Algebra Examples

$\frac{3}{2} [\begin{array}{c} x & 6 \\ 8 & 4 \end{array}] + y [\begin{array}{c} 1 & 4 \\ 3 & 2 \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{3}{2}$ by each element of the matrix.

$[\begin{array}{c} \frac{3}{2} x & \frac{3}{2} \cdot 6 \\ \frac{3}{2} \cdot 8 & \frac{3}{2} \cdot 4 \end{array}] + y [\begin{array}{c} 1 & 4 \\ 3 & 2 \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 1.2

Simplify each element in the matrix.

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Step 1.2.1

Combine $\frac{3}{2}$ and $x$ .

$[\begin{array}{c} \frac{3 x}{2} & \frac{3}{2} \cdot 6 \\ \frac{3}{2} \cdot 8 & \frac{3}{2} \cdot 4 \end{array}] + y [\begin{array}{c} 1 & 4 \\ 3 & 2 \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 1.2.2

Cancel the common factor of $2$ .

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Step 1.2.2.1

Factor $2$ out of $6$ .

$[\begin{array}{c} \frac{3 x}{2} & \frac{3}{2} \cdot (2 (3)) \\ \frac{3}{2} \cdot 8 & \frac{3}{2} \cdot 4 \end{array}] + y [\begin{array}{c} 1 & 4 \\ 3 & 2 \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 1.2.2.2

Cancel the common factor.

$[\begin{array}{c} \frac{3 x}{2} & \frac{3}{2} \cdot (2 \cdot 3) \\ \frac{3}{2} \cdot 8 & \frac{3}{2} \cdot 4 \end{array}] + y [\begin{array}{c} 1 & 4 \\ 3 & 2 \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 1.2.2.3

Rewrite the expression.

$[\begin{array}{c} \frac{3 x}{2} & 3 \cdot 3 \\ \frac{3}{2} \cdot 8 & \frac{3}{2} \cdot 4 \end{array}] + y [\begin{array}{c} 1 & 4 \\ 3 & 2 \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 1.2.3

Multiply $3$ by $3$ .

$[\begin{array}{c} \frac{3 x}{2} & 9 \\ \frac{3}{2} \cdot 8 & \frac{3}{2} \cdot 4 \end{array}] + y [\begin{array}{c} 1 & 4 \\ 3 & 2 \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 1.2.4

Cancel the common factor of $2$ .

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Step 1.2.4.1

Factor $2$ out of $8$ .

$[\begin{array}{c} \frac{3 x}{2} & 9 \\ \frac{3}{2} \cdot (2 (4)) & \frac{3}{2} \cdot 4 \end{array}] + y [\begin{array}{c} 1 & 4 \\ 3 & 2 \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 1.2.4.2

Cancel the common factor.

$[\begin{array}{c} \frac{3 x}{2} & 9 \\ \frac{3}{2} \cdot (2 \cdot 4) & \frac{3}{2} \cdot 4 \end{array}] + y [\begin{array}{c} 1 & 4 \\ 3 & 2 \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 1.2.4.3

Rewrite the expression.

$[\begin{array}{c} \frac{3 x}{2} & 9 \\ 3 \cdot 4 & \frac{3}{2} \cdot 4 \end{array}] + y [\begin{array}{c} 1 & 4 \\ 3 & 2 \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 1.2.5

Multiply $3$ by $4$ .

$[\begin{array}{c} \frac{3 x}{2} & 9 \\ 12 & \frac{3}{2} \cdot 4 \end{array}] + y [\begin{array}{c} 1 & 4 \\ 3 & 2 \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 1.2.6

Cancel the common factor of $2$ .

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Step 1.2.6.1

Factor $2$ out of $4$ .

$[\begin{array}{c} \frac{3 x}{2} & 9 \\ 12 & \frac{3}{2} \cdot (2 (2)) \end{array}] + y [\begin{array}{c} 1 & 4 \\ 3 & 2 \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 1.2.6.2

Cancel the common factor.

