Finite Math Examples

$4 [\begin{array}{c} x - 2 & 4 \\ - 2 & z \end{array}] + 3 [\begin{array}{c} 6 & - 2 \\ - w & 3 \end{array}] = [\begin{array}{c} 26 & y \\ 7 & 13 \end{array}]$

Step 1

Simplify each term.

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Step 1.1

Multiply $4$ by each element of the matrix.

$[\begin{array}{c} 4 (x - 2) & 4 \cdot 4 \\ 4 \cdot - 2 & 4 z \end{array}] + 3 [\begin{array}{c} 6 & - 2 \\ - w & 3 \end{array}] = [\begin{array}{c} 26 & y \\ 7 & 13 \end{array}]$

Step 1.2

Simplify each element in the matrix.

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Step 1.2.1

Apply the distributive property.

$[\begin{array}{c} 4 x + 4 \cdot - 2 & 4 \cdot 4 \\ 4 \cdot - 2 & 4 z \end{array}] + 3 [\begin{array}{c} 6 & - 2 \\ - w & 3 \end{array}] = [\begin{array}{c} 26 & y \\ 7 & 13 \end{array}]$

Step 1.2.2

Multiply $4$ by $- 2$ .

$[\begin{array}{c} 4 x - 8 & 4 \cdot 4 \\ 4 \cdot - 2 & 4 z \end{array}] + 3 [\begin{array}{c} 6 & - 2 \\ - w & 3 \end{array}] = [\begin{array}{c} 26 & y \\ 7 & 13 \end{array}]$

Step 1.2.3

Multiply $4$ by $4$ .

$[\begin{array}{c} 4 x - 8 & 16 \\ 4 \cdot - 2 & 4 z \end{array}] + 3 [\begin{array}{c} 6 & - 2 \\ - w & 3 \end{array}] = [\begin{array}{c} 26 & y \\ 7 & 13 \end{array}]$

Step 1.2.4

Multiply $4$ by $- 2$ .

$[\begin{array}{c} 4 x - 8 & 16 \\ - 8 & 4 z \end{array}] + 3 [\begin{array}{c} 6 & - 2 \\ - w & 3 \end{array}] = [\begin{array}{c} 26 & y \\ 7 & 13 \end{array}]$

Step 1.3

Multiply $3$ by each element of the matrix.

$[\begin{array}{c} 4 x - 8 & 16 \\ - 8 & 4 z \end{array}] + [\begin{array}{c} 3 \cdot 6 & 3 \cdot - 2 \\ 3 (- w) & 3 \cdot 3 \end{array}] = [\begin{array}{c} 26 & y \\ 7 & 13 \end{array}]$

Step 1.4

Simplify each element in the matrix.

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Step 1.4.1

Multiply $3$ by $6$ .

$[\begin{array}{c} 4 x - 8 & 16 \\ - 8 & 4 z \end{array}] + [\begin{array}{c} 18 & 3 \cdot - 2 \\ 3 (- w) & 3 \cdot 3 \end{array}] = [\begin{array}{c} 26 & y \\ 7 & 13 \end{array}]$

Step 1.4.2

Multiply $3$ by $- 2$ .

$[\begin{array}{c} 4 x - 8 & 16 \\ - 8 & 4 z \end{array}] + [\begin{array}{c} 18 & - 6 \\ 3 (- w) & 3 \cdot 3 \end{array}] = [\begin{array}{c} 26 & y \\ 7 & 13 \end{array}]$

Step 1.4.3

Multiply $- 1$ by $3$ .

$[\begin{array}{c} 4 x - 8 & 16 \\ - 8 & 4 z \end{array}] + [\begin{array}{c} 18 & - 6 \\ - 3 w & 3 \cdot 3 \end{array}] = [\begin{array}{c} 26 & y \\ 7 & 13 \end{array}]$

Step 1.4.4

Multiply $3$ by $3$ .

$[\begin{array}{c} 4 x - 8 & 16 \\ - 8 & 4 z \end{array}] + [\begin{array}{c} 18 & - 6 \\ - 3 w & 9 \end{array}] = [\begin{array}{c} 26 & y \\ 7 & 13 \end{array}]$

Step 2

Add the corresponding elements.

$[\begin{array}{c} 4 x - 8 + 18 & 16 - 6 \\ - 8 - 3 w & 4 z + 9 \end{array}] = [\begin{array}{c} 26 & y \\ 7 & 13 \end{array}]$

Step 3

Simplify each element.

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Step 3.1

Add $- 8$ and $18$ .

$[\begin{array}{c} 4 x + 10 & 16 - 6 \\ - 8 - 3 w & 4 z + 9 \end{array}] = [\begin{array}{c} 26 & y \\ 7 & 13 \end{array}]$

Step 3.2

Subtract $6$ from $16$ .

$[\begin{array}{c} 4 x + 10 & 10 \\ - 8 - 3 w & 4 z + 9 \end{array}] = [\begin{array}{c} 26 & y \\ 7 & 13 \end{array}]$

Step 4

Write as a linear system of equations.

$4 x + 10 = 26$

$10 = y$

$- 8 - 3 w = 7$

$4 z + 9 = 13$

Step 5

Solve the system of equations.

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Step 5.1

Solve for $x$ in $4 x + 10 = 26$ .

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Step 5.1.1

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

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Step 5.1.1.1

Subtract $10$ from both sides of the equation.

$4 x = 26 - 10$

$10 = y$

$- 8 - 3 w = 7$

$4 z + 9 = 13$

Step 5.1.1.2

Subtract $10$ from $26$ .

