Linear Algebra Examples

⎡ ⎢ ⎣ \begin{matrix} 679 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 3 a + 2 b + c 7 a + b - c 9 a - b - c \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 6 \\ 7 \\ 9 \end{array}] = [\begin{array}{c} 3 a + 2 b + c \\ 7 a + b - c \\ 9 a - b - c \end{array}]$

Step 1

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

6 = 3 a + 2 b + c

$6 = 3 a + 2 b + c$

7 = 7 a + b - c

$7 = 7 a + b - c$

9 = 9 a - b - c

$9 = 9 a - b - c$

Step 2

Solve for

c

$c$ in

6 = 3 a + 2 b + c

$6 = 3 a + 2 b + c$ .

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Step 2.1

Rewrite the equation as

3 a + 2 b + c = 6

$3 a + 2 b + c = 6$ .

3 a + 2 b + c = 6

$3 a + 2 b + c = 6$

7 = 7 a + b - c

$7 = 7 a + b - c$

9 = 9 a - b - c

$9 = 9 a - b - c$

Step 2.2

Move all terms not containing

c

$c$ to the right side of the equation.

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Step 2.2.1

Subtract

3 a

$3 a$ from both sides of the equation.

2 b + c = 6 - 3 a

$2 b + c = 6 - 3 a$

7 = 7 a + b - c

$7 = 7 a + b - c$

9 = 9 a - b - c

$9 = 9 a - b - c$

Step 2.2.2

Subtract

2 b

$2 b$ from both sides of the equation.

c = 6 - 3 a - 2 b

$c = 6 - 3 a - 2 b$

7 = 7 a + b - c

$7 = 7 a + b - c$

9 = 9 a - b - c

$9 = 9 a - b - c$

c = 6 - 3 a - 2 b

$c = 6 - 3 a - 2 b$

7 = 7 a + b - c

$7 = 7 a + b - c$

9 = 9 a - b - c

$9 = 9 a - b - c$

c = 6 - 3 a - 2 b

$c = 6 - 3 a - 2 b$

7 = 7 a + b - c

$7 = 7 a + b - c$

9 = 9 a - b - c

$9 = 9 a - b - c$

Step 3

Replace all occurrences of

c

$c$ with

6 - 3 a - 2 b

$6 - 3 a - 2 b$ in each equation.

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Step 3.1

Replace all occurrences of

c

$c$ in

7 = 7 a + b - c

$7 = 7 a + b - c$ with

6 - 3 a - 2 b

$6 - 3 a - 2 b$ .

7 = 7 a + b - (6 - 3 a - 2 b)

$7 = 7 a + b - (6 - 3 a - 2 b)$

$c = 6 - 3 a - 2 b$

$9 = 9 a - b - c$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify

$7 a + b - (6 - 3 a - 2 b)$ .

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Step 3.2.1.1

Simplify each term.

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Step 3.2.1.1.1

Apply the distributive property.

$7 = 7 a + b - 1 \cdot 6 - (- 3 a) - (- 2 b)$

$c = 6 - 3 a - 2 b$

$9 = 9 a - b - c$

Step 3.2.1.1.2

Simplify.

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Step 3.2.1.1.2.1

Multiply

$- 1$ by

$6$ .

$7 = 7 a + b - 6 - (- 3 a) - (- 2 b)$

$c = 6 - 3 a - 2 b$

$9 = 9 a - b - c$

Step 3.2.1.1.2.2

Multiply

$- 3$ by

$- 1$ .

$7 = 7 a + b - 6 + 3 a - (- 2 b)$

$c = 6 - 3 a - 2 b$

$9 = 9 a - b - c$

Step 3.2.1.1.2.3

Multiply

$- 2$ by

$- 1$ .

$7 = 7 a + b - 6 + 3 a + 2 b$

$c = 6 - 3 a - 2 b$

$9 = 9 a - b - c$

$7 = 7 a + b - 6 + 3 a + 2 b$

$c = 6 - 3 a - 2 b$

$9 = 9 a - b - c$

$7 = 7 a + b - 6 + 3 a + 2 b$

$c = 6 - 3 a - 2 b$

$9 = 9 a - b - c$

Step 3.2.1.2

Simplify by adding terms.