$[\begin{array}{c} \frac{3 x}{2} & 9 \\ 12 & \frac{3}{2} \cdot (2 \cdot 2) \end{array}] + y [\begin{array}{c} 1 & 4 \\ 3 & 2 \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 1.2.6.3

Rewrite the expression.

$[\begin{array}{c} \frac{3 x}{2} & 9 \\ 12 & 3 \cdot 2 \end{array}] + y [\begin{array}{c} 1 & 4 \\ 3 & 2 \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 1.2.7

Multiply $3$ by $2$ .

$[\begin{array}{c} \frac{3 x}{2} & 9 \\ 12 & 6 \end{array}] + y [\begin{array}{c} 1 & 4 \\ 3 & 2 \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 1.3

Multiply $y$ by each element of the matrix.

$[\begin{array}{c} \frac{3 x}{2} & 9 \\ 12 & 6 \end{array}] + [\begin{array}{c} y \cdot 1 & y \cdot 4 \\ y \cdot 3 & y \cdot 2 \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 1.4

Simplify each element in the matrix.

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Step 1.4.1

Multiply $y$ by $1$ .

$[\begin{array}{c} \frac{3 x}{2} & 9 \\ 12 & 6 \end{array}] + [\begin{array}{c} y & y \cdot 4 \\ y \cdot 3 & y \cdot 2 \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 1.4.2

Move $4$ to the left of $y$ .

$[\begin{array}{c} \frac{3 x}{2} & 9 \\ 12 & 6 \end{array}] + [\begin{array}{c} y & 4 y \\ y \cdot 3 & y \cdot 2 \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 1.4.3

Move $3$ to the left of $y$ .

$[\begin{array}{c} \frac{3 x}{2} & 9 \\ 12 & 6 \end{array}] + [\begin{array}{c} y & 4 y \\ 3 y & y \cdot 2 \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 1.4.4

Move $2$ to the left of $y$ .

$[\begin{array}{c} \frac{3 x}{2} & 9 \\ 12 & 6 \end{array}] + [\begin{array}{c} y & 4 y \\ 3 y & 2 y \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 2

Add the corresponding elements.

$[\begin{array}{c} \frac{3 x}{2} + y & 9 + 4 y \\ 12 + 3 y & 6 + 2 y \end{array}] = [\begin{array}{c} z & z \\ 6 z & 2 \end{array}]$

Step 3

Write as a linear system of equations.

$\frac{3 x}{2} + y = z$

$9 + 4 y = z$

$12 + 3 y = 6 z$

$6 + 2 y = 2$

Step 4

Solve the system of equations.

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Step 4.1

Solve for $y$ in $6 + 2 y = 2$ .

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Step 4.1.1

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

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Step 4.1.1.1

Subtract $6$ from both sides of the equation.

$2 y = 2 - 6$

$\frac{3 x}{2} + y = z$

$9 + 4 y = z$

$12 + 3 y = 6 z$

Step 4.1.1.2

Subtract $6$ from $2$ .

$2 y = - 4$

$\frac{3 x}{2} + y = z$

$9 + 4 y = z$

$12 + 3 y = 6 z$

$2 y = - 4$

$\frac{3 x}{2} + y = z$

$9 + 4 y = z$

$12 + 3 y = 6 z$

Step 4.1.2

Divide each term in $2 y = - 4$ by $2$ and simplify.

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Step 4.1.2.1

Divide each term in $2 y = - 4$ by $2$ .

$\frac{2 y}{2} = \frac{- 4}{2}$

$\frac{3 x}{2} + y = z$

$9 + 4 y = z$

$12 + 3 y = 6 z$

Step 4.1.2.2

Simplify the left side.

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Step 4.1.2.2.1

Cancel the common factor of $2$ .

