Finite Math Examples

$[\begin{array}{c} x & - 4 \\ 4 & y \end{array}] + [\begin{array}{c} - 8 & z \\ - 2 & 8 \end{array}] = [\begin{array}{c} - 6 & - 4 \\ 2 u & 14 \end{array}]$

Step 1

Add the corresponding elements.

$[\begin{array}{c} x - 8 & - 4 + z \\ 4 - 2 & y + 8 \end{array}] = [\begin{array}{c} - 6 & - 4 \\ 2 u & 14 \end{array}]$

Step 2

Subtract $2$ from $4$ .

$[\begin{array}{c} x - 8 & - 4 + z \\ 2 & y + 8 \end{array}] = [\begin{array}{c} - 6 & - 4 \\ 2 u & 14 \end{array}]$

Step 3

Write as a linear system of equations.

$x - 8 = - 6$

$- 4 + z = - 4$

$2 = 2 u$

$y + 8 = 14$

Step 4

Solve the system of equations.

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Step 4.1

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

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Step 4.1.1

Add $8$ to both sides of the equation.

$x = - 6 + 8$

$- 4 + z = - 4$

$2 = 2 u$

$y + 8 = 14$

Step 4.1.2

Add $- 6$ and $8$ .

$x = 2$

$- 4 + z = - 4$

$2 = 2 u$

$y + 8 = 14$

$x = 2$

$- 4 + z = - 4$

$2 = 2 u$

$y + 8 = 14$

Step 4.2

Replace all occurrences of $x$ with $2$ in each equation.

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Step 4.2.1

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

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Step 4.2.1.1

Add $4$ to both sides of the equation.

$z = - 4 + 4$

$x = 2$

$2 = 2 u$

$y + 8 = 14$

Step 4.2.1.2

Add $- 4$ and $4$ .

$z = 0$

$x = 2$

$2 = 2 u$

$y + 8 = 14$

$z = 0$

$x = 2$

$2 = 2 u$

$y + 8 = 14$

Step 4.2.2

Rewrite the equation as $2 u = 2$ .

$2 u = 2$

$z = 0$

$x = 2$

$y + 8 = 14$

Step 4.2.3

Divide each term in $2 u = 2$ by $2$ and simplify.

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Step 4.2.3.1

Divide each term in $2 u = 2$ by $2$ .

$\frac{2 u}{2} = \frac{2}{2}$

$z = 0$

$x = 2$

$y + 8 = 14$

Step 4.2.3.2

Simplify the left side.

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Step 4.2.3.2.1

Cancel the common factor of $2$ .

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Step 4.2.3.2.1.1

Cancel the common factor.

$\frac{2 u}{2} = \frac{2}{2}$

$z = 0$

$x = 2$

$y + 8 = 14$

Step 4.2.3.2.1.2

Divide $u$ by $1$ .

$u = \frac{2}{2}$

$z = 0$

$x = 2$

$y + 8 = 14$

$u = \frac{2}{2}$

$z = 0$

$x = 2$

$y + 8 = 14$

$u = \frac{2}{2}$

$z = 0$

$x = 2$

$y + 8 = 14$

Step 4.2.3.3

Simplify the right side.

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Step 4.2.3.3.1

Divide $2$ by $2$ .

$u = 1$

$z = 0$

$x = 2$

$y + 8 = 14$

$u = 1$

$z = 0$

$x = 2$

$y + 8 = 14$

$u = 1$

$z = 0$

$x = 2$

$y + 8 = 14$

Step 4.2.4

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

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Step 4.2.4.1

Subtract $8$ from both sides of the equation.

$y = 14 - 8$

$u = 1$

$z = 0$

$x = 2$

Step 4.2.4.2

Subtract $8$ from $14$ .

$y = 6$

$u = 1$

$z = 0$

$x = 2$

$y = 6$

$u = 1$

$z = 0$

$x = 2$

$y = 6$

$u = 1$

$z = 0$

$x = 2$

Step 4.3

List all of the solutions.

$y = 6, u = 1, z = 0, x = 2$