Precalculus Examples

$[\begin{array}{c} 16 & 4 & 16 & 4 \\ - 3 & 13 & 15 & 16 \\ 0 & 2 & 22 & 0 \end{array}] = [\begin{array}{c} 16 & 4 & 7 x + 2 & 4 \\ - 3 & 13 & 15 & 8 x \\ 0 & 2 & 3 y - 5 & 0 \end{array}]$

Step 1

The matrix equation can be written as a set of equations.

$16 = 16$

$4 = 4$

$16 = 7 x + 2$

$4 = 4$

$- 3 = - 3$

$13 = 13$

$15 = 15$

$16 = 8 x$

$0 = 0$

$2 = 2$

$22 = 3 y - 5$

$0 = 0$

Step 2

Solve for $x$ in $16 = 7 x + 2$ .

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Step 2.1

Rewrite the equation as $7 x + 2 = 16$ .

$7 x + 2 = 16$

$16 = 8 x$

$22 = 3 y - 5$

Step 2.2

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

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Step 2.2.1

Subtract $2$ from both sides of the equation.

$7 x = 16 - 2$

$16 = 8 x$

$22 = 3 y - 5$

Step 2.2.2

Subtract $2$ from $16$ .

$7 x = 14$

$16 = 8 x$

$22 = 3 y - 5$

$7 x = 14$

$16 = 8 x$

$22 = 3 y - 5$

Step 2.3

Divide each term in $7 x = 14$ by $7$ and simplify.

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Step 2.3.1

Divide each term in $7 x = 14$ by $7$ .

$\frac{7 x}{7} = \frac{14}{7}$

$16 = 8 x$

$22 = 3 y - 5$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $7$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{14}{7}$

$16 = 8 x$

$22 = 3 y - 5$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{14}{7}$

$16 = 8 x$

$22 = 3 y - 5$

$x = \frac{14}{7}$

$16 = 8 x$

$22 = 3 y - 5$

$x = \frac{14}{7}$

$16 = 8 x$

$22 = 3 y - 5$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $14$ by $7$ .

$x = 2$

$16 = 8 x$

$22 = 3 y - 5$

$x = 2$

$16 = 8 x$

$22 = 3 y - 5$

$x = 2$

$16 = 8 x$

$22 = 3 y - 5$

$x = 2$

$16 = 8 x$

$22 = 3 y - 5$

Step 3

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

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Step 3.1

Replace all occurrences of $x$ in $16 = 8 x$ with $2$ .

$16 = 8 (2)$

$x = 2$

$22 = 3 y - 5$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $8$ by $2$ .

$16 = 16$

$x = 2$

$22 = 3 y - 5$

$16 = 16$

$x = 2$

$22 = 3 y - 5$

Step 3.3

Rewrite the equation as $3 y - 5 = 22$ .

$3 y - 5 = 22$

$16 = 16$

$x = 2$

Step 3.4

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

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Step 3.4.1

Add $5$ to both sides of the equation.

$3 y = 22 + 5$

$16 = 16$

$x = 2$

Step 3.4.2

Add $22$ and $5$ .

$3 y = 27$

$16 = 16$

$x = 2$

$3 y = 27$

$16 = 16$

$x = 2$

Step 3.5

Divide each term in $3 y = 27$ by $3$ and simplify.

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Step 3.5.1

Divide each term in $3 y = 27$ by $3$ .

$\frac{3 y}{3} = \frac{27}{3}$

$16 = 16$

$x = 2$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $3$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{27}{3}$

$16 = 16$

$x = 2$

Step 3.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{27}{3}$

$16 = 16$

$x = 2$

$y = \frac{27}{3}$

$16 = 16$

$x = 2$

$y = \frac{27}{3}$

$16 = 16$

$x = 2$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Divide $27$ by $3$ .

$y = 9$

$16 = 16$

$x = 2$

$y = 9$

$16 = 16$

$x = 2$

$y = 9$

$16 = 16$

$x = 2$

$y = 9$

$16 = 16$

$x = 2$

Step 4

List all of the solutions.

$y = 9, x = 2$