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Step 3.2.1.2.1

Add

$7 a$ and

$3 a$ .

$7 = 10 a + b - 6 + 2 b$

$c = 6 - 3 a - 2 b$

$9 = 9 a - b - c$

Step 3.2.1.2.2

Add

$b$ and

$2 b$ .

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

$9 = 9 a - b - c$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

$9 = 9 a - b - c$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

$9 = 9 a - b - c$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

$9 = 9 a - b - c$

Step 3.3

Replace all occurrences of

$c$ in

$9 = 9 a - b - c$ with

$6 - 3 a - 2 b$ .

$9 = 9 a - b - (6 - 3 a - 2 b)$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

Step 3.4

Simplify the right side.

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Step 3.4.1

Simplify

$9 a - b - (6 - 3 a - 2 b)$ .

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Step 3.4.1.1

Simplify each term.

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Step 3.4.1.1.1

Apply the distributive property.

$9 = 9 a - b - 1 \cdot 6 - (- 3 a) - (- 2 b)$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

Step 3.4.1.1.2

Simplify.

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Step 3.4.1.1.2.1

Multiply

$- 1$ by

$6$ .

$9 = 9 a - b - 6 - (- 3 a) - (- 2 b)$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

Step 3.4.1.1.2.2

Multiply

$- 3$ by

$- 1$ .

$9 = 9 a - b - 6 + 3 a - (- 2 b)$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

Step 3.4.1.1.2.3

Multiply

$- 2$ by

$- 1$ .

$9 = 9 a - b - 6 + 3 a + 2 b$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

$9 = 9 a - b - 6 + 3 a + 2 b$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

$9 = 9 a - b - 6 + 3 a + 2 b$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

Step 3.4.1.2

Simplify by adding terms.

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Step 3.4.1.2.1

Add

$9 a$ and

$3 a$ .

$9 = 12 a - b - 6 + 2 b$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

Step 3.4.1.2.2

Add

$- b$ and

$2 b$ .

$9 = 12 a + b - 6$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

$9 = 12 a + b - 6$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

$9 = 12 a + b - 6$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

$9 = 12 a + b - 6$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

$9 = 12 a + b - 6$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

Step 4

Solve for

$b$ in

$9 = 12 a + b - 6$ .

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Step 4.1

Rewrite the equation as

$12 a + b - 6 = 9$ .

$12 a + b - 6 = 9$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

Step 4.2

Move all terms not containing

$b$ to the right side of the equation.

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Step 4.2.1

Subtract

$12 a$ from both sides of the equation.

$b - 6 = 9 - 12 a$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

Step 4.2.2

Add

$6$ to both sides of the equation.

$b = 9 - 12 a + 6$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

Step 4.2.3

Add

$9$ and

$6$ .

$b = - 12 a + 15$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

$b = - 12 a + 15$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

$b = - 12 a + 15$

$7 = 10 a + 3 b - 6$

$c = 6 - 3 a - 2 b$

Step 5

Replace all occurrences of

$b$ with

$- 12 a + 15$ in each equation.

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Step 5.1

Replace all occurrences of

$b$ in

$7 = 10 a + 3 b - 6$ with

$- 12 a + 15$ .

$7 = 10 a + 3 (- 12 a + 15) - 6$

$b = - 12 a + 15$

$c = 6 - 3 a - 2 b$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify

$10 a + 3 (- 12 a + 15) - 6$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Apply the distributive property.

$7 = 10 a + 3 (- 12 a) + 3 \cdot 15 - 6$

$b = - 12 a + 15$

$c = 6 - 3 a - 2 b$

Step 5.2.1.1.2

Multiply

$- 12$ by

$3$ .

