Finite Math Examples

$[\begin{array}{c} 1 & x \\ 2 y & - 4 \end{array}] - 6 [\begin{array}{c} 2 & - 2 \\ 0 & 3 \end{array}] = [\begin{array}{c} 3 z & 16 \\ 6 & - u \end{array}]$

Step 1

Simplify each term.

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Step 1.1

Multiply $- 6$ by each element of the matrix.

$[\begin{array}{c} 1 & x \\ 2 y & - 4 \end{array}] + [\begin{array}{c} - 6 \cdot 2 & - 6 \cdot - 2 \\ - 6 \cdot 0 & - 6 \cdot 3 \end{array}] = [\begin{array}{c} 3 z & 16 \\ 6 & - u \end{array}]$

Step 1.2

Simplify each element in the matrix.

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Step 1.2.1

Multiply $- 6$ by $2$ .

$[\begin{array}{c} 1 & x \\ 2 y & - 4 \end{array}] + [\begin{array}{c} - 12 & - 6 \cdot - 2 \\ - 6 \cdot 0 & - 6 \cdot 3 \end{array}] = [\begin{array}{c} 3 z & 16 \\ 6 & - u \end{array}]$

Step 1.2.2

Multiply $- 6$ by $- 2$ .

$[\begin{array}{c} 1 & x \\ 2 y & - 4 \end{array}] + [\begin{array}{c} - 12 & 12 \\ - 6 \cdot 0 & - 6 \cdot 3 \end{array}] = [\begin{array}{c} 3 z & 16 \\ 6 & - u \end{array}]$

Step 1.2.3

Multiply $- 6$ by $0$ .

$[\begin{array}{c} 1 & x \\ 2 y & - 4 \end{array}] + [\begin{array}{c} - 12 & 12 \\ 0 & - 6 \cdot 3 \end{array}] = [\begin{array}{c} 3 z & 16 \\ 6 & - u \end{array}]$

Step 1.2.4

Multiply $- 6$ by $3$ .

$[\begin{array}{c} 1 & x \\ 2 y & - 4 \end{array}] + [\begin{array}{c} - 12 & 12 \\ 0 & - 18 \end{array}] = [\begin{array}{c} 3 z & 16 \\ 6 & - u \end{array}]$

Step 2

Add the corresponding elements.

$[\begin{array}{c} 1 - 12 & x + 12 \\ 2 y + 0 & - 4 - 18 \end{array}] = [\begin{array}{c} 3 z & 16 \\ 6 & - u \end{array}]$

Step 3

Simplify each element.

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Step 3.1

Subtract $12$ from $1$ .

$[\begin{array}{c} - 11 & x + 12 \\ 2 y + 0 & - 4 - 18 \end{array}] = [\begin{array}{c} 3 z & 16 \\ 6 & - u \end{array}]$

Step 3.2

Add $2 y$ and $0$ .

$[\begin{array}{c} - 11 & x + 12 \\ 2 y & - 4 - 18 \end{array}] = [\begin{array}{c} 3 z & 16 \\ 6 & - u \end{array}]$

Step 3.3

Subtract $18$ from $- 4$ .

$[\begin{array}{c} - 11 & x + 12 \\ 2 y & - 22 \end{array}] = [\begin{array}{c} 3 z & 16 \\ 6 & - u \end{array}]$

Step 4

Write as a linear system of equations.

$- 11 = 3 z$

$x + 12 = 16$

$2 y = 6$

$- 22 = - u$

Step 5

Solve the system of equations.

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Step 5.1

Solve for $z$ in $- 11 = 3 z$ .

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Step 5.1.1

Rewrite the equation as $3 z = - 11$ .

$3 z = - 11$

$x + 12 = 16$

$2 y = 6$

$- 22 = - u$

Step 5.1.2

Divide each term in $3 z = - 11$ by $3$ and simplify.

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Step 5.1.2.1

Divide each term in $3 z = - 11$ by $3$ .

$\frac{3 z}{3} = \frac{- 11}{3}$

$x + 12 = 16$

$2 y = 6$

$- 22 = - u$

Step 5.1.2.2

Simplify the left side.

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Step 5.1.2.2.1

Cancel the common factor of $3$ .

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Step 5.1.2.2.1.1

Cancel the common factor.

