Algebra Examples

$[\begin{array}{c} 15 & 8 \\ 4 x & 0 \end{array}] - [\begin{array}{c} 2 & 5 \\ x & 3 y + 6 \end{array}] = [\begin{array}{c} 13 & 3 \\ 12 & 4 y + 8 \end{array}]$

Step 1

Simplify.

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Step 1.1

Subtract the corresponding elements.

$[\begin{array}{c} 15 - 2 & 8 - 5 \\ 4 x - x & 0 - (3 y + 6) \end{array}] = [\begin{array}{c} 13 & 3 \\ 12 & 4 y + 8 \end{array}]$

Step 1.2

Simplify each element.

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Step 1.2.1

Subtract $2$ from $15$ .

$[\begin{array}{c} 13 & 8 - 5 \\ 4 x - x & 0 - (3 y + 6) \end{array}] = [\begin{array}{c} 13 & 3 \\ 12 & 4 y + 8 \end{array}]$

Step 1.2.2

Subtract $5$ from $8$ .

$[\begin{array}{c} 13 & 3 \\ 4 x - x & 0 - (3 y + 6) \end{array}] = [\begin{array}{c} 13 & 3 \\ 12 & 4 y + 8 \end{array}]$

Step 1.2.3

Subtract $x$ from $4 x$ .

$[\begin{array}{c} 13 & 3 \\ 3 x & 0 - (3 y + 6) \end{array}] = [\begin{array}{c} 13 & 3 \\ 12 & 4 y + 8 \end{array}]$

Step 1.2.4

Subtract $3 y + 6$ from $0$ .

$[\begin{array}{c} 13 & 3 \\ 3 x & - (3 y + 6) \end{array}] = [\begin{array}{c} 13 & 3 \\ 12 & 4 y + 8 \end{array}]$

Step 1.2.5

Apply the distributive property.

$[\begin{array}{c} 13 & 3 \\ 3 x & - (3 y) - 1 \cdot 6 \end{array}] = [\begin{array}{c} 13 & 3 \\ 12 & 4 y + 8 \end{array}]$

Step 1.2.6

Multiply $3$ by $- 1$ .

$[\begin{array}{c} 13 & 3 \\ 3 x & - 3 y - 1 \cdot 6 \end{array}] = [\begin{array}{c} 13 & 3 \\ 12 & 4 y + 8 \end{array}]$

Step 1.2.7

Multiply $- 1$ by $6$ .

$[\begin{array}{c} 13 & 3 \\ 3 x & - 3 y - 6 \end{array}] = [\begin{array}{c} 13 & 3 \\ 12 & 4 y + 8 \end{array}]$

Step 2

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

$13 = 13$

$3 = 3$

$3 x = 12$

$- 3 y - 6 = 4 y + 8$

Step 3

Divide each term in $3 x = 12$ by $3$ and simplify.

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Step 3.1

Divide each term in $3 x = 12$ by $3$ .

$\frac{3 x}{3} = \frac{12}{3}$

$- 3 y - 6 = 4 y + 8$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $3$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{12}{3}$

$- 3 y - 6 = 4 y + 8$

Step 3.2.1.2

Divide $x$ by $1$ .

$x = \frac{12}{3}$

$- 3 y - 6 = 4 y + 8$

$x = \frac{12}{3}$

$- 3 y - 6 = 4 y + 8$

$x = \frac{12}{3}$

$- 3 y - 6 = 4 y + 8$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $12$ by $3$ .

$x = 4$

$- 3 y - 6 = 4 y + 8$

$x = 4$

$- 3 y - 6 = 4 y + 8$

$x = 4$

$- 3 y - 6 = 4 y + 8$

Step 4

Replace all occurrences of $x$ with $4$ in each equation.

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Step 4.1

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

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Step 4.1.1

Subtract $4 y$ from both sides of the equation.

$- 3 y - 6 - 4 y = 8$

$x = 4$

Step 4.1.2

Subtract $4 y$ from $- 3 y$ .

$- 7 y - 6 = 8$

$x = 4$

$- 7 y - 6 = 8$

$x = 4$

Step 4.2

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

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Step 4.2.1

Add $6$ to both sides of the equation.

$- 7 y = 8 + 6$

$x = 4$

Step 4.2.2

Add $8$ and $6$ .

$- 7 y = 14$

$x = 4$

$- 7 y = 14$

$x = 4$

Step 4.3

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

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Step 4.3.1

Divide each term in $- 7 y = 14$ by $- 7$ .

$\frac{- 7 y}{- 7} = \frac{14}{- 7}$

$x = 4$

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.

$\frac{- 7 y}{- 7} = \frac{14}{- 7}$

$x = 4$

Step 4.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{14}{- 7}$

$x = 4$

$y = \frac{14}{- 7}$

$x = 4$

$y = \frac{14}{- 7}$

$x = 4$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $14$ by $- 7$ .

$y = - 2$

$x = 4$

$y = - 2$

$x = 4$

$y = - 2$

$x = 4$

$y = - 2$

$x = 4$

Step 5

List all of the solutions.

$y = - 2, x = 4$