Finite Math Examples

2 [\begin{matrix} a & 3 b c & 4 d \end{matrix}] + 3 [\begin{matrix} 2 a & b c & 3 d \end{matrix}] = [\begin{matrix} a & - 54 - 50 & 51 \end{matrix}]

$2 [\begin{array}{c} a & 3 b \\ c & 4 d \end{array}] + 3 [\begin{array}{c} 2 a & b \\ c & 3 d \end{array}] = [\begin{array}{c} a & - 54 \\ - 50 & 51 \end{array}]$

Step 1

Simplify each term.

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Step 1.1

Multiply

2

$2$ by each element of the matrix.

[\begin{matrix} 2 a & 2 (3 b) 2 c & 2 (4 d) \end{matrix}] + 3 [\begin{matrix} 2 a & b c & 3 d \end{matrix}] = [\begin{matrix} a & - 54 - 50 & 51 \end{matrix}]

$[\begin{array}{c} 2 a & 2 (3 b) \\ 2 c & 2 (4 d) \end{array}] + 3 [\begin{array}{c} 2 a & b \\ c & 3 d \end{array}] = [\begin{array}{c} a & - 54 \\ - 50 & 51 \end{array}]$

Step 1.2

Simplify each element in the matrix.

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Step 1.2.1

Multiply

3

$3$ by

2

$2$ .

[\begin{matrix} 2 a & 6 b 2 c & 2 (4 d) \end{matrix}] + 3 [\begin{matrix} 2 a & b c & 3 d \end{matrix}] = [\begin{matrix} a & - 54 - 50 & 51 \end{matrix}]

$[\begin{array}{c} 2 a & 6 b \\ 2 c & 2 (4 d) \end{array}] + 3 [\begin{array}{c} 2 a & b \\ c & 3 d \end{array}] = [\begin{array}{c} a & - 54 \\ - 50 & 51 \end{array}]$

Step 1.2.2

Multiply

4

$4$ by

2

$2$ .

[\begin{matrix} 2 a & 6 b 2 c & 8 d \end{matrix}] + 3 [\begin{matrix} 2 a & b c & 3 d \end{matrix}] = [\begin{matrix} a & - 54 - 50 & 51 \end{matrix}]

$[\begin{array}{c} 2 a & 6 b \\ 2 c & 8 d \end{array}] + 3 [\begin{array}{c} 2 a & b \\ c & 3 d \end{array}] = [\begin{array}{c} a & - 54 \\ - 50 & 51 \end{array}]$

[\begin{matrix} 2 a & 6 b 2 c & 8 d \end{matrix}] + 3 [\begin{matrix} 2 a & b c & 3 d \end{matrix}] = [\begin{matrix} a & - 54 - 50 & 51 \end{matrix}]

$[\begin{array}{c} 2 a & 6 b \\ 2 c & 8 d \end{array}] + 3 [\begin{array}{c} 2 a & b \\ c & 3 d \end{array}] = [\begin{array}{c} a & - 54 \\ - 50 & 51 \end{array}]$

Step 1.3

Multiply

3

$3$ by each element of the matrix.

[\begin{matrix} 2 a & 6 b 2 c & 8 d \end{matrix}] + [\begin{matrix} 3 (2 a) & 3 b 3 c & 3 (3 d) \end{matrix}] = [\begin{matrix} a & - 54 - 50 & 51 \end{matrix}]

$[\begin{array}{c} 2 a & 6 b \\ 2 c & 8 d \end{array}] + [\begin{array}{c} 3 (2 a) & 3 b \\ 3 c & 3 (3 d) \end{array}] = [\begin{array}{c} a & - 54 \\ - 50 & 51 \end{array}]$

Step 1.4

Simplify each element in the matrix.

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Step 1.4.1

Multiply

2

$2$ by

3

$3$ .

