Finite Math Examples

$[\begin{array}{c} 2 x & 4 y - 3 \\ z & 5 w \end{array}] = [\begin{array}{c} 10 & 10 \\ 7 & 15 \end{array}]$

Step 1

Write as a linear system of equations.

$2 x = 10$

$4 y - 3 = 10$

$z = 7$

$5 w = 15$

Step 2

Solve the system of equations.

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Step 2.1

Divide each term in $2 x = 10$ by $2$ and simplify.

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Step 2.1.1

Divide each term in $2 x = 10$ by $2$ .

$\frac{2 x}{2} = \frac{10}{2}$

$4 y - 3 = 10$

$z = 7$

$5 w = 15$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $2$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{10}{2}$

$4 y - 3 = 10$

$z = 7$

$5 w = 15$

Step 2.1.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{2}$

$4 y - 3 = 10$

$z = 7$

$5 w = 15$

$x = \frac{10}{2}$

$4 y - 3 = 10$

$z = 7$

$5 w = 15$

$x = \frac{10}{2}$

$4 y - 3 = 10$

$z = 7$

$5 w = 15$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $10$ by $2$ .

$x = 5$

$4 y - 3 = 10$

$z = 7$

$5 w = 15$

$x = 5$

$4 y - 3 = 10$

$z = 7$

$5 w = 15$

$x = 5$

$4 y - 3 = 10$

$z = 7$

$5 w = 15$

Step 2.2

Replace all occurrences of $x$ with $5$ 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 $3$ to both sides of the equation.

$4 y = 10 + 3$

$x = 5$

$z = 7$

$5 w = 15$

Step 2.2.1.2

Add $10$ and $3$ .

$4 y = 13$

$x = 5$

$z = 7$

$5 w = 15$

$4 y = 13$

$x = 5$

$z = 7$

$5 w = 15$

Step 2.2.2

Divide each term in $4 y = 13$ by $4$ and simplify.

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Step 2.2.2.1

Divide each term in $4 y = 13$ by $4$ .

$\frac{4 y}{4} = \frac{13}{4}$

$x = 5$

$z = 7$

$5 w = 15$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $4$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{13}{4}$

$x = 5$

$z = 7$

$5 w = 15$

Step 2.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{13}{4}$

$x = 5$

$z = 7$

$5 w = 15$

$y = \frac{13}{4}$

$x = 5$

$z = 7$

$5 w = 15$

$y = \frac{13}{4}$

$x = 5$

$z = 7$

$5 w = 15$

$y = \frac{13}{4}$

$x = 5$

$z = 7$

$5 w = 15$

Step 2.2.3

Divide each term in $5 w = 15$ by $5$ and simplify.

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Step 2.2.3.1

Divide each term in $5 w = 15$ by $5$ .

$\frac{5 w}{5} = \frac{15}{5}$

$y = \frac{13}{4}$

$x = 5$

$z = 7$

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Cancel the common factor of $5$ .

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Step 2.2.3.2.1.1

Cancel the common factor.

$\frac{5 w}{5} = \frac{15}{5}$

$y = \frac{13}{4}$

$x = 5$

$z = 7$

Step 2.2.3.2.1.2

Divide $w$ by $1$ .

$w = \frac{15}{5}$

$y = \frac{13}{4}$

$x = 5$

$z = 7$

$w = \frac{15}{5}$

$y = \frac{13}{4}$

$x = 5$

$z = 7$

$w = \frac{15}{5}$

$y = \frac{13}{4}$

$x = 5$

$z = 7$

Step 2.2.3.3

Simplify the right side.

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Step 2.2.3.3.1

Divide $15$ by $5$ .

$w = 3$

$y = \frac{13}{4}$

$x = 5$

$z = 7$

$w = 3$

$y = \frac{13}{4}$

$x = 5$

$z = 7$

$w = 3$

$y = \frac{13}{4}$

$x = 5$

$z = 7$

$w = 3$

$y = \frac{13}{4}$

$x = 5$

$z = 7$

Step 2.3

List all of the solutions.

$w = 3, y = \frac{13}{4}, x = 5, z = 7$