Algebra Examples

$3 x + 2 y - z = 17$ $x - 2 y - 4 z = 5$ $2 x + y + 2 z = 5$

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.

$x - 4 z = 5 + 2 y$

$3 x + 2 y - z = 17$

$2 x + y + 2 z = 5$

Step 1.2

Add $4 z$ to both sides of the equation.

$x = 5 + 2 y + 4 z$

$3 x + 2 y - z = 17$

$2 x + y + 2 z = 5$

$x = 5 + 2 y + 4 z$

$3 x + 2 y - z = 17$

$2 x + y + 2 z = 5$

Step 2

Replace all occurrences of $x$ with $5 + 2 y + 4 z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 x + 2 y - z = 17$ with $5 + 2 y + 4 z$ .

$3 (5 + 2 y + 4 z) + 2 y - z = 17$

$x = 5 + 2 y + 4 z$

$2 x + y + 2 z = 5$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (5 + 2 y + 4 z) + 2 y - z$ .

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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.

$3 \cdot 5 + 3 (2 y) + 3 (4 z) + 2 y - z = 17$

$x = 5 + 2 y + 4 z$

$2 x + y + 2 z = 5$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $3$ by $5$ .

$15 + 3 (2 y) + 3 (4 z) + 2 y - z = 17$

$x = 5 + 2 y + 4 z$

$2 x + y + 2 z = 5$

Step 2.2.1.1.2.2

Multiply $2$ by $3$ .

$15 + 6 y + 3 (4 z) + 2 y - z = 17$

$x = 5 + 2 y + 4 z$

$2 x + y + 2 z = 5$

Step 2.2.1.1.2.3

Multiply $4$ by $3$ .

$15 + 6 y + 12 z + 2 y - z = 17$

$x = 5 + 2 y + 4 z$

$2 x + y + 2 z = 5$

$15 + 6 y + 12 z + 2 y - z = 17$

$x = 5 + 2 y + 4 z$

$2 x + y + 2 z = 5$

$15 + 6 y + 12 z + 2 y - z = 17$

$x = 5 + 2 y + 4 z$

$2 x + y + 2 z = 5$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $6 y$ and $2 y$ .

$15 + 8 y + 12 z - z = 17$

$x = 5 + 2 y + 4 z$

$2 x + y + 2 z = 5$

Step 2.2.1.2.2

Subtract $z$ from $12 z$ .

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

$2 x + y + 2 z = 5$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

$2 x + y + 2 z = 5$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

$2 x + y + 2 z = 5$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

$2 x + y + 2 z = 5$

Step 2.3

Replace all occurrences of $x$ in $2 x + y + 2 z = 5$ with $5 + 2 y + 4 z$ .

$2 (5 + 2 y + 4 z) + y + 2 z = 5$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $2 (5 + 2 y + 4 z) + y + 2 z$ .

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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.

$2 \cdot 5 + 2 (2 y) + 2 (4 z) + y + 2 z = 5$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $2$ by $5$ .

$10 + 2 (2 y) + 2 (4 z) + y + 2 z = 5$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

Step 2.4.1.1.2.2

Multiply $2$ by $2$ .

$10 + 4 y + 2 (4 z) + y + 2 z = 5$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

Step 2.4.1.1.2.3

Multiply $4$ by $2$ .

$10 + 4 y + 8 z + y + 2 z = 5$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

$10 + 4 y + 8 z + y + 2 z = 5$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

$10 + 4 y + 8 z + y + 2 z = 5$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Add $4 y$ and $y$ .

$10 + 5 y + 8 z + 2 z = 5$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

Step 2.4.1.2.2

Add $8 z$ and $2 z$ .

$10 + 5 y + 10 z = 5$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

$10 + 5 y + 10 z = 5$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

$10 + 5 y + 10 z = 5$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

$10 + 5 y + 10 z = 5$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

$10 + 5 y + 10 z = 5$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

Step 3

Solve for $y$ in $10 + 5 y + 10 z = 5$ .

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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

Subtract $10$ from both sides of the equation.

$5 y + 10 z = 5 - 10$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

Step 3.1.2

Subtract $10 z$ from both sides of the equation.

