Linear Algebra Examples

a [\begin{matrix} 1 - 2 \end{matrix}] + b [\begin{matrix} 3 - 2 \end{matrix}] = [\begin{matrix} - 2 1 \end{matrix}]

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

Step 1

Simplify each term.

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Step 1.1

Multiply

a

$a$ by each element of the matrix.

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

$a$ by

1

$1$ .

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

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

Step 1.2.2

Move

- 2

$- 2$ to the left of

a

$a$ .

[\begin{matrix} a - 2 a \end{matrix}] + b [\begin{matrix} 3 - 2 \end{matrix}] = [\begin{matrix} - 2 1 \end{matrix}]

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

[\begin{matrix} a - 2 a \end{matrix}] + b [\begin{matrix} 3 - 2 \end{matrix}] = [\begin{matrix} - 2 1 \end{matrix}]

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

Step 1.3

Multiply

b

$b$ by each element of the matrix.

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

$3$ to the left of

b

$b$ .

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

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

Step 1.4.2

Move

- 2

$- 2$ to the left of

b

$b$ .

[\begin{matrix} a - 2 a \end{matrix}] + [\begin{matrix} 3 b - 2 b \end{matrix}] = [\begin{matrix} - 2 1 \end{matrix}]

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

[\begin{matrix} a - 2 a \end{matrix}] + [\begin{matrix} 3 b - 2 b \end{matrix}] = [\begin{matrix} - 2 1 \end{matrix}]

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

[\begin{matrix} a - 2 a \end{matrix}] + [\begin{matrix} 3 b - 2 b \end{matrix}] = [\begin{matrix} - 2 1 \end{matrix}]

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

Step 2

Add the corresponding elements.

[\begin{matrix} a + 3 b - 2 a - 2 b \end{matrix}] = [\begin{matrix} - 2 1 \end{matrix}]

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

Step 3

Write as a linear system of equations.

a + 3 b = - 2

$a + 3 b = - 2$

- 2 a - 2 b = 1

$- 2 a - 2 b = 1$

Step 4

Solve the system of equations.

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Step 4.1

Subtract

3 b

$3 b$ from both sides of the equation.

a = - 2 - 3 b

$a = - 2 - 3 b$

- 2 a - 2 b = 1

$- 2 a - 2 b = 1$

Step 4.2

Replace all occurrences of

a

$a$ with

- 2 - 3 b

$- 2 - 3 b$ in each equation.

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Step 4.2.1

Replace all occurrences of

a

$a$ in

- 2 a - 2 b = 1

$- 2 a - 2 b = 1$ with

- 2 - 3 b

$- 2 - 3 b$ .

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

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

a = - 2 - 3 b

$a = - 2 - 3 b$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify

- 2 (- 2 - 3 b) - 2 b

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

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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

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

a = - 2 - 3 b

$a = - 2 - 3 b$

Step 4.2.2.1.1.2

Multiply

- 2

$- 2$ by

- 2

$- 2$ .

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

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

a = - 2 - 3 b

$a = - 2 - 3 b$

Step 4.2.2.1.1.3

Multiply

- 3

$- 3$ by

- 2

$- 2$ .

4 + 6 b - 2 b = 1

$4 + 6 b - 2 b = 1$

a = - 2 - 3 b

$a = - 2 - 3 b$

4 + 6 b - 2 b = 1

$4 + 6 b - 2 b = 1$

a = - 2 - 3 b

$a = - 2 - 3 b$

Step 4.2.2.1.2

Subtract

2 b

$2 b$ from

6 b

$6 b$ .

4 + 4 b = 1

$4 + 4 b = 1$

a = - 2 - 3 b

$a = - 2 - 3 b$

4 + 4 b = 1

$4 + 4 b = 1$

a = - 2 - 3 b

$a = - 2 - 3 b$

4 + 4 b = 1

$4 + 4 b = 1$

a = - 2 - 3 b

$a = - 2 - 3 b$

4 + 4 b = 1

$4 + 4 b = 1$

a = - 2 - 3 b

$a = - 2 - 3 b$

Step 4.3

Solve for

b

$b$ in

4 + 4 b = 1

$4 + 4 b = 1$ .

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Step 4.3.1

Move all terms not containing

b

$b$ to the right side of the equation.

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Step 4.3.1.1

Subtract

4

$4$ from both sides of the equation.

4 b = 1 - 4

$4 b = 1 - 4$

a = - 2 - 3 b

$a = - 2 - 3 b$

Step 4.3.1.2

Subtract

4

$4$ from

1

$1$ .

4 b = - 3

$4 b = - 3$

a = - 2 - 3 b

$a = - 2 - 3 b$

4 b = - 3

$4 b = - 3$

a = - 2 - 3 b

$a = - 2 - 3 b$

Step 4.3.2

Divide each term in

4 b = - 3

$4 b = - 3$ by

4

$4$ and simplify.

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Step 4.3.2.1

Divide each term in

4 b = - 3

$4 b = - 3$ by

4

$4$ .

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

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

a = - 2 - 3 b

$a = - 2 - 3 b$

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

$4$ .

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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.

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