Algebra Examples

$[\begin{array}{c} 6 & y \\ 18 & z \end{array}] - 3 [\begin{array}{c} x + 1 & 2 \\ 0 & 1 \end{array}] = [\begin{array}{c} 18 & 17 \\ 9 w & 4 z \end{array}]$

Step 1

Simplify each term.

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Step 1.1

Multiply $- 3$ by each element of the matrix.

$[\begin{array}{c} 6 & y \\ 18 & z \end{array}] + [\begin{array}{c} - 3 (x + 1) & - 3 \cdot 2 \\ - 3 \cdot 0 & - 3 \cdot 1 \end{array}] = [\begin{array}{c} 18 & 17 \\ 9 w & 4 z \end{array}]$

Step 1.2

Simplify each element in the matrix.

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Step 1.2.1

Apply the distributive property.

$[\begin{array}{c} 6 & y \\ 18 & z \end{array}] + [\begin{array}{c} - 3 x - 3 \cdot 1 & - 3 \cdot 2 \\ - 3 \cdot 0 & - 3 \cdot 1 \end{array}] = [\begin{array}{c} 18 & 17 \\ 9 w & 4 z \end{array}]$

Step 1.2.2

Multiply $- 3$ by $1$ .

$[\begin{array}{c} 6 & y \\ 18 & z \end{array}] + [\begin{array}{c} - 3 x - 3 & - 3 \cdot 2 \\ - 3 \cdot 0 & - 3 \cdot 1 \end{array}] = [\begin{array}{c} 18 & 17 \\ 9 w & 4 z \end{array}]$

Step 1.2.3

Multiply $- 3$ by $2$ .

$[\begin{array}{c} 6 & y \\ 18 & z \end{array}] + [\begin{array}{c} - 3 x - 3 & - 6 \\ - 3 \cdot 0 & - 3 \cdot 1 \end{array}] = [\begin{array}{c} 18 & 17 \\ 9 w & 4 z \end{array}]$

Step 1.2.4

Multiply $- 3$ by $0$ .

$[\begin{array}{c} 6 & y \\ 18 & z \end{array}] + [\begin{array}{c} - 3 x - 3 & - 6 \\ 0 & - 3 \cdot 1 \end{array}] = [\begin{array}{c} 18 & 17 \\ 9 w & 4 z \end{array}]$

Step 1.2.5

Multiply $- 3$ by $1$ .

$[\begin{array}{c} 6 & y \\ 18 & z \end{array}] + [\begin{array}{c} - 3 x - 3 & - 6 \\ 0 & - 3 \end{array}] = [\begin{array}{c} 18 & 17 \\ 9 w & 4 z \end{array}]$

Step 2

Add the corresponding elements.

$[\begin{array}{c} 6 - 3 x - 3 & y - 6 \\ 18 + 0 & z - 3 \end{array}] = [\begin{array}{c} 18 & 17 \\ 9 w & 4 z \end{array}]$

Step 3

Simplify each element.

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Step 3.1

Subtract $3$ from $6$ .

$[\begin{array}{c} - 3 x + 3 & y - 6 \\ 18 + 0 & z - 3 \end{array}] = [\begin{array}{c} 18 & 17 \\ 9 w & 4 z \end{array}]$

Step 3.2

Add $18$ and $0$ .

$[\begin{array}{c} - 3 x + 3 & y - 6 \\ 18 & z - 3 \end{array}] = [\begin{array}{c} 18 & 17 \\ 9 w & 4 z \end{array}]$

Step 4

Write as a linear system of equations.

$- 3 x + 3 = 18$

$y - 6 = 17$

$18 = 9 w$

$z - 3 = 4 z$

Step 5

Solve the system of equations.

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Step 5.1

Solve for $x$ in $- 3 x + 3 = 18$ .

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Step 5.1.1

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

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Step 5.1.1.1

Subtract $3$ from both sides of the equation.

$- 3 x = 18 - 3$

$y - 6 = 17$

$18 = 9 w$

$z - 3 = 4 z$

Step 5.1.1.2

Subtract $3$ from $18$ .

$- 3 x = 15$

$y - 6 = 17$

$18 = 9 w$

$z - 3 = 4 z$

$- 3 x = 15$

$y - 6 = 17$

$18 = 9 w$

$z - 3 = 4 z$

Step 5.1.2

Divide each term in $- 3 x = 15$ by $- 3$ and simplify.

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Step 5.1.2.1

Divide each term in $- 3 x = 15$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{15}{- 3}$

$y - 6 = 17$

$18 = 9 w$

$z - 3 = 4 z$

Step 5.1.2.2

Simplify the left side.

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Step 5.1.2.2.1

Cancel the common factor of $- 3$ .