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Step 4.1.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{- 4}{2}$

$\frac{3 x}{2} + y = z$

$9 + 4 y = z$

$12 + 3 y = 6 z$

Step 4.1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 4}{2}$

$\frac{3 x}{2} + y = z$

$9 + 4 y = z$

$12 + 3 y = 6 z$

$y = \frac{- 4}{2}$

$\frac{3 x}{2} + y = z$

$9 + 4 y = z$

$12 + 3 y = 6 z$

$y = \frac{- 4}{2}$

$\frac{3 x}{2} + y = z$

$9 + 4 y = z$

$12 + 3 y = 6 z$

Step 4.1.2.3

Simplify the right side.

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Step 4.1.2.3.1

Divide $- 4$ by $2$ .

$y = - 2$

$\frac{3 x}{2} + y = z$

$9 + 4 y = z$

$12 + 3 y = 6 z$

$y = - 2$

$\frac{3 x}{2} + y = z$

$9 + 4 y = z$

$12 + 3 y = 6 z$

$y = - 2$

$\frac{3 x}{2} + y = z$

$9 + 4 y = z$

$12 + 3 y = 6 z$

$y = - 2$

$\frac{3 x}{2} + y = z$

$9 + 4 y = z$

$12 + 3 y = 6 z$

Step 4.2

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

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Step 4.2.1

Replace all occurrences of $y$ in $\frac{3 x}{2} + y = z$ with $- 2$ .

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

$y = - 2$

$9 + 4 y = z$

$12 + 3 y = 6 z$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Remove parentheses.

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

$y = - 2$

$9 + 4 y = z$

$12 + 3 y = 6 z$

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

$y = - 2$

$9 + 4 y = z$

$12 + 3 y = 6 z$

Step 4.2.3

Replace all occurrences of $y$ in $9 + 4 y = z$ with $- 2$ .

$9 + 4 (- 2) = z$

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

$y = - 2$

$12 + 3 y = 6 z$

Step 4.2.4

Simplify the left side.

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Step 4.2.4.1

Simplify $9 + 4 (- 2)$ .

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Step 4.2.4.1.1

Multiply $4$ by $- 2$ .

$9 - 8 = z$

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

$y = - 2$

$12 + 3 y = 6 z$

Step 4.2.4.1.2

Subtract $8$ from $9$ .

$1 = z$

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

$y = - 2$

$12 + 3 y = 6 z$

$1 = z$

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

$y = - 2$

$12 + 3 y = 6 z$

$1 = z$

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

$y = - 2$

$12 + 3 y = 6 z$

Step 4.2.5

Replace all occurrences of $y$ in $12 + 3 y = 6 z$ with $- 2$ .

$12 + 3 (- 2) = 6 z$

$1 = z$

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

$y = - 2$

Step 4.2.6

Simplify the left side.

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Step 4.2.6.1

Simplify $12 + 3 (- 2)$ .

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Step 4.2.6.1.1

Multiply $3$ by $- 2$ .

$12 - 6 = 6 z$

$1 = z$

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

$y = - 2$

Step 4.2.6.1.2

Subtract $6$ from $12$ .

$6 = 6 z$

$1 = z$

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

$y = - 2$

$6 = 6 z$

$1 = z$

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

$y = - 2$

$6 = 6 z$

$1 = z$

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

$y = - 2$

$6 = 6 z$

$1 = z$

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

$y = - 2$

Step 4.3

Solve for $z$ in $6 = 6 z$ .

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Step 4.3.1

Rewrite the equation as $6 z = 6$ .

$6 z = 6$

$1 = z$

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

$y = - 2$

Step 4.3.2

Divide each term in $6 z = 6$ by $6$ and simplify.

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Step 4.3.2.1

Divide each term in $6 z = 6$ by $6$ .

$\frac{6 z}{6} = \frac{6}{6}$

$1 = z$

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

$y = - 2$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Cancel the common factor of $6$ .

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Step 4.3.2.2.1.1

Cancel the common factor.

