Linear Algebra Examples

$x [\begin{array}{c} 2 \\ - 3 \end{array}] + y [\begin{array}{c} - 4 \\ - 2 \end{array}] = [\begin{array}{c} - 4 \\ - 1 \end{array}]$

Step 1

Simplify.

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Step 1.1

Simplify each term.

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Step 1.1.1

Multiply $x$ by each element of the matrix.

$[\begin{array}{c} x \cdot 2 \\ x \cdot - 3 \end{array}] + y [\begin{array}{c} - 4 \\ - 2 \end{array}] = [\begin{array}{c} - 4 \\ - 1 \end{array}]$

Step 1.1.2

Simplify each element in the matrix.

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Step 1.1.2.1

Move $2$ to the left of $x$ .

$[\begin{array}{c} 2 x \\ x \cdot - 3 \end{array}] + y [\begin{array}{c} - 4 \\ - 2 \end{array}] = [\begin{array}{c} - 4 \\ - 1 \end{array}]$

Step 1.1.2.2

Move $- 3$ to the left of $x$ .

$[\begin{array}{c} 2 x \\ - 3 x \end{array}] + y [\begin{array}{c} - 4 \\ - 2 \end{array}] = [\begin{array}{c} - 4 \\ - 1 \end{array}]$

Step 1.1.3

Multiply $y$ by each element of the matrix.

$[\begin{array}{c} 2 x \\ - 3 x \end{array}] + [\begin{array}{c} y \cdot - 4 \\ y \cdot - 2 \end{array}] = [\begin{array}{c} - 4 \\ - 1 \end{array}]$

Step 1.1.4

Simplify each element in the matrix.

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Step 1.1.4.1

Move $- 4$ to the left of $y$ .

$[\begin{array}{c} 2 x \\ - 3 x \end{array}] + [\begin{array}{c} - 4 y \\ y \cdot - 2 \end{array}] = [\begin{array}{c} - 4 \\ - 1 \end{array}]$

Step 1.1.4.2

Move $- 2$ to the left of $y$ .

$[\begin{array}{c} 2 x \\ - 3 x \end{array}] + [\begin{array}{c} - 4 y \\ - 2 y \end{array}] = [\begin{array}{c} - 4 \\ - 1 \end{array}]$

Step 1.2

Add the corresponding elements.

$[\begin{array}{c} 2 x - 4 y \\ - 3 x - 2 y \end{array}] = [\begin{array}{c} - 4 \\ - 1 \end{array}]$

Step 2

The matrix equation can be written as a set of equations.

$2 x - 4 y = - 4$

$- 3 x - 2 y = - 1$

Step 3

Solve for $x$ in $2 x - 4 y = - 4$ .

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Step 3.1

Add $4 y$ to both sides of the equation.

$2 x = - 4 + 4 y$

$- 3 x - 2 y = - 1$

Step 3.2

Divide each term in $2 x = - 4 + 4 y$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 x = - 4 + 4 y$ by $2$ .

$\frac{2 x}{2} = \frac{- 4}{2} + \frac{4 y}{2}$

$- 3 x - 2 y = - 1$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 4}{2} + \frac{4 y}{2}$

$- 3 x - 2 y = - 1$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{2} + \frac{4 y}{2}$

$- 3 x - 2 y = - 1$

$x = \frac{- 4}{2} + \frac{4 y}{2}$

$- 3 x - 2 y = - 1$

$x = \frac{- 4}{2} + \frac{4 y}{2}$

$- 3 x - 2 y = - 1$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Divide $- 4$ by $2$ .

$x = - 2 + \frac{4 y}{2}$

$- 3 x - 2 y = - 1$

Step 3.2.3.1.2

Cancel the common factor of $4$ and $2$ .

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Step 3.2.3.1.2.1

Factor $2$ out of $4 y$ .

$x = - 2 + \frac{2 (2 y)}{2}$

$- 3 x - 2 y = - 1$

Step 3.2.3.1.2.2

Cancel the common factors.

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Step 3.2.3.1.2.2.1

Factor $2$ out of $2$ .

$x = - 2 + \frac{2 (2 y)}{2 (1)}$

$- 3 x - 2 y = - 1$

Step 3.2.3.1.2.2.2

Cancel the common factor.

$x = - 2 + \frac{2 (2 y)}{2 \cdot 1}$

$- 3 x - 2 y = - 1$

Step 3.2.3.1.2.2.3

Rewrite the expression.

$x = - 2 + \frac{2 y}{1}$

$- 3 x - 2 y = - 1$

Step 3.2.3.1.2.2.4

Divide $2 y$ by $1$ .

$x = - 2 + 2 y$

$- 3 x - 2 y = - 1$

$x = - 2 + 2 y$

$- 3 x - 2 y = - 1$

$x = - 2 + 2 y$

$- 3 x - 2 y = - 1$

$x = - 2 + 2 y$

$- 3 x - 2 y = - 1$

$x = - 2 + 2 y$

$- 3 x - 2 y = - 1$

$x = - 2 + 2 y$

$- 3 x - 2 y = - 1$

$x = - 2 + 2 y$

$- 3 x - 2 y = - 1$

Step 4

Replace all occurrences of $x$ with $- 2 + 2 y$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $- 3 x - 2 y = - 1$ with $- 2 + 2 y$ .