[\begin{matrix} 2 a & 6 b 2 c & 8 d \end{matrix}] + [\begin{matrix} 6 a & 3 b 3 c & 3 (3 d) \end{matrix}] = [\begin{matrix} a & - 54 - 50 & 51 \end{matrix}]

$[\begin{array}{c} 2 a & 6 b \\ 2 c & 8 d \end{array}] + [\begin{array}{c} 6 a & 3 b \\ 3 c & 3 (3 d) \end{array}] = [\begin{array}{c} a & - 54 \\ - 50 & 51 \end{array}]$

Step 1.4.2

Multiply

3

$3$ by

3

$3$ .

[\begin{matrix} 2 a & 6 b 2 c & 8 d \end{matrix}] + [\begin{matrix} 6 a & 3 b 3 c & 9 d \end{matrix}] = [\begin{matrix} a & - 54 - 50 & 51 \end{matrix}]

$[\begin{array}{c} 2 a & 6 b \\ 2 c & 8 d \end{array}] + [\begin{array}{c} 6 a & 3 b \\ 3 c & 9 d \end{array}] = [\begin{array}{c} a & - 54 \\ - 50 & 51 \end{array}]$

[\begin{matrix} 2 a & 6 b 2 c & 8 d \end{matrix}] + [\begin{matrix} 6 a & 3 b 3 c & 9 d \end{matrix}] = [\begin{matrix} a & - 54 - 50 & 51 \end{matrix}]

$[\begin{array}{c} 2 a & 6 b \\ 2 c & 8 d \end{array}] + [\begin{array}{c} 6 a & 3 b \\ 3 c & 9 d \end{array}] = [\begin{array}{c} a & - 54 \\ - 50 & 51 \end{array}]$

[\begin{matrix} 2 a & 6 b 2 c & 8 d \end{matrix}] + [\begin{matrix} 6 a & 3 b 3 c & 9 d \end{matrix}] = [\begin{matrix} a & - 54 - 50 & 51 \end{matrix}]

$[\begin{array}{c} 2 a & 6 b \\ 2 c & 8 d \end{array}] + [\begin{array}{c} 6 a & 3 b \\ 3 c & 9 d \end{array}] = [\begin{array}{c} a & - 54 \\ - 50 & 51 \end{array}]$

Step 2

Add the corresponding elements.

[\begin{matrix} 2 a + 6 a & 6 b + 3 b 2 c + 3 c & 8 d + 9 d \end{matrix}] = [\begin{matrix} a & - 54 - 50 & 51 \end{matrix}]

$[\begin{array}{c} 2 a + 6 a & 6 b + 3 b \\ 2 c + 3 c & 8 d + 9 d \end{array}] = [\begin{array}{c} a & - 54 \\ - 50 & 51 \end{array}]$

Step 3

Simplify each element.

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Step 3.1

Add

2 a

$2 a$ and

6 a

$6 a$ .

[\begin{matrix} 8 a & 6 b + 3 b 2 c + 3 c & 8 d + 9 d \end{matrix}] = [\begin{matrix} a & - 54 - 50 & 51 \end{matrix}]

$[\begin{array}{c} 8 a & 6 b + 3 b \\ 2 c + 3 c & 8 d + 9 d \end{array}] = [\begin{array}{c} a & - 54 \\ - 50 & 51 \end{array}]$

Step 3.2

Add

6 b

$6 b$ and

3 b

$3 b$ .

[\begin{matrix} 8 a & 9 b 2 c + 3 c & 8 d + 9 d \end{matrix}] = [\begin{matrix} a & - 54 - 50 & 51 \end{matrix}]

$[\begin{array}{c} 8 a & 9 b \\ 2 c + 3 c & 8 d + 9 d \end{array}] = [\begin{array}{c} a & - 54 \\ - 50 & 51 \end{array}]$

Step 3.3

Add

2 c

$2 c$ and

3 c

$3 c$ .

