Algebra Examples

$3 x + 9 y + z = - 2$ $x + 9 y + z = 4$ $9 x - 9 y + 5 z = 8$

Step 1

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

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Step 1.1

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

$9 y + z = - 2 - 3 x$

$x + 9 y + z = 4$

$9 x - 9 y + 5 z = 8$

Step 1.2

Subtract $9 y$ from both sides of the equation.

$z = - 2 - 3 x - 9 y$

$x + 9 y + z = 4$

$9 x - 9 y + 5 z = 8$

$z = - 2 - 3 x - 9 y$

$x + 9 y + z = 4$

$9 x - 9 y + 5 z = 8$

Step 2

Replace all occurrences of $z$ with $- 2 - 3 x - 9 y$ in each equation.

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Step 2.1

Replace all occurrences of $z$ in $x + 9 y + z = 4$ with $- 2 - 3 x - 9 y$ .

$x + 9 y - 2 - 3 x - 9 y = 4$

$z = - 2 - 3 x - 9 y$

$9 x - 9 y + 5 z = 8$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $x + 9 y - 2 - 3 x - 9 y$ .

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Step 2.2.1.1

Remove parentheses.

$x + 9 y - 2 - 3 x - 9 y = 4$

$z = - 2 - 3 x - 9 y$

$9 x - 9 y + 5 z = 8$

Step 2.2.1.2

Combine the opposite terms in $x + 9 y - 2 - 3 x - 9 y$ .

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Step 2.2.1.2.1

Subtract $9 y$ from $9 y$ .

$x - 2 - 3 x + 0 = 4$

$z = - 2 - 3 x - 9 y$

$9 x - 9 y + 5 z = 8$

Step 2.2.1.2.2

Add $x - 2 - 3 x$ and $0$ .

$x - 2 - 3 x = 4$

$z = - 2 - 3 x - 9 y$

$9 x - 9 y + 5 z = 8$

$x - 2 - 3 x = 4$

$z = - 2 - 3 x - 9 y$

$9 x - 9 y + 5 z = 8$

Step 2.2.1.3

Subtract $3 x$ from $x$ .

$- 2 x - 2 = 4$

$z = - 2 - 3 x - 9 y$

$9 x - 9 y + 5 z = 8$

$- 2 x - 2 = 4$

$z = - 2 - 3 x - 9 y$

$9 x - 9 y + 5 z = 8$

$- 2 x - 2 = 4$

$z = - 2 - 3 x - 9 y$

$9 x - 9 y + 5 z = 8$

Step 2.3

Replace all occurrences of $z$ in $9 x - 9 y + 5 z = 8$ with $- 2 - 3 x - 9 y$ .

$9 x - 9 y + 5 (- 2 - 3 x - 9 y) = 8$

$- 2 x - 2 = 4$

$z = - 2 - 3 x - 9 y$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $9 x - 9 y + 5 (- 2 - 3 x - 9 y)$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Apply the distributive property.

$9 x - 9 y + 5 \cdot - 2 + 5 (- 3 x) + 5 (- 9 y) = 8$

$- 2 x - 2 = 4$

$z = - 2 - 3 x - 9 y$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $5$ by $- 2$ .

$9 x - 9 y - 10 + 5 (- 3 x) + 5 (- 9 y) = 8$

$- 2 x - 2 = 4$

$z = - 2 - 3 x - 9 y$

Step 2.4.1.1.2.2

Multiply $- 3$ by $5$ .

$9 x - 9 y - 10 - 15 x + 5 (- 9 y) = 8$

$- 2 x - 2 = 4$

$z = - 2 - 3 x - 9 y$

Step 2.4.1.1.2.3

Multiply $- 9$ by $5$ .

$9 x - 9 y - 10 - 15 x - 45 y = 8$

$- 2 x - 2 = 4$

$z = - 2 - 3 x - 9 y$

$9 x - 9 y - 10 - 15 x - 45 y = 8$

$- 2 x - 2 = 4$

$z = - 2 - 3 x - 9 y$

$9 x - 9 y - 10 - 15 x - 45 y = 8$

$- 2 x - 2 = 4$

$z = - 2 - 3 x - 9 y$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $15 x$ from $9 x$ .

$- 6 x - 9 y - 10 - 45 y = 8$

$- 2 x - 2 = 4$

$z = - 2 - 3 x - 9 y$

Step 2.4.1.2.2

Subtract $45 y$ from $- 9 y$ .

$- 6 x - 54 y - 10 = 8$

$- 2 x - 2 = 4$

$z = - 2 - 3 x - 9 y$

$- 6 x - 54 y - 10 = 8$

$- 2 x - 2 = 4$

$z = - 2 - 3 x - 9 y$

$- 6 x - 54 y - 10 = 8$

$- 2 x - 2 = 4$

$z = - 2 - 3 x - 9 y$

$- 6 x - 54 y - 10 = 8$

$- 2 x - 2 = 4$

$z = - 2 - 3 x - 9 y$

$- 6 x - 54 y - 10 = 8$

$- 2 x - 2 = 4$

$z = - 2 - 3 x - 9 y$

Step 3

Solve for $x$ in $- 2 x - 2 = 4$ .

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Step 3.1

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

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Step 3.1.1

Add $2$ to both sides of the equation.

$- 2 x = 4 + 2$

$- 6 x - 54 y - 10 = 8$

$z = - 2 - 3 x - 9 y$

Step 3.1.2

Add $4$ and $2$ .