$- 3 (- 2 + 2 y) - 2 y = - 1$

$x = - 2 + 2 y$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 3 (- 2 + 2 y) - 2 y$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$- 3 \cdot - 2 - 3 (2 y) - 2 y = - 1$

$x = - 2 + 2 y$

Step 4.2.1.1.2

Multiply $- 3$ by $- 2$ .

$6 - 3 (2 y) - 2 y = - 1$

$x = - 2 + 2 y$

Step 4.2.1.1.3

Multiply $2$ by $- 3$ .

$6 - 6 y - 2 y = - 1$

$x = - 2 + 2 y$

$6 - 6 y - 2 y = - 1$

$x = - 2 + 2 y$

Step 4.2.1.2

Subtract $2 y$ from $- 6 y$ .

$6 - 8 y = - 1$

$x = - 2 + 2 y$

$6 - 8 y = - 1$

$x = - 2 + 2 y$

$6 - 8 y = - 1$

$x = - 2 + 2 y$

$6 - 8 y = - 1$

$x = - 2 + 2 y$

Step 5

Solve for $y$ in $6 - 8 y = - 1$ .

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Step 5.1

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

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Step 5.1.1

Subtract $6$ from both sides of the equation.

$- 8 y = - 1 - 6$

$x = - 2 + 2 y$

Step 5.1.2

Subtract $6$ from $- 1$ .

$- 8 y = - 7$

$x = - 2 + 2 y$

$- 8 y = - 7$

$x = - 2 + 2 y$

Step 5.2

Divide each term in $- 8 y = - 7$ by $- 8$ and simplify.

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Step 5.2.1

Divide each term in $- 8 y = - 7$ by $- 8$ .

$\frac{- 8 y}{- 8} = \frac{- 7}{- 8}$

$x = - 2 + 2 y$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 8$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 8 y}{- 8} = \frac{- 7}{- 8}$

$x = - 2 + 2 y$

Step 5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 7}{- 8}$

$x = - 2 + 2 y$

$y = \frac{- 7}{- 8}$

$x = - 2 + 2 y$

$y = \frac{- 7}{- 8}$

$x = - 2 + 2 y$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Dividing two negative values results in a positive value.

$y = \frac{7}{8}$

$x = - 2 + 2 y$

$y = \frac{7}{8}$

$x = - 2 + 2 y$

$y = \frac{7}{8}$

$x = - 2 + 2 y$

$y = \frac{7}{8}$

$x = - 2 + 2 y$

Step 6

Replace all occurrences of $y$ with $\frac{7}{8}$ in each equation.

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Step 6.1

Replace all occurrences of $y$ in $x = - 2 + 2 y$ with $\frac{7}{8}$ .

$x = - 2 + 2 (\frac{7}{8})$

$y = \frac{7}{8}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- 2 + 2 (\frac{7}{8})$ .

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Step 6.2.1.1

Cancel the common factor of $2$ .

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Step 6.2.1.1.1

Factor $2$ out of $8$ .

$x = - 2 + 2 (\frac{7}{2 (4)})$

$y = \frac{7}{8}$

Step 6.2.1.1.2

Cancel the common factor.

$x = - 2 + 2 (\frac{7}{2 \cdot 4})$

$y = \frac{7}{8}$

Step 6.2.1.1.3

Rewrite the expression.

$x = - 2 + \frac{7}{4}$

$y = \frac{7}{8}$

$x = - 2 + \frac{7}{4}$

$y = \frac{7}{8}$

Step 6.2.1.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = - 2 \cdot \frac{4}{4} + \frac{7}{4}$

$y = \frac{7}{8}$

Step 6.2.1.3

Combine $- 2$ and $\frac{4}{4}$ .

$x = \frac{- 2 \cdot 4}{4} + \frac{7}{4}$

$y = \frac{7}{8}$

Step 6.2.1.4

Combine the numerators over the common denominator.

$x = \frac{- 2 \cdot 4 + 7}{4}$

$y = \frac{7}{8}$

Step 6.2.1.5

Simplify the numerator.

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Step 6.2.1.5.1

Multiply $- 2$ by $4$ .

$x = \frac{- 8 + 7}{4}$

$y = \frac{7}{8}$

Step 6.2.1.5.2

Add $- 8$ and $7$ .

$x = \frac{- 1}{4}$

$y = \frac{7}{8}$

$x = \frac{- 1}{4}$

$y = \frac{7}{8}$

Step 6.2.1.6

Move the negative in front of the fraction.

$x = - \frac{1}{4}$

$y = \frac{7}{8}$

$x = - \frac{1}{4}$

$y = \frac{7}{8}$

$x = - \frac{1}{4}$

$y = \frac{7}{8}$

$x = - \frac{1}{4}$

$y = \frac{7}{8}$

Step 7

List all of the solutions.

$x = - \frac{1}{4}, y = \frac{7}{8}$