Finite Math Examples

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

Step 1

Write as a linear system of equations.

$2 = 2 x$

$4 x = y$

$3 z = z + 6$

$w = - 8$

Step 2

Solve the system of equations.

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Step 2.1

Solve for $x$ in $2 = 2 x$ .

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Step 2.1.1

Rewrite the equation as $2 x = 2$ .

$2 x = 2$

$4 x = y$

$3 z = z + 6$

$w = - 8$

Step 2.1.2

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

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Step 2.1.2.1

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

$\frac{2 x}{2} = \frac{2}{2}$

$4 x = y$

$3 z = z + 6$

$w = - 8$

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $2$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{2}{2}$

$4 x = y$

$3 z = z + 6$

$w = - 8$

Step 2.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{2}$

$4 x = y$

$3 z = z + 6$

$w = - 8$

$x = \frac{2}{2}$

$4 x = y$

$3 z = z + 6$

$w = - 8$

$x = \frac{2}{2}$

$4 x = y$

$3 z = z + 6$

$w = - 8$

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Divide $2$ by $2$ .

$x = 1$

$4 x = y$

$3 z = z + 6$

$w = - 8$

$x = 1$

$4 x = y$

$3 z = z + 6$

$w = - 8$

$x = 1$

$4 x = y$

$3 z = z + 6$

$w = - 8$

$x = 1$

$4 x = y$

$3 z = z + 6$

$w = - 8$

Step 2.2

Replace all occurrences of $x$ with $1$ in each equation.

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Step 2.2.1

Replace all occurrences of $x$ in $4 x = y$ with $1$ .

$4 (1) = y$

$x = 1$

$3 z = z + 6$

$w = - 8$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Multiply $4$ by $1$ .

$4 = y$

$x = 1$

$3 z = z + 6$

$w = - 8$

$4 = y$

$x = 1$

$3 z = z + 6$

$w = - 8$

Step 2.2.3

Move all terms containing $z$ to the left side of the equation.

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Step 2.2.3.1

Subtract $z$ from both sides of the equation.

$3 z - z = 6$

$4 = y$

$x = 1$

$w = - 8$

Step 2.2.3.2

Subtract $z$ from $3 z$ .

$2 z = 6$

$4 = y$

$x = 1$

$w = - 8$

$2 z = 6$

$4 = y$

$x = 1$

$w = - 8$

Step 2.2.4

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

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Step 2.2.4.1

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

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

$4 = y$

$x = 1$

$w = - 8$

Step 2.2.4.2

Simplify the left side.

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Step 2.2.4.2.1

Cancel the common factor of $2$ .

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Step 2.2.4.2.1.1

Cancel the common factor.

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

$4 = y$

$x = 1$

$w = - 8$

Step 2.2.4.2.1.2

Divide $z$ by $1$ .

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

$4 = y$

$x = 1$

$w = - 8$

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

$4 = y$

$x = 1$

$w = - 8$

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

$4 = y$

$x = 1$

$w = - 8$

Step 2.2.4.3

Simplify the right side.

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Step 2.2.4.3.1

Divide $6$ by $2$ .

$z = 3$

$4 = y$

$x = 1$

$w = - 8$

$z = 3$

$4 = y$

$x = 1$

$w = - 8$

$z = 3$

$4 = y$

$x = 1$

$w = - 8$

$z = 3$

$4 = y$

$x = 1$

$w = - 8$

Step 2.3

Rewrite the equation as $y = 4$ .

$y = 4$

$z = 3$

$x = 1$

$w = - 8$

Step 2.4

Solve the system of equations.

$y = 4 z = 3 x = 1 w = - 8$

Step 2.5

List all of the solutions.

$y = 4, z = 3, x = 1, w = - 8$