$- 2 x = 6$

$- 6 x - 54 y - 10 = 8$

$z = - 2 - 3 x - 9 y$

$- 2 x = 6$

$- 6 x - 54 y - 10 = 8$

$z = - 2 - 3 x - 9 y$

Step 3.2

Divide each term in $- 2 x = 6$ by $- 2$ and simplify.

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Step 3.2.1

Divide each term in $- 2 x = 6$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{6}{- 2}$

$- 6 x - 54 y - 10 = 8$

$z = - 2 - 3 x - 9 y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{6}{- 2}$

$- 6 x - 54 y - 10 = 8$

$z = - 2 - 3 x - 9 y$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{- 2}$

$- 6 x - 54 y - 10 = 8$

$z = - 2 - 3 x - 9 y$

$x = \frac{6}{- 2}$

$- 6 x - 54 y - 10 = 8$

$z = - 2 - 3 x - 9 y$

$x = \frac{6}{- 2}$

$- 6 x - 54 y - 10 = 8$

$z = - 2 - 3 x - 9 y$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $6$ by $- 2$ .

$x = - 3$

$- 6 x - 54 y - 10 = 8$

$z = - 2 - 3 x - 9 y$

$x = - 3$

$- 6 x - 54 y - 10 = 8$

$z = - 2 - 3 x - 9 y$

$x = - 3$

$- 6 x - 54 y - 10 = 8$

$z = - 2 - 3 x - 9 y$

$x = - 3$

$- 6 x - 54 y - 10 = 8$

$z = - 2 - 3 x - 9 y$

Step 4

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

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Step 4.1

Replace all occurrences of $x$ in $- 6 x - 54 y - 10 = 8$ with $- 3$ .

$- 6 \cdot - 3 - 54 y - 10 = 8$

$x = - 3$

$z = - 2 - 3 x - 9 y$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 6 \cdot - 3 - 54 y - 10$ .

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Step 4.2.1.1

Multiply $- 6$ by $- 3$ .

$18 - 54 y - 10 = 8$

$x = - 3$

$z = - 2 - 3 x - 9 y$

Step 4.2.1.2

Subtract $10$ from $18$ .

$- 54 y + 8 = 8$

$x = - 3$

$z = - 2 - 3 x - 9 y$

$- 54 y + 8 = 8$

$x = - 3$

$z = - 2 - 3 x - 9 y$

$- 54 y + 8 = 8$

$x = - 3$

$z = - 2 - 3 x - 9 y$

Step 4.3

Replace all occurrences of $x$ in $z = - 2 - 3 x - 9 y$ with $- 3$ .

$z = - 2 - 3 \cdot - 3 - 9 y$

$- 54 y + 8 = 8$

$x = - 3$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $- 2 - 3 \cdot - 3 - 9 y$ .

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Step 4.4.1.1

Multiply $- 3$ by $- 3$ .

$z = - 2 + 9 - 9 y$

$- 54 y + 8 = 8$

$x = - 3$

Step 4.4.1.2

Add $- 2$ and $9$ .

$z = 7 - 9 y$

$- 54 y + 8 = 8$

$x = - 3$

$z = 7 - 9 y$

$- 54 y + 8 = 8$

$x = - 3$

$z = 7 - 9 y$

$- 54 y + 8 = 8$

$x = - 3$

$z = 7 - 9 y$

$- 54 y + 8 = 8$

$x = - 3$

Step 5

Solve for $y$ in $- 54 y + 8 = 8$ .

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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 $8$ from both sides of the equation.

$- 54 y = 8 - 8$

$z = 7 - 9 y$

$x = - 3$

Step 5.1.2

Subtract $8$ from $8$ .

$- 54 y = 0$

$z = 7 - 9 y$

$x = - 3$

$- 54 y = 0$

$z = 7 - 9 y$

$x = - 3$

Step 5.2

Divide each term in $- 54 y = 0$ by $- 54$ and simplify.

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Step 5.2.1

Divide each term in $- 54 y = 0$ by $- 54$ .

$\frac{- 54 y}{- 54} = \frac{0}{- 54}$

$z = 7 - 9 y$

$x = - 3$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 54$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 54 y}{- 54} = \frac{0}{- 54}$

$z = 7 - 9 y$

$x = - 3$

Step 5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{0}{- 54}$

$z = 7 - 9 y$

$x = - 3$

$y = \frac{0}{- 54}$

$z = 7 - 9 y$

$x = - 3$

$y = \frac{0}{- 54}$

$z = 7 - 9 y$

$x = - 3$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $0$ by $- 54$ .

$y = 0$

$z = 7 - 9 y$

$x = - 3$

$y = 0$

$z = 7 - 9 y$

$x = - 3$

$y = 0$

$z = 7 - 9 y$

$x = - 3$

$y = 0$

$z = 7 - 9 y$

$x = - 3$

Step 6

Replace all occurrences of $y$ with $0$ in each equation.

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Step 6.1

Replace all occurrences of $y$ in $z = 7 - 9 y$ with $0$ .

$z = 7 - 9 \cdot 0$

$y = 0$

$x = - 3$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $7 - 9 \cdot 0$ .

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Step 6.2.1.1

Multiply $- 9$ by $0$ .

$z = 7 + 0$

$y = 0$

$x = - 3$

Step 6.2.1.2

Add $7$ and $0$ .

$z = 7$

$y = 0$

$x = - 3$

$z = 7$

$y = 0$

$x = - 3$

$z = 7$

$y = 0$

$x = - 3$

$z = 7$

$y = 0$

$x = - 3$

Step 7

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(- 3, 0, 7)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 3, 0, 7)$

Equation Form:

$x = - 3, y = 0, z = 7$