Algebra Examples

$x - 2 y + 2 z = 4$ $2 y - 4 z = 8$ $- x + 4 z = 2$

Step 1

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

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Step 1.1

Add $2 y$ to both sides of the equation.

$x + 2 z = 4 + 2 y$

$2 y - 4 z = 8$

$- x + 4 z = 2$

Step 1.2

Subtract $2 z$ from both sides of the equation.

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

$- x + 4 z = 2$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

$- x + 4 z = 2$

Step 2

Replace all occurrences of $x$ with $4 + 2 y - 2 z$ in each equation.

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Step 2.1

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

$- (4 + 2 y - 2 z) + 4 z = 2$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- (4 + 2 y - 2 z) + 4 z$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$- 1 \cdot 4 - (2 y) - (- 2 z) + 4 z = 2$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 1$ by $4$ .

$- 4 - (2 y) - (- 2 z) + 4 z = 2$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

Step 2.2.1.1.2.2

Multiply $2$ by $- 1$ .

$- 4 - 2 y - (- 2 z) + 4 z = 2$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

Step 2.2.1.1.2.3

Multiply $- 2$ by $- 1$ .

$- 4 - 2 y + 2 z + 4 z = 2$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

$- 4 - 2 y + 2 z + 4 z = 2$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

$- 4 - 2 y + 2 z + 4 z = 2$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

Step 2.2.1.2

Add $2 z$ and $4 z$ .

$- 4 - 2 y + 6 z = 2$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

$- 4 - 2 y + 6 z = 2$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

$- 4 - 2 y + 6 z = 2$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

$- 4 - 2 y + 6 z = 2$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

Step 3

Solve for $y$ in $- 4 - 2 y + 6 z = 2$ .

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Step 3.1

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

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Step 3.1.1

Add $4$ to both sides of the equation.

$- 2 y + 6 z = 2 + 4$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

Step 3.1.2

Subtract $6 z$ from both sides of the equation.

$- 2 y = 2 + 4 - 6 z$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

Step 3.1.3

Add $2$ and $4$ .

$- 2 y = 6 - 6 z$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

$- 2 y = 6 - 6 z$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

Step 3.2

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

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Step 3.2.1

Divide each term in $- 2 y = 6 - 6 z$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{6}{- 2} + \frac{- 6 z}{- 2}$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

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.

$\frac{- 2 y}{- 2} = \frac{6}{- 2} + \frac{- 6 z}{- 2}$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{6}{- 2} + \frac{- 6 z}{- 2}$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

$y = \frac{6}{- 2} + \frac{- 6 z}{- 2}$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

$y = \frac{6}{- 2} + \frac{- 6 z}{- 2}$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Divide $6$ by $- 2$ .

$y = - 3 + \frac{- 6 z}{- 2}$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

Step 3.2.3.1.2

Cancel the common factor of $- 6$ and $- 2$ .

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Step 3.2.3.1.2.1

Factor $- 2$ out of $- 6 z$ .

$y = - 3 + \frac{- 2 (3 z)}{- 2}$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

Step 3.2.3.1.2.2

Cancel the common factors.

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Step 3.2.3.1.2.2.1

Factor $- 2$ out of $- 2$ .

$y = - 3 + \frac{- 2 (3 z)}{- 2 \cdot 1}$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

Step 3.2.3.1.2.2.2

Cancel the common factor.

$y = - 3 + \frac{- 2 (3 z)}{- 2 \cdot 1}$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

Step 3.2.3.1.2.2.3

Rewrite the expression.

$y = - 3 + \frac{3 z}{1}$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

Step 3.2.3.1.2.2.4

Divide $3 z$ by $1$ .

$y = - 3 + 3 z$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

$y = - 3 + 3 z$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

$y = - 3 + 3 z$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

$y = - 3 + 3 z$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

$y = - 3 + 3 z$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

$y = - 3 + 3 z$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

$y = - 3 + 3 z$

$x = 4 + 2 y - 2 z$

$2 y - 4 z = 8$

Step 4

Replace all occurrences of $y$ with $- 3 + 3 z$ in each equation.

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Step 4.1

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

$x = 4 + 2 (- 3 + 3 z) - 2 z$

$y = - 3 + 3 z$

$2 y - 4 z = 8$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $4 + 2 (- 3 + 3 z) - 2 z$ .

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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.

