Finite Math Examples

$[\begin{array}{c} x + 1 & y - 6 \\ z - 3 & w + 5 \end{array}] = [\begin{array}{c} - 1 & 7 \\ 2 & 3 \end{array}]$

Step 1

Write as a linear system of equations.

$x + 1 = - 1$

$y - 6 = 7$

$z - 3 = 2$

$w + 5 = 3$

Step 2

Solve the system of equations.

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Step 2.1

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

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Step 2.1.1

Subtract $1$ from both sides of the equation.

$x = - 1 - 1$

$y - 6 = 7$

$z - 3 = 2$

$w + 5 = 3$

Step 2.1.2

Subtract $1$ from $- 1$ .

$x = - 2$

$y - 6 = 7$

$z - 3 = 2$

$w + 5 = 3$

$x = - 2$

$y - 6 = 7$

$z - 3 = 2$

$w + 5 = 3$

Step 2.2

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

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Step 2.2.1

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

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Step 2.2.1.1

Add $6$ to both sides of the equation.

$y = 7 + 6$

$x = - 2$

$z - 3 = 2$

$w + 5 = 3$

Step 2.2.1.2

Add $7$ and $6$ .

$y = 13$

$x = - 2$

$z - 3 = 2$

$w + 5 = 3$

$y = 13$

$x = - 2$

$z - 3 = 2$

$w + 5 = 3$

Step 2.2.2

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

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Step 2.2.2.1

Add $3$ to both sides of the equation.

$z = 2 + 3$

$y = 13$

$x = - 2$

$w + 5 = 3$

Step 2.2.2.2

Add $2$ and $3$ .

$z = 5$

$y = 13$

$x = - 2$

$w + 5 = 3$

$z = 5$

$y = 13$

$x = - 2$

$w + 5 = 3$

Step 2.2.3

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

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Step 2.2.3.1

Subtract $5$ from both sides of the equation.

$w = 3 - 5$

$z = 5$

$y = 13$

$x = - 2$

Step 2.2.3.2

Subtract $5$ from $3$ .

$w = - 2$

$z = 5$

$y = 13$

$x = - 2$

$w = - 2$

$z = 5$

$y = 13$

$x = - 2$

$w = - 2$

$z = 5$

$y = 13$

$x = - 2$

Step 2.3

List all of the solutions.

$w = - 2, z = 5, y = 13, x = - 2$