$7 = 10 a - 36 a + 3 \cdot 15 - 6$

$b = - 12 a + 15$

$c = 6 - 3 a - 2 b$

Step 5.2.1.1.3

Multiply

$3$ by

$15$ .

$7 = 10 a - 36 a + 45 - 6$

$b = - 12 a + 15$

$c = 6 - 3 a - 2 b$

$7 = 10 a - 36 a + 45 - 6$

$b = - 12 a + 15$

$c = 6 - 3 a - 2 b$

Step 5.2.1.2

Simplify by adding terms.

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Step 5.2.1.2.1

Subtract

$36 a$ from

$10 a$ .

$7 = - 26 a + 45 - 6$

$b = - 12 a + 15$

$c = 6 - 3 a - 2 b$

Step 5.2.1.2.2

Subtract

$6$ from

$45$ .

$7 = - 26 a + 39$

$b = - 12 a + 15$

$c = 6 - 3 a - 2 b$

$7 = - 26 a + 39$

$b = - 12 a + 15$

$c = 6 - 3 a - 2 b$

$7 = - 26 a + 39$

$b = - 12 a + 15$

$c = 6 - 3 a - 2 b$

$7 = - 26 a + 39$

$b = - 12 a + 15$

$c = 6 - 3 a - 2 b$

Step 5.3

Replace all occurrences of

$b$ in

$c = 6 - 3 a - 2 b$ with

$- 12 a + 15$ .

$c = 6 - 3 a - 2 (- 12 a + 15)$

$7 = - 26 a + 39$

$b = - 12 a + 15$

Step 5.4

Simplify the right side.

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Step 5.4.1

Simplify

$6 - 3 a - 2 (- 12 a + 15)$ .

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Step 5.4.1.1

Simplify each term.

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Step 5.4.1.1.1

Apply the distributive property.

$c = 6 - 3 a - 2 (- 12 a) - 2 \cdot 15$

$7 = - 26 a + 39$

$b = - 12 a + 15$

Step 5.4.1.1.2

Multiply

$- 12$ by

$- 2$ .

$c = 6 - 3 a + 24 a - 2 \cdot 15$

$7 = - 26 a + 39$

$b = - 12 a + 15$

Step 5.4.1.1.3

Multiply

$- 2$ by

$15$ .

$c = 6 - 3 a + 24 a - 30$

$7 = - 26 a + 39$

$b = - 12 a + 15$

$c = 6 - 3 a + 24 a - 30$

$7 = - 26 a + 39$

$b = - 12 a + 15$

Step 5.4.1.2

Simplify by adding terms.

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Step 5.4.1.2.1

Subtract

$30$ from

$6$ .

$c = - 3 a + 24 a - 24$

$7 = - 26 a + 39$

$b = - 12 a + 15$

Step 5.4.1.2.2

Add

$- 3 a$ and

$24 a$ .

$c = 21 a - 24$

$7 = - 26 a + 39$

$b = - 12 a + 15$

$c = 21 a - 24$

$7 = - 26 a + 39$

$b = - 12 a + 15$

$c = 21 a - 24$

$7 = - 26 a + 39$

$b = - 12 a + 15$

$c = 21 a - 24$

$7 = - 26 a + 39$

$b = - 12 a + 15$

$c = 21 a - 24$

$7 = - 26 a + 39$

$b = - 12 a + 15$

Step 6

Solve for

$a$ in

$7 = - 26 a + 39$ .

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Step 6.1

Rewrite the equation as

$- 26 a + 39 = 7$ .

$- 26 a + 39 = 7$

$c = 21 a - 24$

$b = - 12 a + 15$

Step 6.2

Move all terms not containing

$a$ to the right side of the equation.

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Step 6.2.1

Subtract

$39$ from both sides of the equation.

$- 26 a = 7 - 39$

$c = 21 a - 24$

$b = - 12 a + 15$

Step 6.2.2

Subtract

$39$ from

$7$ .