$x = 4 + 2 \cdot - 3 + 2 (3 z) - 2 z$

$y = - 3 + 3 z$

$2 y - 4 z = 8$

Step 4.2.1.1.2

Multiply $2$ by $- 3$ .

$x = 4 - 6 + 2 (3 z) - 2 z$

$y = - 3 + 3 z$

$2 y - 4 z = 8$

Step 4.2.1.1.3

Multiply $3$ by $2$ .

$x = 4 - 6 + 6 z - 2 z$

$y = - 3 + 3 z$

$2 y - 4 z = 8$

$x = 4 - 6 + 6 z - 2 z$

$y = - 3 + 3 z$

$2 y - 4 z = 8$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Subtract $6$ from $4$ .

$x = - 2 + 6 z - 2 z$

$y = - 3 + 3 z$

$2 y - 4 z = 8$

Step 4.2.1.2.2

Subtract $2 z$ from $6 z$ .

$x = - 2 + 4 z$

$y = - 3 + 3 z$

$2 y - 4 z = 8$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

$2 y - 4 z = 8$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

$2 y - 4 z = 8$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

$2 y - 4 z = 8$

Step 4.3

Replace all occurrences of $y$ in $2 y - 4 z = 8$ with $- 3 + 3 z$ .

$2 (- 3 + 3 z) - 4 z = 8$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

Step 4.4

Simplify the left side.

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Step 4.4.1

Simplify $2 (- 3 + 3 z) - 4 z$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$2 \cdot - 3 + 2 (3 z) - 4 z = 8$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

Step 4.4.1.1.2

Multiply $2$ by $- 3$ .

$- 6 + 2 (3 z) - 4 z = 8$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

Step 4.4.1.1.3

Multiply $3$ by $2$ .

$- 6 + 6 z - 4 z = 8$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

$- 6 + 6 z - 4 z = 8$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

Step 4.4.1.2

Subtract $4 z$ from $6 z$ .

$- 6 + 2 z = 8$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

$- 6 + 2 z = 8$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

$- 6 + 2 z = 8$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

$- 6 + 2 z = 8$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

Step 5

Solve for $z$ in $- 6 + 2 z = 8$ .

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Step 5.1

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

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Step 5.1.1

Add $6$ to both sides of the equation.

$2 z = 8 + 6$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

Step 5.1.2

Add $8$ and $6$ .

$2 z = 14$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

$2 z = 14$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

Step 5.2

Divide each term in $2 z = 14$ by $2$ and simplify.

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Step 5.2.1

Divide each term in $2 z = 14$ by $2$ .

$\frac{2 z}{2} = \frac{14}{2}$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{2 z}{2} = \frac{14}{2}$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

Step 5.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{14}{2}$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

$z = \frac{14}{2}$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

$z = \frac{14}{2}$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $14$ by $2$ .

$z = 7$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

$z = 7$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

$z = 7$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

$z = 7$

$x = - 2 + 4 z$

$y = - 3 + 3 z$

Step 6

Replace all occurrences of $z$ with $7$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $x = - 2 + 4 z$ with $7$ .

$x = - 2 + 4 (7)$

$z = 7$

$y = - 3 + 3 z$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- 2 + 4 (7)$ .

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Step 6.2.1.1

Multiply $4$ by $7$ .

$x = - 2 + 28$

$z = 7$

$y = - 3 + 3 z$

Step 6.2.1.2

Add $- 2$ and $28$ .

$x = 26$

$z = 7$

$y = - 3 + 3 z$

$x = 26$

$z = 7$

$y = - 3 + 3 z$

$x = 26$

$z = 7$

$y = - 3 + 3 z$

Step 6.3

Replace all occurrences of $z$ in $y = - 3 + 3 z$ with $7$ .

$y = - 3 + 3 (7)$

$x = 26$

$z = 7$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- 3 + 3 (7)$ .

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Step 6.4.1.1

Multiply $3$ by $7$ .

$y = - 3 + 21$

$x = 26$

$z = 7$

Step 6.4.1.2

Add $- 3$ and $21$ .

$y = 18$

$x = 26$

$z = 7$

$y = 18$

$x = 26$

$z = 7$

$y = 18$

$x = 26$

$z = 7$

$y = 18$

$x = 26$

$z = 7$

Step 7

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

$(26, 18, 7)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(26, 18, 7)$

Equation Form:

$x = 26, y = 18, z = 7$