[\begin{matrix} 8 a & 9 b 5 c & 8 d + 9 d \end{matrix}] = [\begin{matrix} a & - 54 - 50 & 51 \end{matrix}]

$[\begin{array}{c} 8 a & 9 b \\ 5 c & 8 d + 9 d \end{array}] = [\begin{array}{c} a & - 54 \\ - 50 & 51 \end{array}]$

Step 3.4

Add

8 d

$8 d$ and

9 d

$9 d$ .

[\begin{matrix} 8 a & 9 b 5 c & 17 d \end{matrix}] = [\begin{matrix} a & - 54 - 50 & 51 \end{matrix}]

$[\begin{array}{c} 8 a & 9 b \\ 5 c & 17 d \end{array}] = [\begin{array}{c} a & - 54 \\ - 50 & 51 \end{array}]$

[\begin{matrix} 8 a & 9 b 5 c & 17 d \end{matrix}] = [\begin{matrix} a & - 54 - 50 & 51 \end{matrix}]

$[\begin{array}{c} 8 a & 9 b \\ 5 c & 17 d \end{array}] = [\begin{array}{c} a & - 54 \\ - 50 & 51 \end{array}]$

Step 4

Write as a linear system of equations.

8 a = a

$8 a = a$

9 b = - 54

$9 b = - 54$

5 c = - 50

$5 c = - 50$

17 d = 51

$17 d = 51$

Step 5

Solve the system of equations.

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Step 5.1

Solve for

a

$a$ in

8 a = a

$8 a = a$ .

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Step 5.1.1

Move all terms containing

a

$a$ to the left side of the equation.

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Step 5.1.1.1

Subtract

a

$a$ from both sides of the equation.

8 a - a = 0

$8 a - a = 0$

9 b = - 54

$9 b = - 54$

5 c = - 50

$5 c = - 50$

17 d = 51

$17 d = 51$

Step 5.1.1.2

Subtract

a

$a$ from

8 a

$8 a$ .

7 a = 0

$7 a = 0$

9 b = - 54

$9 b = - 54$

5 c = - 50

$5 c = - 50$

17 d = 51

$17 d = 51$

7 a = 0

$7 a = 0$

9 b = - 54

$9 b = - 54$

5 c = - 50

$5 c = - 50$

17 d = 51

$17 d = 51$

Step 5.1.2

Divide each term in

7 a = 0

$7 a = 0$ by

7

$7$ and simplify.

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Step 5.1.2.1

Divide each term in

7 a = 0

$7 a = 0$ by

7

$7$ .

\frac{7 a}{7} = \frac{0}{7}

$\frac{7 a}{7} = \frac{0}{7}$

9 b = - 54

$9 b = - 54$

5 c = - 50

$5 c = - 50$

17 d = 51

$17 d = 51$

Step 5.1.2.2

Simplify the left side.

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Step 5.1.2.2.1

Cancel the common factor of

7

$7$ .

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Step 5.1.2.2.1.1

Cancel the common factor.

$\frac{7 a}{7} = \frac{0}{7}$

$9 b = - 54$

$5 c = - 50$

$17 d = 51$

Step 5.1.2.2.1.2

Divide

$a$ by

$1$ .

$a = \frac{0}{7}$

$9 b = - 54$

$5 c = - 50$

$17 d = 51$

$a = \frac{0}{7}$

$9 b = - 54$

$5 c = - 50$

$17 d = 51$

$a = \frac{0}{7}$

$9 b = - 54$

$5 c = - 50$

$17 d = 51$

Step 5.1.2.3

Simplify the right side.

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Step 5.1.2.3.1

Divide

$0$ by

$7$ .

$a = 0$

$9 b = - 54$

$5 c = - 50$

$17 d = 51$

$a = 0$

$9 b = - 54$

$5 c = - 50$

$17 d = 51$

$a = 0$

$9 b = - 54$

$5 c = - 50$

$17 d = 51$

$a = 0$

$9 b = - 54$

$5 c = - 50$

$17 d = 51$

Step 5.2

Replace all occurrences of

$a$ with

$0$ in each equation.

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Step 5.2.1

Divide each term in

$9 b = - 54$ by

$9$ and simplify.