$- 26 a = - 32$

$c = 21 a - 24$

$b = - 12 a + 15$

$- 26 a = - 32$

$c = 21 a - 24$

$b = - 12 a + 15$

Step 6.3

Divide each term in

$- 26 a = - 32$ by

$- 26$ and simplify.

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Step 6.3.1

Divide each term in

$- 26 a = - 32$ by

$- 26$ .

$\frac{- 26 a}{- 26} = \frac{- 32}{- 26}$

$c = 21 a - 24$

$b = - 12 a + 15$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of

$- 26$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{- 26 a}{- 26} = \frac{- 32}{- 26}$

$c = 21 a - 24$

$b = - 12 a + 15$

Step 6.3.2.1.2

Divide

$a$ by

$1$ .

$a = \frac{- 32}{- 26}$

$c = 21 a - 24$

$b = - 12 a + 15$

$a = \frac{- 32}{- 26}$

$c = 21 a - 24$

$b = - 12 a + 15$

$a = \frac{- 32}{- 26}$

$c = 21 a - 24$

$b = - 12 a + 15$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Cancel the common factor of

$- 32$ and

$- 26$ .

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Step 6.3.3.1.1

Factor

$- 2$ out of

$- 32$ .

$a = \frac{- 2 \cdot 16}{- 26}$

$c = 21 a - 24$

$b = - 12 a + 15$

Step 6.3.3.1.2

Cancel the common factors.

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Step 6.3.3.1.2.1

Factor

$- 2$ out of

$- 26$ .

$a = \frac{- 2 \cdot 16}{- 2 \cdot 13}$

$c = 21 a - 24$

$b = - 12 a + 15$

Step 6.3.3.1.2.2

Cancel the common factor.

$a = \frac{- 2 \cdot 16}{- 2 \cdot 13}$

$c = 21 a - 24$

$b = - 12 a + 15$

Step 6.3.3.1.2.3

Rewrite the expression.

$a = \frac{16}{13}$

$c = 21 a - 24$

$b = - 12 a + 15$

$a = \frac{16}{13}$

$c = 21 a - 24$

$b = - 12 a + 15$

$a = \frac{16}{13}$

$c = 21 a - 24$

$b = - 12 a + 15$

$a = \frac{16}{13}$

$c = 21 a - 24$

$b = - 12 a + 15$

$a = \frac{16}{13}$

$c = 21 a - 24$

$b = - 12 a + 15$

$a = \frac{16}{13}$

$c = 21 a - 24$

$b = - 12 a + 15$

Step 7

Replace all occurrences of

$a$ with

$\frac{16}{13}$ in each equation.

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Step 7.1

Replace all occurrences of

$a$ in

$c = 21 a - 24$ with

$\frac{16}{13}$ .

$c = 21 (\frac{16}{13}) - 24$

$a = \frac{16}{13}$

$b = - 12 a + 15$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify

$21 (\frac{16}{13}) - 24$ .

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Step 7.2.1.1

Multiply

$21 (\frac{16}{13})$ .

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Step 7.2.1.1.1

Combine

$21$ and

$\frac{16}{13}$ .

$c = \frac{21 \cdot 16}{13} - 24$

$a = \frac{16}{13}$

$b = - 12 a + 15$

Step 7.2.1.1.2

Multiply

$21$ by

$16$ .

$c = \frac{336}{13} - 24$

$a = \frac{16}{13}$

$b = - 12 a + 15$

$c = \frac{336}{13} - 24$

$a = \frac{16}{13}$

$b = - 12 a + 15$

Step 7.2.1.2

To write

$- 24$ as a fraction with a common denominator, multiply by

$\frac{13}{13}$ .

$c = \frac{336}{13} - 24 \cdot \frac{13}{13}$

$a = \frac{16}{13}$

$b = - 12 a + 15$

Step 7.2.1.3

Combine

$- 24$ and

$\frac{13}{13}$ .

$c = \frac{336}{13} + \frac{- 24 \cdot 13}{13}$

$a = \frac{16}{13}$

$b = - 12 a + 15$

Step 7.2.1.4

Combine the numerators over the common denominator.