$\frac{3 z}{3} = \frac{- 11}{3}$

$x + 12 = 16$

$2 y = 6$

$- 22 = - u$

Step 5.1.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 11}{3}$

$x + 12 = 16$

$2 y = 6$

$- 22 = - u$

$z = \frac{- 11}{3}$

$x + 12 = 16$

$2 y = 6$

$- 22 = - u$

$z = \frac{- 11}{3}$

$x + 12 = 16$

$2 y = 6$

$- 22 = - u$

Step 5.1.2.3

Simplify the right side.

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Step 5.1.2.3.1

Move the negative in front of the fraction.

$z = - \frac{11}{3}$

$x + 12 = 16$

$2 y = 6$

$- 22 = - u$

$z = - \frac{11}{3}$

$x + 12 = 16$

$2 y = 6$

$- 22 = - u$

$z = - \frac{11}{3}$

$x + 12 = 16$

$2 y = 6$

$- 22 = - u$

$z = - \frac{11}{3}$

$x + 12 = 16$

$2 y = 6$

$- 22 = - u$

Step 5.2

Replace all occurrences of $z$ with $- \frac{11}{3}$ in each equation.

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Step 5.2.1

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

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Step 5.2.1.1

Subtract $12$ from both sides of the equation.

$x = 16 - 12$

$z = - \frac{11}{3}$

$2 y = 6$

$- 22 = - u$

Step 5.2.1.2

Subtract $12$ from $16$ .

$x = 4$

$z = - \frac{11}{3}$

$2 y = 6$

$- 22 = - u$

$x = 4$

$z = - \frac{11}{3}$

$2 y = 6$

$- 22 = - u$

Step 5.2.2

Divide each term in $2 y = 6$ by $2$ and simplify.

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Step 5.2.2.1

Divide each term in $2 y = 6$ by $2$ .

$\frac{2 y}{2} = \frac{6}{2}$

$x = 4$

$z = - \frac{11}{3}$

$- 22 = - u$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{6}{2}$

$x = 4$

$z = - \frac{11}{3}$

$- 22 = - u$

Step 5.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{6}{2}$

$x = 4$

$z = - \frac{11}{3}$

$- 22 = - u$

$y = \frac{6}{2}$

$x = 4$

$z = - \frac{11}{3}$

$- 22 = - u$

$y = \frac{6}{2}$

$x = 4$

$z = - \frac{11}{3}$

$- 22 = - u$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Divide $6$ by $2$ .

$y = 3$

$x = 4$

$z = - \frac{11}{3}$

$- 22 = - u$

$y = 3$

$x = 4$

$z = - \frac{11}{3}$

$- 22 = - u$

$y = 3$

$x = 4$

$z = - \frac{11}{3}$

$- 22 = - u$

Step 5.2.3

Rewrite the equation as $- u = - 22$ .

$- u = - 22$

$y = 3$

$x = 4$

$z = - \frac{11}{3}$

Step 5.2.4

Divide each term in $- u = - 22$ by $- 1$ and simplify.

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Step 5.2.4.1

Divide each term in $- u = - 22$ by $- 1$ .

$\frac{- u}{- 1} = \frac{- 22}{- 1}$

$y = 3$

$x = 4$

$z = - \frac{11}{3}$

Step 5.2.4.2

Simplify the left side.

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Step 5.2.4.2.1

Dividing two negative values results in a positive value.

$\frac{u}{1} = \frac{- 22}{- 1}$

$y = 3$

$x = 4$

$z = - \frac{11}{3}$

Step 5.2.4.2.2

Divide $u$ by $1$ .

$u = \frac{- 22}{- 1}$

$y = 3$

$x = 4$

$z = - \frac{11}{3}$

$u = \frac{- 22}{- 1}$

$y = 3$

$x = 4$

$z = - \frac{11}{3}$

Step 5.2.4.3

Simplify the right side.

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Step 5.2.4.3.1

Divide $- 22$ by $- 1$ .

$u = 22$

$y = 3$

$x = 4$

$z = - \frac{11}{3}$

$u = 22$

$y = 3$

$x = 4$

$z = - \frac{11}{3}$

$u = 22$

$y = 3$

$x = 4$

$z = - \frac{11}{3}$

$u = 22$

$y = 3$

$x = 4$

$z = - \frac{11}{3}$

Step 5.3

List all of the solutions.

$u = 22, y = 3, x = 4, z = - \frac{11}{3}$