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Step 5.2.1.1

Divide each term in

$9 b = - 54$ by

$9$ .

$\frac{9 b}{9} = \frac{- 54}{9}$

$a = 0$

$5 c = - 50$

$17 d = 51$

Step 5.2.1.2

Simplify the left side.

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Step 5.2.1.2.1

Cancel the common factor of

$9$ .

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Step 5.2.1.2.1.1

Cancel the common factor.

$\frac{9 b}{9} = \frac{- 54}{9}$

$a = 0$

$5 c = - 50$

$17 d = 51$

Step 5.2.1.2.1.2

Divide

$b$ by

$1$ .

$b = \frac{- 54}{9}$

$a = 0$

$5 c = - 50$

$17 d = 51$

$b = \frac{- 54}{9}$

$a = 0$

$5 c = - 50$

$17 d = 51$

$b = \frac{- 54}{9}$

$a = 0$

$5 c = - 50$

$17 d = 51$

Step 5.2.1.3

Simplify the right side.

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Step 5.2.1.3.1

Divide

$- 54$ by

$9$ .

$b = - 6$

$a = 0$

$5 c = - 50$

$17 d = 51$

$b = - 6$

$a = 0$

$5 c = - 50$

$17 d = 51$

$b = - 6$

$a = 0$

$5 c = - 50$

$17 d = 51$

Step 5.2.2

Divide each term in

$5 c = - 50$ by

$5$ and simplify.

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Step 5.2.2.1

Divide each term in

$5 c = - 50$ by

$5$ .

$\frac{5 c}{5} = \frac{- 50}{5}$

$b = - 6$

$a = 0$

$17 d = 51$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of

$5$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{5 c}{5} = \frac{- 50}{5}$

$b = - 6$

$a = 0$

$17 d = 51$

Step 5.2.2.2.1.2

Divide

$c$ by

$1$ .

$c = \frac{- 50}{5}$

$b = - 6$

$a = 0$

$17 d = 51$

$c = \frac{- 50}{5}$

$b = - 6$

$a = 0$

$17 d = 51$

$c = \frac{- 50}{5}$

$b = - 6$

$a = 0$

$17 d = 51$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Divide

$- 50$ by

$5$ .

$c = - 10$

$b = - 6$

$a = 0$

$17 d = 51$

$c = - 10$

$b = - 6$

$a = 0$

$17 d = 51$

$c = - 10$

$b = - 6$

$a = 0$

$17 d = 51$

Step 5.2.3

Divide each term in

$17 d = 51$ by

$17$ and simplify.

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Step 5.2.3.1

Divide each term in

$17 d = 51$ by

$17$ .

$\frac{17 d}{17} = \frac{51}{17}$

$c = - 10$

$b = - 6$

$a = 0$

Step 5.2.3.2

Simplify the left side.

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Step 5.2.3.2.1

Cancel the common factor of

$17$ .

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Step 5.2.3.2.1.1

Cancel the common factor.

$\frac{17 d}{17} = \frac{51}{17}$

$c = - 10$

$b = - 6$

$a = 0$

Step 5.2.3.2.1.2

Divide

$d$ by

$1$ .

$d = \frac{51}{17}$

$c = - 10$

$b = - 6$

$a = 0$

$d = \frac{51}{17}$

$c = - 10$

$b = - 6$

$a = 0$

$d = \frac{51}{17}$

$c = - 10$

$b = - 6$

$a = 0$

Step 5.2.3.3

Simplify the right side.

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Step 5.2.3.3.1

Divide

$51$ by

$17$ .

$d = 3$

$c = - 10$

$b = - 6$

$a = 0$

$d = 3$

$c = - 10$

$b = - 6$

$a = 0$

$d = 3$

$c = - 10$

$b = - 6$

$a = 0$

$d = 3$

$c = - 10$

$b = - 6$

$a = 0$

Step 5.3

List all of the solutions.

$d = 3, c = - 10, b = - 6, a = 0$