Algebra Examples

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

Step 1

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

$3 x + y = x + 3$

$x - 2 y = y - 2$

Step 2

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

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Step 2.1

Subtract $3 x$ from both sides of the equation.

$y = x + 3 - 3 x$

$x - 2 y = y - 2$

Step 2.2

Subtract $3 x$ from $x$ .

$y = - 2 x + 3$

$x - 2 y = y - 2$

$y = - 2 x + 3$

$x - 2 y = y - 2$

Step 3

Replace all occurrences of $y$ with $- 2 x + 3$ in each equation.

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Step 3.1

Replace all occurrences of $y$ in $x - 2 y = y - 2$ with $- 2 x + 3$ .

$x - 2 (- 2 x + 3) = (- 2 x + 3) - 2$

$y = - 2 x + 3$

Step 3.2

Simplify $x - 2 (- 2 x + 3) = (- 2 x + 3) - 2$ .

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $x - 2 (- 2 x + 3)$ .

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Step 3.2.1.1.1

Simplify each term.

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Step 3.2.1.1.1.1

Apply the distributive property.

$x - 2 (- 2 x) - 2 \cdot 3 = (- 2 x + 3) - 2$

$y = - 2 x + 3$

Step 3.2.1.1.1.2

Multiply $- 2$ by $- 2$ .

$x + 4 x - 2 \cdot 3 = (- 2 x + 3) - 2$

$y = - 2 x + 3$

Step 3.2.1.1.1.3

Multiply $- 2$ by $3$ .

$x + 4 x - 6 = (- 2 x + 3) - 2$

$y = - 2 x + 3$

$x + 4 x - 6 = (- 2 x + 3) - 2$

$y = - 2 x + 3$

Step 3.2.1.1.2

Add $x$ and $4 x$ .

$5 x - 6 = (- 2 x + 3) - 2$

$y = - 2 x + 3$

$5 x - 6 = (- 2 x + 3) - 2$

$y = - 2 x + 3$

$5 x - 6 = (- 2 x + 3) - 2$

$y = - 2 x + 3$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Subtract $2$ from $3$ .

$5 x - 6 = - 2 x + 1$

$y = - 2 x + 3$

$5 x - 6 = - 2 x + 1$

$y = - 2 x + 3$

$5 x - 6 = - 2 x + 1$

$y = - 2 x + 3$

$5 x - 6 = - 2 x + 1$

$y = - 2 x + 3$

Step 4

Solve for $x$ in $5 x - 6 = - 2 x + 1$ .

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Step 4.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.1.1

Add $2 x$ to both sides of the equation.

$5 x - 6 + 2 x = 1$

$y = - 2 x + 3$

Step 4.1.2

Add $5 x$ and $2 x$ .

$7 x - 6 = 1$

$y = - 2 x + 3$

$7 x - 6 = 1$

$y = - 2 x + 3$

Step 4.2

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

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Step 4.2.1

Add $6$ to both sides of the equation.

$7 x = 1 + 6$

$y = - 2 x + 3$

Step 4.2.2

Add $1$ and $6$ .

$7 x = 7$

$y = - 2 x + 3$

$7 x = 7$

$y = - 2 x + 3$

Step 4.3

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

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Step 4.3.1

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

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

$y = - 2 x + 3$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $7$ .

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Step 4.3.2.1.1

Cancel the common factor.

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

$y = - 2 x + 3$

Step 4.3.2.1.2

Divide $x$ by $1$ .

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

$y = - 2 x + 3$

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

$y = - 2 x + 3$

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

$y = - 2 x + 3$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $7$ by $7$ .

$x = 1$

$y = - 2 x + 3$

$x = 1$

$y = - 2 x + 3$

$x = 1$

$y = - 2 x + 3$

$x = 1$

$y = - 2 x + 3$

Step 5

Replace all occurrences of $x$ with $1$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = - 2 x + 3$ with $1$ .

$y = - 2 \cdot 1 + 3$

$x = 1$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 2 \cdot 1 + 3$ .

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Step 5.2.1.1

Multiply $- 2$ by $1$ .

$y = - 2 + 3$

$x = 1$

Step 5.2.1.2

Add $- 2$ and $3$ .

$y = 1$

$x = 1$

$y = 1$

$x = 1$

$y = 1$

$x = 1$

$y = 1$

$x = 1$

Step 6

List all of the solutions.

$y = 1, x = 1$