$5 y = 5 - 10 - 10 z$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

Step 3.1.3

Subtract $10$ from $5$ .

$5 y = - 5 - 10 z$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

$5 y = - 5 - 10 z$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

Step 3.2

Divide each term in $5 y = - 5 - 10 z$ by $5$ and simplify.

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Step 3.2.1

Divide each term in $5 y = - 5 - 10 z$ by $5$ .

$\frac{5 y}{5} = \frac{- 5}{5} + \frac{- 10 z}{5}$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{- 5}{5} + \frac{- 10 z}{5}$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 5}{5} + \frac{- 10 z}{5}$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

$y = \frac{- 5}{5} + \frac{- 10 z}{5}$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

$y = \frac{- 5}{5} + \frac{- 10 z}{5}$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

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 $- 5$ by $5$ .

$y = - 1 + \frac{- 10 z}{5}$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

Step 3.2.3.1.2

Cancel the common factor of $- 10$ and $5$ .

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Step 3.2.3.1.2.1

Factor $5$ out of $- 10 z$ .

$y = - 1 + \frac{5 (- 2 z)}{5}$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

Step 3.2.3.1.2.2

Cancel the common factors.

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Step 3.2.3.1.2.2.1

Factor $5$ out of $5$ .

$y = - 1 + \frac{5 (- 2 z)}{5 (1)}$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

Step 3.2.3.1.2.2.2

Cancel the common factor.

$y = - 1 + \frac{5 (- 2 z)}{5 \cdot 1}$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

Step 3.2.3.1.2.2.3

Rewrite the expression.

$y = - 1 + \frac{- 2 z}{1}$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

Step 3.2.3.1.2.2.4

Divide $- 2 z$ by $1$ .

$y = - 1 - 2 z$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

$y = - 1 - 2 z$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

$y = - 1 - 2 z$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

$y = - 1 - 2 z$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

$y = - 1 - 2 z$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

$y = - 1 - 2 z$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

$y = - 1 - 2 z$

$15 + 8 y + 11 z = 17$

$x = 5 + 2 y + 4 z$

Step 4

Replace all occurrences of $y$ with $- 1 - 2 z$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $15 + 8 y + 11 z = 17$ with $- 1 - 2 z$ .

$15 + 8 (- 1 - 2 z) + 11 z = 17$

$y = - 1 - 2 z$

$x = 5 + 2 y + 4 z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $15 + 8 (- 1 - 2 z) + 11 z$ .

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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.

$15 + 8 \cdot - 1 + 8 (- 2 z) + 11 z = 17$

$y = - 1 - 2 z$

$x = 5 + 2 y + 4 z$

Step 4.2.1.1.2

Multiply $8$ by $- 1$ .

$15 - 8 + 8 (- 2 z) + 11 z = 17$

$y = - 1 - 2 z$

$x = 5 + 2 y + 4 z$

Step 4.2.1.1.3

Multiply $- 2$ by $8$ .

$15 - 8 - 16 z + 11 z = 17$

$y = - 1 - 2 z$

$x = 5 + 2 y + 4 z$

$15 - 8 - 16 z + 11 z = 17$

$y = - 1 - 2 z$

$x = 5 + 2 y + 4 z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Subtract $8$ from $15$ .

$7 - 16 z + 11 z = 17$

$y = - 1 - 2 z$

$x = 5 + 2 y + 4 z$

Step 4.2.1.2.2

Add $- 16 z$ and $11 z$ .

$7 - 5 z = 17$

$y = - 1 - 2 z$

$x = 5 + 2 y + 4 z$

$7 - 5 z = 17$

$y = - 1 - 2 z$

$x = 5 + 2 y + 4 z$

$7 - 5 z = 17$

$y = - 1 - 2 z$

$x = 5 + 2 y + 4 z$

$7 - 5 z = 17$

$y = - 1 - 2 z$

$x = 5 + 2 y + 4 z$

Step 4.3

Replace all occurrences of $y$ in $x = 5 + 2 y + 4 z$ with $- 1 - 2 z$ .