$c = \frac{336 - 24 \cdot 13}{13}$

$a = \frac{16}{13}$

$b = - 12 a + 15$

Step 7.2.1.5

Simplify the numerator.

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Step 7.2.1.5.1

Multiply

$- 24$ by

$13$ .

$c = \frac{336 - 312}{13}$

$a = \frac{16}{13}$

$b = - 12 a + 15$

Step 7.2.1.5.2

Subtract

$312$ from

$336$ .

$c = \frac{24}{13}$

$a = \frac{16}{13}$

$b = - 12 a + 15$

$c = \frac{24}{13}$

$a = \frac{16}{13}$

$b = - 12 a + 15$

$c = \frac{24}{13}$

$a = \frac{16}{13}$

$b = - 12 a + 15$

$c = \frac{24}{13}$

$a = \frac{16}{13}$

$b = - 12 a + 15$

Step 7.3

Replace all occurrences of

$a$ in

$b = - 12 a + 15$ with

$\frac{16}{13}$ .

$b = - 12 (\frac{16}{13}) + 15$

$c = \frac{24}{13}$

$a = \frac{16}{13}$

Step 7.4

Simplify the right side.

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Step 7.4.1

Simplify

$- 12 (\frac{16}{13}) + 15$ .

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Step 7.4.1.1

Simplify each term.

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Step 7.4.1.1.1

Multiply

$- 12 (\frac{16}{13})$ .

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Step 7.4.1.1.1.1

Combine

$- 12$ and

$\frac{16}{13}$ .

$b = \frac{- 12 \cdot 16}{13} + 15$

$c = \frac{24}{13}$

$a = \frac{16}{13}$

Step 7.4.1.1.1.2

Multiply

$- 12$ by

$16$ .

$b = \frac{- 192}{13} + 15$

$c = \frac{24}{13}$

$a = \frac{16}{13}$

$b = \frac{- 192}{13} + 15$

$c = \frac{24}{13}$

$a = \frac{16}{13}$

Step 7.4.1.1.2

Move the negative in front of the fraction.

$b = - \frac{192}{13} + 15$

$c = \frac{24}{13}$

$a = \frac{16}{13}$

$b = - \frac{192}{13} + 15$

$c = \frac{24}{13}$

$a = \frac{16}{13}$

Step 7.4.1.2

To write

$15$ as a fraction with a common denominator, multiply by

$\frac{13}{13}$ .

$b = - \frac{192}{13} + 15 \cdot \frac{13}{13}$

$c = \frac{24}{13}$

$a = \frac{16}{13}$

Step 7.4.1.3

Combine

$15$ and

$\frac{13}{13}$ .

$b = - \frac{192}{13} + \frac{15 \cdot 13}{13}$

$c = \frac{24}{13}$

$a = \frac{16}{13}$

Step 7.4.1.4

Combine the numerators over the common denominator.

$b = \frac{- 192 + 15 \cdot 13}{13}$

$c = \frac{24}{13}$

$a = \frac{16}{13}$

Step 7.4.1.5

Simplify the numerator.

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Step 7.4.1.5.1

Multiply

$15$ by

$13$ .

$b = \frac{- 192 + 195}{13}$

$c = \frac{24}{13}$

$a = \frac{16}{13}$

Step 7.4.1.5.2

Add

$- 192$ and

$195$ .

$b = \frac{3}{13}$

$c = \frac{24}{13}$

$a = \frac{16}{13}$

$b = \frac{3}{13}$

$c = \frac{24}{13}$

$a = \frac{16}{13}$

$b = \frac{3}{13}$

$c = \frac{24}{13}$

$a = \frac{16}{13}$

$b = \frac{3}{13}$

$c = \frac{24}{13}$

$a = \frac{16}{13}$

$b = \frac{3}{13}$

$c = \frac{24}{13}$

$a = \frac{16}{13}$

Step 8

List all of the solutions.

$b = \frac{3}{13}, c = \frac{24}{13}, a = \frac{16}{13}$

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