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Step 5.1.2.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{15}{- 3}$

$y - 6 = 17$

$18 = 9 w$

$z - 3 = 4 z$

Step 5.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{15}{- 3}$

$y - 6 = 17$

$18 = 9 w$

$z - 3 = 4 z$

$x = \frac{15}{- 3}$

$y - 6 = 17$

$18 = 9 w$

$z - 3 = 4 z$

$x = \frac{15}{- 3}$

$y - 6 = 17$

$18 = 9 w$

$z - 3 = 4 z$

Step 5.1.2.3

Simplify the right side.

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Step 5.1.2.3.1

Divide $15$ by $- 3$ .

$x = - 5$

$y - 6 = 17$

$18 = 9 w$

$z - 3 = 4 z$

$x = - 5$

$y - 6 = 17$

$18 = 9 w$

$z - 3 = 4 z$

$x = - 5$

$y - 6 = 17$

$18 = 9 w$

$z - 3 = 4 z$

$x = - 5$

$y - 6 = 17$

$18 = 9 w$

$z - 3 = 4 z$

Step 5.2

Replace all occurrences of $x$ with $- 5$ in each equation.

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Step 5.2.1

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

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Step 5.2.1.1

Add $6$ to both sides of the equation.

$y = 17 + 6$

$x = - 5$

$18 = 9 w$

$z - 3 = 4 z$

Step 5.2.1.2

Add $17$ and $6$ .

$y = 23$

$x = - 5$

$18 = 9 w$

$z - 3 = 4 z$

$y = 23$

$x = - 5$

$18 = 9 w$

$z - 3 = 4 z$

Step 5.2.2

Rewrite the equation as $9 w = 18$ .

$9 w = 18$

$y = 23$

$x = - 5$

$z - 3 = 4 z$

Step 5.2.3

Divide each term in $9 w = 18$ by $9$ and simplify.

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Step 5.2.3.1

Divide each term in $9 w = 18$ by $9$ .

$\frac{9 w}{9} = \frac{18}{9}$

$y = 23$

$x = - 5$

$z - 3 = 4 z$

Step 5.2.3.2

Simplify the left side.

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Step 5.2.3.2.1

Cancel the common factor of $9$ .

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Step 5.2.3.2.1.1

Cancel the common factor.

$\frac{9 w}{9} = \frac{18}{9}$

$y = 23$

$x = - 5$

$z - 3 = 4 z$

Step 5.2.3.2.1.2

Divide $w$ by $1$ .

$w = \frac{18}{9}$

$y = 23$

$x = - 5$

$z - 3 = 4 z$

$w = \frac{18}{9}$

$y = 23$

$x = - 5$

$z - 3 = 4 z$

$w = \frac{18}{9}$

$y = 23$

$x = - 5$

$z - 3 = 4 z$

Step 5.2.3.3

Simplify the right side.

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Step 5.2.3.3.1

Divide $18$ by $9$ .

$w = 2$

$y = 23$

$x = - 5$

$z - 3 = 4 z$

$w = 2$

$y = 23$

$x = - 5$

$z - 3 = 4 z$

$w = 2$

$y = 23$

$x = - 5$

$z - 3 = 4 z$

Step 5.2.4

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

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Step 5.2.4.1

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

$z - 3 - 4 z = 0$

$w = 2$

$y = 23$

$x = - 5$

Step 5.2.4.2

Subtract $4 z$ from $z$ .

$- 3 z - 3 = 0$

$w = 2$

$y = 23$

$x = - 5$

$- 3 z - 3 = 0$

$w = 2$

$y = 23$

$x = - 5$

Step 5.2.5

Add $3$ to both sides of the equation.

$- 3 z = 3$

$w = 2$

$y = 23$

$x = - 5$

Step 5.2.6

Divide each term in $- 3 z = 3$ by $- 3$ and simplify.

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Step 5.2.6.1

Divide each term in $- 3 z = 3$ by $- 3$ .

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

$w = 2$

$y = 23$

$x = - 5$

Step 5.2.6.2

Simplify the left side.

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Step 5.2.6.2.1

Cancel the common factor of $- 3$ .

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Step 5.2.6.2.1.1

Cancel the common factor.

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

$w = 2$

$y = 23$

$x = - 5$

Step 5.2.6.2.1.2

Divide $z$ by $1$ .

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

$w = 2$

$y = 23$

$x = - 5$

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

$w = 2$

$y = 23$

$x = - 5$

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

$w = 2$

$y = 23$

$x = - 5$

Step 5.2.6.3

Simplify the right side.

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Step 5.2.6.3.1

Divide $3$ by $- 3$ .

$z = - 1$

$w = 2$

$y = 23$

$x = - 5$

$z = - 1$

$w = 2$

$y = 23$

$x = - 5$

$z = - 1$

$w = 2$

$y = 23$

$x = - 5$

$z = - 1$

$w = 2$

$y = 23$

$x = - 5$

Step 5.3

List all of the solutions.

$z = - 1, w = 2, y = 23, x = - 5$