Precalculus Examples

$[\begin{array}{c} 3 y \\ 2 x \\ 6 y \end{array}] = [\begin{array}{c} x + 5 \\ 0 \\ 10 \end{array}]$

Step 1

Write as a linear system of equations.

$3 y = x + 5$

$2 x = 0$

$6 y = 10$

Step 2

Solve the system of equations.

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Step 2.1

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 2.1.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

$3 y = x + 5$

$6 y = 10$

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.

$\frac{2 x}{2} = \frac{0}{2}$

$3 y = x + 5$

$6 y = 10$

Step 2.1.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

$3 y = x + 5$

$6 y = 10$

$x = \frac{0}{2}$

$3 y = x + 5$

$6 y = 10$

$x = \frac{0}{2}$

$3 y = x + 5$

$6 y = 10$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $0$ by $2$ .

$x = 0$

$3 y = x + 5$

$6 y = 10$

$x = 0$

$3 y = x + 5$

$6 y = 10$

$x = 0$

$3 y = x + 5$

$6 y = 10$

Step 2.2

Replace all occurrences of $x$ with $0$ in each equation.

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Step 2.2.1

Replace all occurrences of $x$ in $3 y = x + 5$ with $0$ .

$3 y = (0) + 5$

$x = 0$

$6 y = 10$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Add $0$ and $5$ .

$3 y = 5$

$x = 0$

$6 y = 10$

$3 y = 5$

$x = 0$

$6 y = 10$

Step 2.2.3

Divide each term in $6 y = 10$ by $6$ and simplify.

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Step 2.2.3.1

Divide each term in $6 y = 10$ by $6$ .

$\frac{6 y}{6} = \frac{10}{6}$

$3 y = 5$

$x = 0$

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Cancel the common factor of $6$ .

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Step 2.2.3.2.1.1

Cancel the common factor.

$\frac{6 y}{6} = \frac{10}{6}$

$3 y = 5$

$x = 0$

Step 2.2.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{10}{6}$

$3 y = 5$

$x = 0$

$y = \frac{10}{6}$

$3 y = 5$

$x = 0$

$y = \frac{10}{6}$

$3 y = 5$

$x = 0$

Step 2.2.3.3

Simplify the right side.

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Step 2.2.3.3.1

Cancel the common factor of $10$ and $6$ .

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Step 2.2.3.3.1.1

Factor $2$ out of $10$ .

$y = \frac{2 (5)}{6}$

$3 y = 5$

$x = 0$

Step 2.2.3.3.1.2

Cancel the common factors.

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Step 2.2.3.3.1.2.1

Factor $2$ out of $6$ .

$y = \frac{2 \cdot 5}{2 \cdot 3}$

$3 y = 5$

$x = 0$

Step 2.2.3.3.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot 5}{2 \cdot 3}$

$3 y = 5$

$x = 0$

Step 2.2.3.3.1.2.3

Rewrite the expression.

$y = \frac{5}{3}$

$3 y = 5$

$x = 0$

$y = \frac{5}{3}$

$3 y = 5$

$x = 0$

$y = \frac{5}{3}$

$3 y = 5$

$x = 0$

$y = \frac{5}{3}$

$3 y = 5$

$x = 0$

$y = \frac{5}{3}$

$3 y = 5$

$x = 0$

$y = \frac{5}{3}$

$3 y = 5$

$x = 0$

Step 2.3

Replace all occurrences of $y$ with $\frac{5}{3}$ in each equation.

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Step 2.3.1

Replace all occurrences of $y$ in $3 y = 5$ with $\frac{5}{3}$ .

$3 (\frac{5}{3}) = 5$

$y = \frac{5}{3}$

$x = 0$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $3$ .

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Step 2.3.2.1.1

Cancel the common factor.

$3 (\frac{5}{3}) = 5$

$y = \frac{5}{3}$

$x = 0$

Step 2.3.2.1.2

Rewrite the expression.

$5 = 5$

$y = \frac{5}{3}$

$x = 0$

$5 = 5$

$y = \frac{5}{3}$

$x = 0$

$5 = 5$

$y = \frac{5}{3}$

$x = 0$

$5 = 5$

$y = \frac{5}{3}$

$x = 0$

Step 2.4

Remove any equations from the system that are always true.

$y = \frac{5}{3}$

$x = 0$

Step 2.5

List all of the solutions.

$y = \frac{5}{3}, x = 0$