Linear Algebra Examples

$[\begin{array}{c} - 2 & - 5 \\ - 2 & - 1 \end{array}] [\begin{array}{c} a \\ b \end{array}] + [\begin{array}{c} - 19 \\ 11 \end{array}] = [\begin{array}{c} - 43 \\ - 5 \end{array}]$

Step 1

Simplify.

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Step 1.1

Multiply $[\begin{array}{c} - 2 & - 5 \\ - 2 & - 1 \end{array}] [\begin{array}{c} a \\ b \end{array}]$ .

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Step 1.1.1

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 1.1.2

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

$[\begin{array}{c} - 2 a - 5 b \\ - 2 a - b \end{array}] + [\begin{array}{c} - 19 \\ 11 \end{array}] = [\begin{array}{c} - 43 \\ - 5 \end{array}]$

Step 1.2

Add the corresponding elements.

$[\begin{array}{c} - 2 a - 5 b - 19 \\ - 2 a - b + 11 \end{array}] = [\begin{array}{c} - 43 \\ - 5 \end{array}]$

Step 2

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

$- 2 a - 5 b - 19 = - 43$

$- 2 a - b + 11 = - 5$

Step 3

Solve for $a$ in $- 2 a - 5 b - 19 = - 43$ .

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Step 3.1

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

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Step 3.1.1

Add $5 b$ to both sides of the equation.

$- 2 a - 19 = - 43 + 5 b$

$- 2 a - b + 11 = - 5$

Step 3.1.2

Add $19$ to both sides of the equation.

$- 2 a = - 43 + 5 b + 19$

$- 2 a - b + 11 = - 5$

Step 3.1.3

Add $- 43$ and $19$ .

$- 2 a = 5 b - 24$

$- 2 a - b + 11 = - 5$

$- 2 a = 5 b - 24$

$- 2 a - b + 11 = - 5$

Step 3.2

Divide each term in $- 2 a = 5 b - 24$ by $- 2$ and simplify.

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Step 3.2.1

Divide each term in $- 2 a = 5 b - 24$ by $- 2$ .

$\frac{- 2 a}{- 2} = \frac{5 b}{- 2} + \frac{- 24}{- 2}$

$- 2 a - b + 11 = - 5$

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.

$\frac{- 2 a}{- 2} = \frac{5 b}{- 2} + \frac{- 24}{- 2}$

$- 2 a - b + 11 = - 5$

Step 3.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{5 b}{- 2} + \frac{- 24}{- 2}$

$- 2 a - b + 11 = - 5$

$a = \frac{5 b}{- 2} + \frac{- 24}{- 2}$

$- 2 a - b + 11 = - 5$

$a = \frac{5 b}{- 2} + \frac{- 24}{- 2}$

$- 2 a - b + 11 = - 5$

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

Move the negative in front of the fraction.

$a = - \frac{5 b}{2} + \frac{- 24}{- 2}$

$- 2 a - b + 11 = - 5$

Step 3.2.3.1.2

Divide $- 24$ by $- 2$ .

$a = - \frac{5 b}{2} + 12$

$- 2 a - b + 11 = - 5$

$a = - \frac{5 b}{2} + 12$

$- 2 a - b + 11 = - 5$

$a = - \frac{5 b}{2} + 12$

$- 2 a - b + 11 = - 5$

$a = - \frac{5 b}{2} + 12$

$- 2 a - b + 11 = - 5$

$a = - \frac{5 b}{2} + 12$

$- 2 a - b + 11 = - 5$

Step 4

Replace all occurrences of $a$ with $- \frac{5 b}{2} + 12$ in each equation.

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Step 4.1

Replace all occurrences of $a$ in $- 2 a - b + 11 = - 5$ with $- \frac{5 b}{2} + 12$ .

$- 2 (- \frac{5 b}{2} + 12) - b + 11 = - 5$

$a = - \frac{5 b}{2} + 12$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 2 (- \frac{5 b}{2} + 12) - b + 11$ .

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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.

$- 2 (- \frac{5 b}{2}) - 2 \cdot 12 - b + 11 = - 5$

$a = - \frac{5 b}{2} + 12$

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Move the leading negative in $- \frac{5 b}{2}$ into the numerator.

$- 2 \frac{- 5 b}{2} - 2 \cdot 12 - b + 11 = - 5$

$a = - \frac{5 b}{2} + 12$

Step 4.2.1.1.2.2

Factor $2$ out of $- 2$ .

