Linear Algebra Examples

$[\begin{array}{c} 1 \\ 2 \end{array}] = [\begin{array}{c} 2 x \\ 3 y \end{array}]$

Step 1

Write as a linear system of equations.

$1 = 2 x$

$2 = 3 y$

Step 2

Solve the system of equations.

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Step 2.1

Solve for $x$ in $1 = 2 x$ .

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Step 2.1.1

Rewrite the equation as $2 x = 1$ .

$2 x = 1$

$2 = 3 y$

Step 2.1.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 2.1.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

$2 = 3 y$

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.

$\frac{2 x}{2} = \frac{1}{2}$

$2 = 3 y$

Step 2.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

$2 = 3 y$

$x = \frac{1}{2}$

$2 = 3 y$

$x = \frac{1}{2}$

$2 = 3 y$

$x = \frac{1}{2}$

$2 = 3 y$

$x = \frac{1}{2}$

$2 = 3 y$

Step 2.2

Replace all occurrences of $x$ with $\frac{1}{2}$ in each equation.

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Step 2.2.1

Rewrite the equation as $3 y = 2$ .

$3 y = 2$

$x = \frac{1}{2}$

Step 2.2.2

Divide each term in $3 y = 2$ by $3$ and simplify.

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Step 2.2.2.1

Divide each term in $3 y = 2$ by $3$ .

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

$x = \frac{1}{2}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

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

$x = \frac{1}{2}$

Step 2.2.2.2.1.2

Divide $y$ by $1$ .

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

$x = \frac{1}{2}$

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

$x = \frac{1}{2}$

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

$x = \frac{1}{2}$

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

$x = \frac{1}{2}$

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

$x = \frac{1}{2}$

Step 2.3

List all of the solutions.

$y = \frac{2}{3}, x = \frac{1}{2}$

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