$\frac{6 z}{6} = \frac{6}{6}$

$1 = z$

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

$y = - 2$

Step 4.3.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{6}{6}$

$1 = z$

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

$y = - 2$

$z = \frac{6}{6}$

$1 = z$

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

$y = - 2$

$z = \frac{6}{6}$

$1 = z$

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

$y = - 2$

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

Divide $6$ by $6$ .

$z = 1$

$1 = z$

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

$y = - 2$

$z = 1$

$1 = z$

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

$y = - 2$

$z = 1$

$1 = z$

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

$y = - 2$

$z = 1$

$1 = z$

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

$y = - 2$

Step 4.4

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

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Step 4.4.1

Replace all occurrences of $z$ in $\frac{3 x}{2} - 2 = z$ with $1$ .

$\frac{3 x}{2} - 2 = 1$

$z = 1$

$1 = z$

$y = - 2$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Remove parentheses.

$\frac{3 x}{2} - 2 = 1$

$z = 1$

$1 = z$

$y = - 2$

$\frac{3 x}{2} - 2 = 1$

$z = 1$

$1 = z$

$y = - 2$

$\frac{3 x}{2} - 2 = 1$

$z = 1$

$1 = z$

$y = - 2$

Step 4.5

Solve for $x$ in $\frac{3 x}{2} - 2 = 1$ .

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Step 4.5.1

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

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Step 4.5.1.1

Add $2$ to both sides of the equation.

$\frac{3 x}{2} = 1 + 2$

$z = 1$

$y = - 2$

Step 4.5.1.2

Add $1$ and $2$ .

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

$z = 1$

$y = - 2$

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

$z = 1$

$y = - 2$

Step 4.5.2

Multiply both sides of the equation by $\frac{2}{3}$ .

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

$z = 1$

$y = - 2$

Step 4.5.3

Simplify both sides of the equation.

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Step 4.5.3.1

Simplify the left side.

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Step 4.5.3.1.1

Simplify $\frac{2}{3} \cdot \frac{3 x}{2}$ .

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Step 4.5.3.1.1.1

Cancel the common factor of $2$ .

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Step 4.5.3.1.1.1.1

Cancel the common factor.

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

$z = 1$

$y = - 2$

Step 4.5.3.1.1.1.2

Rewrite the expression.

$\frac{1}{3} \cdot (3 x) = \frac{2}{3} \cdot 3$

$z = 1$

$y = - 2$

$\frac{1}{3} \cdot (3 x) = \frac{2}{3} \cdot 3$

$z = 1$

$y = - 2$

Step 4.5.3.1.1.2

Cancel the common factor of $3$ .

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Step 4.5.3.1.1.2.1

Factor $3$ out of $3 x$ .

$\frac{1}{3} \cdot (3 (x)) = \frac{2}{3} \cdot 3$

$z = 1$

$y = - 2$

Step 4.5.3.1.1.2.2

Cancel the common factor.

$\frac{1}{3} \cdot (3 x) = \frac{2}{3} \cdot 3$

$z = 1$

$y = - 2$

Step 4.5.3.1.1.2.3

Rewrite the expression.

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

$z = 1$

$y = - 2$

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

$z = 1$

$y = - 2$

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

$z = 1$

$y = - 2$

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

$z = 1$

$y = - 2$

Step 4.5.3.2

Simplify the right side.

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Step 4.5.3.2.1

Cancel the common factor of $3$ .

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Step 4.5.3.2.1.1

Cancel the common factor.

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

$z = 1$

$y = - 2$

Step 4.5.3.2.1.2

Rewrite the expression.

$x = 2$

$z = 1$

$y = - 2$

$x = 2$

$z = 1$

$y = - 2$

$x = 2$

$z = 1$

$y = - 2$

$x = 2$

$z = 1$

$y = - 2$

$x = 2$

$z = 1$

$y = - 2$

Step 4.6

Solve the system of equations.

$x = 2 z = 1 y = - 2$

Step 4.7

List all of the solutions.

$x = 2, z = 1, y = - 2$