Algebra Examples

$[\begin{array}{c} 2 x \\ x \end{array}] - [\begin{array}{c} 8 y \\ y \end{array}] = [\begin{array}{c} 12 \\ 1 \end{array}]$

Step 1

Subtract the corresponding elements.

$[\begin{array}{c} 2 x - 8 y \\ x - y \end{array}] = [\begin{array}{c} 12 \\ 1 \end{array}]$

Step 2

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

$2 x - 8 y = 12$

$x - y = 1$

Step 3

Add $y$ to both sides of the equation.

$x = 1 + y$

$2 x - 8 y = 12$

Step 4

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

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Step 4.1

Replace all occurrences of $x$ in $2 x - 8 y = 12$ with $1 + y$ .

$2 (1 + y) - 8 y = 12$

$x = 1 + y$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $2 (1 + y) - 8 y$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$2 \cdot 1 + 2 y - 8 y = 12$

$x = 1 + y$

Step 4.2.1.1.2

Multiply $2$ by $1$ .

$2 + 2 y - 8 y = 12$

$x = 1 + y$

$2 + 2 y - 8 y = 12$

$x = 1 + y$

Step 4.2.1.2

Subtract $8 y$ from $2 y$ .

$2 - 6 y = 12$

$x = 1 + y$

$2 - 6 y = 12$

$x = 1 + y$

$2 - 6 y = 12$

$x = 1 + y$

$2 - 6 y = 12$

$x = 1 + y$

Step 5

Solve for $y$ in $2 - 6 y = 12$ .

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Step 5.1

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

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Step 5.1.1

Subtract $2$ from both sides of the equation.

$- 6 y = 12 - 2$

$x = 1 + y$

Step 5.1.2

Subtract $2$ from $12$ .

$- 6 y = 10$

$x = 1 + y$

$- 6 y = 10$

$x = 1 + y$

Step 5.2

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

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Step 5.2.1

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

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

$x = 1 + y$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 6$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

$x = 1 + y$

Step 5.2.2.1.2

Divide $y$ by $1$ .

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

$x = 1 + y$

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

$x = 1 + y$

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

$x = 1 + y$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

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

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Step 5.2.3.1.1

Factor $2$ out of $10$ .

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

$x = 1 + y$

Step 5.2.3.1.2

Cancel the common factors.

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Step 5.2.3.1.2.1

Factor $2$ out of $- 6$ .

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

$x = 1 + y$

Step 5.2.3.1.2.2

Cancel the common factor.

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

$x = 1 + y$

Step 5.2.3.1.2.3

Rewrite the expression.

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

$x = 1 + y$

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

$x = 1 + y$

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

$x = 1 + y$

Step 5.2.3.2

Move the negative in front of the fraction.

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

$x = 1 + y$

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

$x = 1 + y$

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

$x = 1 + y$

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

$x = 1 + y$

Step 6

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

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Step 6.1

Replace all occurrences of $y$ in $x = 1 + y$ with $- \frac{5}{3}$ .

$x = 1 - \frac{5}{3}$

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

Step 6.2

Simplify $x = 1 - \frac{5}{3}$ .

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Remove parentheses.

$x = 1 - \frac{5}{3}$

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

$x = 1 - \frac{5}{3}$

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

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $1 - \frac{5}{3}$ .

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Step 6.2.2.1.1

Write $1$ as a fraction with a common denominator.

$x = \frac{3}{3} - \frac{5}{3}$

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

Step 6.2.2.1.2

Combine the numerators over the common denominator.

$x = \frac{3 - 5}{3}$

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

Step 6.2.2.1.3

Subtract $5$ from $3$ .

$x = \frac{- 2}{3}$

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

Step 6.2.2.1.4

Move the negative in front of the fraction.

$x = - \frac{2}{3}$

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

$x = - \frac{2}{3}$

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

$x = - \frac{2}{3}$

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

$x = - \frac{2}{3}$

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

$x = - \frac{2}{3}$

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

Step 7

List all of the solutions.

$x = - \frac{2}{3}, y = - \frac{5}{3}$