Step 2.2.2.1.1

Simplify each term.

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Step 2.2.2.1.1.1

Apply the distributive property.

$3 (\frac{7}{2}) + 3 (- \frac{3 y}{2}) - y = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.2.2.1.1.2

Multiply $3 (\frac{7}{2})$ .

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Step 2.2.2.1.1.2.1

Combine $3$ and $\frac{7}{2}$ .

$\frac{3 \cdot 7}{2} + 3 (- \frac{3 y}{2}) - y = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.2.2.1.1.2.2

Multiply $3$ by $7$ .

$\frac{21}{2} + 3 (- \frac{3 y}{2}) - y = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

$\frac{21}{2} + 3 (- \frac{3 y}{2}) - y = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.2.2.1.1.3

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

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Step 2.2.2.1.1.3.1

Multiply $- 1$ by $3$ .

$\frac{21}{2} - 3 \frac{3 y}{2} - y = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.2.2.1.1.3.2

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

$\frac{21}{2} + \frac{- 3 (3 y)}{2} - y = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.2.2.1.1.3.3

Multiply $3$ by $- 3$ .

$\frac{21}{2} + \frac{- 9 y}{2} - y = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

$\frac{21}{2} + \frac{- 9 y}{2} - y = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.2.2.1.1.4

Move the negative in front of the fraction.

$\frac{21}{2} - \frac{9 y}{2} - y = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

$\frac{21}{2} - \frac{9 y}{2} - y = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.2.2.1.2

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

$\frac{21}{2} - \frac{9 y}{2} - y \cdot \frac{2}{2} = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.2.2.1.3

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

$\frac{21}{2} - \frac{9 y}{2} + \frac{- y \cdot 2}{2} = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.2.2.1.4

Combine the numerators over the common denominator.

$\frac{21}{2} + \frac{- 9 y - y \cdot 2}{2} = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.2.2.1.5

Combine the numerators over the common denominator.

$\frac{21 - 9 y - y \cdot 2}{2} = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.2.2.1.6

Multiply $2$ by $- 1$ .

$\frac{21 - 9 y - 2 y}{2} = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.2.2.1.7

Subtract $2 y$ from $- 9 y$ .

$\frac{21 - 11 y}{2} = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

$\frac{21 - 11 y}{2} = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

$\frac{21 - 11 y}{2} = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

$\frac{21 - 11 y}{2} = 16$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.3

Solve for $y$ in $\frac{21 - 11 y}{2} = 16$ .

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Step 2.3.1

Multiply both sides by $2$ .

$\frac{21 - 11 y}{2} \cdot 2 = 16 \cdot 2$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Simplify the left side.

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Step 2.3.2.1.1

Simplify $\frac{21 - 11 y}{2} \cdot 2$ .

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Step 2.3.2.1.1.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1.1.1

Cancel the common factor.

$\frac{21 - 11 y}{2} \cdot 2 = 16 \cdot 2$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.3.2.1.1.1.2

Rewrite the expression.

$21 - 11 y = 16 \cdot 2$

$x = \frac{7}{2} - \frac{3 y}{2}$

$21 - 11 y = 16 \cdot 2$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.3.2.1.1.2

Reorder $21$ and $- 11 y$ .

$- 11 y + 21 = 16 \cdot 2$

$x = \frac{7}{2} - \frac{3 y}{2}$

$- 11 y + 21 = 16 \cdot 2$

$x = \frac{7}{2} - \frac{3 y}{2}$

$- 11 y + 21 = 16 \cdot 2$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.3.2.2

Simplify the right side.

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Step 2.3.2.2.1

Multiply $16$ by $2$ .

$- 11 y + 21 = 32$

$x = \frac{7}{2} - \frac{3 y}{2}$

$- 11 y + 21 = 32$

$x = \frac{7}{2} - \frac{3 y}{2}$

$- 11 y + 21 = 32$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.3.3

Solve for $y$ .

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Step 2.3.3.1

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

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Step 2.3.3.1.1

Subtract $21$ from both sides of the equation.

$- 11 y = 32 - 21$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.3.3.1.2

Subtract $21$ from $32$ .

$- 11 y = 11$

$x = \frac{7}{2} - \frac{3 y}{2}$

$- 11 y = 11$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.3.3.2

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

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Step 2.3.3.2.1

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

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

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.3.3.2.2

Simplify the left side.

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Step 2.3.3.2.2.1

Cancel the common factor of $- 11$ .

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Step 2.3.3.2.2.1.1

Cancel the common factor.

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

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.3.3.2.2.1.2

Divide $y$ by $1$ .

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

$x = \frac{7}{2} - \frac{3 y}{2}$

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

$x = \frac{7}{2} - \frac{3 y}{2}$

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

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.3.3.2.3

Simplify the right side.

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Step 2.3.3.2.3.1

Divide $11$ by $- 11$ .

$y = - 1$

$x = \frac{7}{2} - \frac{3 y}{2}$

$y = - 1$

$x = \frac{7}{2} - \frac{3 y}{2}$

$y = - 1$

$x = \frac{7}{2} - \frac{3 y}{2}$

$y = - 1$

$x = \frac{7}{2} - \frac{3 y}{2}$

$y = - 1$

$x = \frac{7}{2} - \frac{3 y}{2}$

Step 2.4

Replace all occurrences of $y$ with $- 1$ in each equation.

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Step 2.4.1

Replace all occurrences of $y$ in $x = \frac{7}{2} - \frac{3 y}{2}$ with $- 1$ .

$x = \frac{7}{2} - \frac{3 (- 1)}{2}$

$y = - 1$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\frac{7}{2} - \frac{3 (- 1)}{2}$ .

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Step 2.4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{7 - 3 \cdot - 1}{2}$

$y = - 1$

Step 2.4.2.1.2

Simplify the expression.

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Step 2.4.2.1.2.1

Multiply $- 3$ by $- 1$ .

$x = \frac{7 + 3}{2}$

$y = - 1$

Step 2.4.2.1.2.2

Add $7$ and $3$ .

$x = \frac{10}{2}$

$y = - 1$

Step 2.4.2.1.2.3

Divide $10$ by $2$ .

$x = 5$

$y = - 1$

$x = 5$

$y = - 1$

$x = 5$

$y = - 1$

$x = 5$

$y = - 1$

$x = 5$

$y = - 1$

Step 2.5

List all of the solutions.

$x = 5, y = - 1$