Linear Algebra Examples

$a [\begin{array}{c} 1 \\ - 2 \end{array}] + b [\begin{array}{c} 3 \\ - 2 \end{array}] = [\begin{array}{c} - 2 \\ 1 \end{array}]$

Step 1

Simplify each term.

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Step 1.1

Multiply $a$ by each element of the matrix.

$[\begin{array}{c} a \cdot 1 \\ a \cdot - 2 \end{array}] + b [\begin{array}{c} 3 \\ - 2 \end{array}] = [\begin{array}{c} - 2 \\ 1 \end{array}]$

Step 1.2

Simplify each element in the matrix.

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Step 1.2.1

Multiply $a$ by $1$ .

$[\begin{array}{c} a \\ a \cdot - 2 \end{array}] + b [\begin{array}{c} 3 \\ - 2 \end{array}] = [\begin{array}{c} - 2 \\ 1 \end{array}]$

Step 1.2.2

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

$[\begin{array}{c} a \\ - 2 a \end{array}] + b [\begin{array}{c} 3 \\ - 2 \end{array}] = [\begin{array}{c} - 2 \\ 1 \end{array}]$

Step 1.3

Multiply $b$ by each element of the matrix.

$[\begin{array}{c} a \\ - 2 a \end{array}] + [\begin{array}{c} b \cdot 3 \\ b \cdot - 2 \end{array}] = [\begin{array}{c} - 2 \\ 1 \end{array}]$

Step 1.4

Simplify each element in the matrix.

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Step 1.4.1

Move $3$ to the left of $b$ .

$[\begin{array}{c} a \\ - 2 a \end{array}] + [\begin{array}{c} 3 b \\ b \cdot - 2 \end{array}] = [\begin{array}{c} - 2 \\ 1 \end{array}]$

Step 1.4.2

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

$[\begin{array}{c} a \\ - 2 a \end{array}] + [\begin{array}{c} 3 b \\ - 2 b \end{array}] = [\begin{array}{c} - 2 \\ 1 \end{array}]$

Step 2

Add the corresponding elements.

$[\begin{array}{c} a + 3 b \\ - 2 a - 2 b \end{array}] = [\begin{array}{c} - 2 \\ 1 \end{array}]$

Step 3

Write as a linear system of equations.

$a + 3 b = - 2$

$- 2 a - 2 b = 1$

Step 4

Solve the system of equations.

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Step 4.1

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

$a = - 2 - 3 b$

$- 2 a - 2 b = 1$

Step 4.2

Replace all occurrences of $a$ with $- 2 - 3 b$ in each equation.

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Step 4.2.1

Replace all occurrences of $a$ in $- 2 a - 2 b = 1$ with $- 2 - 3 b$ .

$- 2 (- 2 - 3 b) - 2 b = 1$

$a = - 2 - 3 b$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify $- 2 (- 2 - 3 b) - 2 b$ .

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Step 4.2.2.1.1

Simplify each term.

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Step 4.2.2.1.1.1

Apply the distributive property.

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

$a = - 2 - 3 b$

Step 4.2.2.1.1.2

Multiply $- 2$ by $- 2$ .

$4 - 2 (- 3 b) - 2 b = 1$

$a = - 2 - 3 b$

Step 4.2.2.1.1.3

Multiply $- 3$ by $- 2$ .

$4 + 6 b - 2 b = 1$

$a = - 2 - 3 b$

$4 + 6 b - 2 b = 1$

$a = - 2 - 3 b$

Step 4.2.2.1.2

Subtract $2 b$ from $6 b$ .

$4 + 4 b = 1$

$a = - 2 - 3 b$

$4 + 4 b = 1$

$a = - 2 - 3 b$

$4 + 4 b = 1$

$a = - 2 - 3 b$

$4 + 4 b = 1$

$a = - 2 - 3 b$

Step 4.3

Solve for $b$ in $4 + 4 b = 1$ .

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Step 4.3.1

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

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Step 4.3.1.1

Subtract $4$ from both sides of the equation.

$4 b = 1 - 4$

$a = - 2 - 3 b$

Step 4.3.1.2

Subtract $4$ from $1$ .

$4 b = - 3$

$a = - 2 - 3 b$

$4 b = - 3$

$a = - 2 - 3 b$

Step 4.3.2

Divide each term in $4 b = - 3$ by $4$ and simplify.

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Step 4.3.2.1

Divide each term in $4 b = - 3$ by $4$ .

$\frac{4 b}{4} = \frac{- 3}{4}$

$a = - 2 - 3 b$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Cancel the common factor of $4$ .

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Step 4.3.2.2.1.1

Cancel the common factor.

$\frac{4 b}{4} = \frac{- 3}{4}$

$a = - 2 - 3 b$

Step 4.3.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 3}{4}$

$a = - 2 - 3 b$

$b = \frac{- 3}{4}$

$a = - 2 - 3 b$

$b = \frac{- 3}{4}$

$a = - 2 - 3 b$

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

Move the negative in front of the fraction.

$b = - \frac{3}{4}$

$a = - 2 - 3 b$

$b = - \frac{3}{4}$

$a = - 2 - 3 b$

$b = - \frac{3}{4}$

$a = - 2 - 3 b$

$b = - \frac{3}{4}$

$a = - 2 - 3 b$

Step 4.4

Replace all occurrences of $b$ with $- \frac{3}{4}$ in each equation.

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Step 4.4.1

Replace all occurrences of $b$ in $a = - 2 - 3 b$ with $- \frac{3}{4}$ .

$a = - 2 - 3 (- \frac{3}{4})$

$b = - \frac{3}{4}$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

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

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Step 4.4.2.1.1

Multiply $- 3 (- \frac{3}{4})$ .

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Step 4.4.2.1.1.1

Multiply $- 1$ by $- 3$ .

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

$b = - \frac{3}{4}$

Step 4.4.2.1.1.2

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

$a = - 2 + \frac{3 \cdot 3}{4}$

$b = - \frac{3}{4}$

Step 4.4.2.1.1.3

Multiply $3$ by $3$ .

$a = - 2 + \frac{9}{4}$

$b = - \frac{3}{4}$

$a = - 2 + \frac{9}{4}$

$b = - \frac{3}{4}$

Step 4.4.2.1.2

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

$a = - 2 \cdot \frac{4}{4} + \frac{9}{4}$

$b = - \frac{3}{4}$

Step 4.4.2.1.3

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

$a = \frac{- 2 \cdot 4}{4} + \frac{9}{4}$

$b = - \frac{3}{4}$

Step 4.4.2.1.4

Combine the numerators over the common denominator.

$a = \frac{- 2 \cdot 4 + 9}{4}$

$b = - \frac{3}{4}$

Step 4.4.2.1.5

Simplify the numerator.

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Step 4.4.2.1.5.1

Multiply $- 2$ by $4$ .

$a = \frac{- 8 + 9}{4}$

$b = - \frac{3}{4}$

Step 4.4.2.1.5.2

Add $- 8$ and $9$ .

$a = \frac{1}{4}$

$b = - \frac{3}{4}$

$a = \frac{1}{4}$

$b = - \frac{3}{4}$

$a = \frac{1}{4}$

$b = - \frac{3}{4}$

$a = \frac{1}{4}$

$b = - \frac{3}{4}$

$a = \frac{1}{4}$

$b = - \frac{3}{4}$

Step 4.5

List all of the solutions.

$a = \frac{1}{4}, b = - \frac{3}{4}$