$2 (- 1) (\frac{- 5 b}{2}) - 2 \cdot 12 - b + 11 = - 5$

$a = - \frac{5 b}{2} + 12$

Step 4.2.1.1.2.3

Cancel the common factor.

$2 \cdot (- 1 \frac{- 5 b}{2}) - 2 \cdot 12 - b + 11 = - 5$

$a = - \frac{5 b}{2} + 12$

Step 4.2.1.1.2.4

Rewrite the expression.

$- 1 (- 5 b) - 2 \cdot 12 - b + 11 = - 5$

$a = - \frac{5 b}{2} + 12$

$- 1 (- 5 b) - 2 \cdot 12 - b + 11 = - 5$

$a = - \frac{5 b}{2} + 12$

Step 4.2.1.1.3

Multiply $- 5$ by $- 1$ .

$5 b - 2 \cdot 12 - b + 11 = - 5$

$a = - \frac{5 b}{2} + 12$

Step 4.2.1.1.4

Multiply $- 2$ by $12$ .

$5 b - 24 - b + 11 = - 5$

$a = - \frac{5 b}{2} + 12$

$5 b - 24 - b + 11 = - 5$

$a = - \frac{5 b}{2} + 12$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Subtract $b$ from $5 b$ .

$4 b - 24 + 11 = - 5$

$a = - \frac{5 b}{2} + 12$

Step 4.2.1.2.2

Add $- 24$ and $11$ .

$4 b - 13 = - 5$

$a = - \frac{5 b}{2} + 12$

$4 b - 13 = - 5$

$a = - \frac{5 b}{2} + 12$

$4 b - 13 = - 5$

$a = - \frac{5 b}{2} + 12$

$4 b - 13 = - 5$

$a = - \frac{5 b}{2} + 12$

$4 b - 13 = - 5$

$a = - \frac{5 b}{2} + 12$

Step 5

Solve for $b$ in $4 b - 13 = - 5$ .

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Step 5.1

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

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Step 5.1.1

Add $13$ to both sides of the equation.

$4 b = - 5 + 13$

$a = - \frac{5 b}{2} + 12$

Step 5.1.2

Add $- 5$ and $13$ .

$4 b = 8$

$a = - \frac{5 b}{2} + 12$

$4 b = 8$

$a = - \frac{5 b}{2} + 12$

Step 5.2

Divide each term in $4 b = 8$ by $4$ and simplify.

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Step 5.2.1

Divide each term in $4 b = 8$ by $4$ .

$\frac{4 b}{4} = \frac{8}{4}$

$a = - \frac{5 b}{2} + 12$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $4$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{4 b}{4} = \frac{8}{4}$

$a = - \frac{5 b}{2} + 12$

Step 5.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{8}{4}$

$a = - \frac{5 b}{2} + 12$

$b = \frac{8}{4}$

$a = - \frac{5 b}{2} + 12$

$b = \frac{8}{4}$

$a = - \frac{5 b}{2} + 12$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $8$ by $4$ .

$b = 2$

$a = - \frac{5 b}{2} + 12$

$b = 2$

$a = - \frac{5 b}{2} + 12$

$b = 2$

$a = - \frac{5 b}{2} + 12$

$b = 2$

$a = - \frac{5 b}{2} + 12$

Step 6

Replace all occurrences of $b$ with $2$ in each equation.

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Step 6.1

Replace all occurrences of $b$ in $a = - \frac{5 b}{2} + 12$ with $2$ .

$a = - \frac{5 (2)}{2} + 12$

$b = 2$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- \frac{5 (2)}{2} + 12$ .

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Step 6.2.1.1

Simplify each term.

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Step 6.2.1.1.1

Cancel the common factor of $2$ .

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Step 6.2.1.1.1.1

Cancel the common factor.

$a = - \frac{5 \cdot 2}{2} + 12$

$b = 2$

Step 6.2.1.1.1.2

Divide $5$ by $1$ .

$a = - 1 \cdot 5 + 12$

$b = 2$

$a = - 1 \cdot 5 + 12$

$b = 2$

Step 6.2.1.1.2

Multiply $- 1$ by $5$ .

$a = - 5 + 12$

$b = 2$

$a = - 5 + 12$

$b = 2$

Step 6.2.1.2

Add $- 5$ and $12$ .

$a = 7$

$b = 2$

$a = 7$

$b = 2$

$a = 7$

$b = 2$

$a = 7$

$b = 2$

Step 7

List all of the solutions.

$a = 7, b = 2$