$4 x = 16$

$10 = y$

$- 8 - 3 w = 7$

$4 z + 9 = 13$

$4 x = 16$

$10 = y$

$- 8 - 3 w = 7$

$4 z + 9 = 13$

Step 5.1.2

Divide each term in $4 x = 16$ by $4$ and simplify.

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Step 5.1.2.1

Divide each term in $4 x = 16$ by $4$ .

$\frac{4 x}{4} = \frac{16}{4}$

$10 = y$

$- 8 - 3 w = 7$

$4 z + 9 = 13$

Step 5.1.2.2

Simplify the left side.

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Step 5.1.2.2.1

Cancel the common factor of $4$ .

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Step 5.1.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{16}{4}$

$10 = y$

$- 8 - 3 w = 7$

$4 z + 9 = 13$

Step 5.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{16}{4}$

$10 = y$

$- 8 - 3 w = 7$

$4 z + 9 = 13$

$x = \frac{16}{4}$

$10 = y$

$- 8 - 3 w = 7$

$4 z + 9 = 13$

$x = \frac{16}{4}$

$10 = y$

$- 8 - 3 w = 7$

$4 z + 9 = 13$

Step 5.1.2.3

Simplify the right side.

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Step 5.1.2.3.1

Divide $16$ by $4$ .

$x = 4$

$10 = y$

$- 8 - 3 w = 7$

$4 z + 9 = 13$

$x = 4$

$10 = y$

$- 8 - 3 w = 7$

$4 z + 9 = 13$

$x = 4$

$10 = y$

$- 8 - 3 w = 7$

$4 z + 9 = 13$

$x = 4$

$10 = y$

$- 8 - 3 w = 7$

$4 z + 9 = 13$

Step 5.2

Replace all occurrences of $x$ with $4$ in each equation.

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Step 5.2.1

Rewrite the equation as $y = 10$ .

$y = 10$

$x = 4$

$- 8 - 3 w = 7$

$4 z + 9 = 13$

Step 5.2.2

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

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Step 5.2.2.1

Add $8$ to both sides of the equation.

$- 3 w = 7 + 8$

$y = 10$

$x = 4$

$4 z + 9 = 13$

Step 5.2.2.2

Add $7$ and $8$ .

$- 3 w = 15$

$y = 10$

$x = 4$

$4 z + 9 = 13$

$- 3 w = 15$

$y = 10$

$x = 4$

$4 z + 9 = 13$

Step 5.2.3

Divide each term in $- 3 w = 15$ by $- 3$ and simplify.

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Step 5.2.3.1

Divide each term in $- 3 w = 15$ by $- 3$ .

$\frac{- 3 w}{- 3} = \frac{15}{- 3}$

$y = 10$

$x = 4$

$4 z + 9 = 13$

Step 5.2.3.2

Simplify the left side.

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Step 5.2.3.2.1

Cancel the common factor of $- 3$ .

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Step 5.2.3.2.1.1

Cancel the common factor.

$\frac{- 3 w}{- 3} = \frac{15}{- 3}$

$y = 10$

$x = 4$

$4 z + 9 = 13$

Step 5.2.3.2.1.2

Divide $w$ by $1$ .

$w = \frac{15}{- 3}$

$y = 10$

$x = 4$

$4 z + 9 = 13$

$w = \frac{15}{- 3}$

$y = 10$

$x = 4$

$4 z + 9 = 13$

$w = \frac{15}{- 3}$

$y = 10$

$x = 4$

$4 z + 9 = 13$

Step 5.2.3.3

Simplify the right side.

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Step 5.2.3.3.1

Divide $15$ by $- 3$ .

$w = - 5$

$y = 10$

$x = 4$

$4 z + 9 = 13$

$w = - 5$

$y = 10$

$x = 4$

$4 z + 9 = 13$

$w = - 5$

$y = 10$

$x = 4$

$4 z + 9 = 13$

Step 5.2.4

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

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Step 5.2.4.1

Subtract $9$ from both sides of the equation.

$4 z = 13 - 9$

$w = - 5$

$y = 10$

$x = 4$

Step 5.2.4.2

Subtract $9$ from $13$ .

$4 z = 4$

$w = - 5$

$y = 10$

$x = 4$

$4 z = 4$

$w = - 5$

$y = 10$

$x = 4$

Step 5.2.5

Divide each term in $4 z = 4$ by $4$ and simplify.

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Step 5.2.5.1

Divide each term in $4 z = 4$ by $4$ .

$\frac{4 z}{4} = \frac{4}{4}$

$w = - 5$

$y = 10$

$x = 4$

Step 5.2.5.2

Simplify the left side.

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Step 5.2.5.2.1

Cancel the common factor of $4$ .

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Step 5.2.5.2.1.1

Cancel the common factor.

$\frac{4 z}{4} = \frac{4}{4}$

$w = - 5$

$y = 10$

$x = 4$

Step 5.2.5.2.1.2

Divide $z$ by $1$ .

$z = \frac{4}{4}$

$w = - 5$

$y = 10$

$x = 4$

$z = \frac{4}{4}$

$w = - 5$

$y = 10$

$x = 4$

$z = \frac{4}{4}$

$w = - 5$

$y = 10$

$x = 4$

Step 5.2.5.3

Simplify the right side.

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Step 5.2.5.3.1

Divide $4$ by $4$ .

$z = 1$

$w = - 5$

$y = 10$

$x = 4$

$z = 1$

$w = - 5$

$y = 10$

$x = 4$

$z = 1$

$w = - 5$

$y = 10$

$x = 4$

$z = 1$

$w = - 5$

$y = 10$

$x = 4$

Step 5.3

List all of the solutions.

$z = 1, w = - 5, y = 10, x = 4$