$x = 5 + 2 (- 1 - 2 z) + 4 z$

$7 - 5 z = 17$

$y = - 1 - 2 z$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $5 + 2 (- 1 - 2 z) + 4 z$ .

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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.

$x = 5 + 2 \cdot - 1 + 2 (- 2 z) + 4 z$

$7 - 5 z = 17$

$y = - 1 - 2 z$

Step 4.4.1.1.2

Multiply $2$ by $- 1$ .

$x = 5 - 2 + 2 (- 2 z) + 4 z$

$7 - 5 z = 17$

$y = - 1 - 2 z$

Step 4.4.1.1.3

Multiply $- 2$ by $2$ .

$x = 5 - 2 - 4 z + 4 z$

$7 - 5 z = 17$

$y = - 1 - 2 z$

$x = 5 - 2 - 4 z + 4 z$

$7 - 5 z = 17$

$y = - 1 - 2 z$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Combine the opposite terms in $5 - 2 - 4 z + 4 z$ .

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Step 4.4.1.2.1.1

Add $- 4 z$ and $4 z$ .

$x = 5 - 2 + 0$

$7 - 5 z = 17$

$y = - 1 - 2 z$

Step 4.4.1.2.1.2

Add $5 - 2$ and $0$ .

$x = 5 - 2$

$7 - 5 z = 17$

$y = - 1 - 2 z$

$x = 5 - 2$

$7 - 5 z = 17$

$y = - 1 - 2 z$

Step 4.4.1.2.2

Subtract $2$ from $5$ .

$x = 3$

$7 - 5 z = 17$

$y = - 1 - 2 z$

$x = 3$

$7 - 5 z = 17$

$y = - 1 - 2 z$

$x = 3$

$7 - 5 z = 17$

$y = - 1 - 2 z$

$x = 3$

$7 - 5 z = 17$

$y = - 1 - 2 z$

$x = 3$

$7 - 5 z = 17$

$y = - 1 - 2 z$

Step 5

Solve for $z$ in $7 - 5 z = 17$ .

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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

Subtract $7$ from both sides of the equation.

$- 5 z = 17 - 7$

$x = 3$

$y = - 1 - 2 z$

Step 5.1.2

Subtract $7$ from $17$ .

$- 5 z = 10$

$x = 3$

$y = - 1 - 2 z$

$- 5 z = 10$

$x = 3$

$y = - 1 - 2 z$

Step 5.2

Divide each term in $- 5 z = 10$ by $- 5$ and simplify.

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Step 5.2.1

Divide each term in $- 5 z = 10$ by $- 5$ .

$\frac{- 5 z}{- 5} = \frac{10}{- 5}$

$x = 3$

$y = - 1 - 2 z$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 5$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 5 z}{- 5} = \frac{10}{- 5}$

$x = 3$

$y = - 1 - 2 z$

Step 5.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{10}{- 5}$

$x = 3$

$y = - 1 - 2 z$

$z = \frac{10}{- 5}$

$x = 3$

$y = - 1 - 2 z$

$z = \frac{10}{- 5}$

$x = 3$

$y = - 1 - 2 z$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $10$ by $- 5$ .

$z = - 2$

$x = 3$

$y = - 1 - 2 z$

$z = - 2$

$x = 3$

$y = - 1 - 2 z$

$z = - 2$

$x = 3$

$y = - 1 - 2 z$

$z = - 2$

$x = 3$

$y = - 1 - 2 z$

Step 6

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

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Step 6.1

Replace all occurrences of $z$ in $y = - 1 - 2 z$ with $- 2$ .

$y = - 1 - 2 \cdot - 2$

$z = - 2$

$x = 3$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- 1 - 2 \cdot - 2$ .

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Step 6.2.1.1

Multiply $- 2$ by $- 2$ .

$y = - 1 + 4$

$z = - 2$

$x = 3$

Step 6.2.1.2

Add $- 1$ and $4$ .

$y = 3$

$z = - 2$

$x = 3$

$y = 3$

$z = - 2$

$x = 3$

$y = 3$

$z = - 2$

$x = 3$

$y = 3$

$z = - 2$

$x = 3$

Step 7

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

$(3, 3, - 2)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(3, 3, - 2)$

Equation Form:

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