Algebra Examples

$x + 2 y - z = 3$ $- 2 x - y + z = - 1$ $6 x - 3 y - z = - 7$

Step 1

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

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Step 1.1

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

$x - z = 3 - 2 y$

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

$6 x - 3 y - z = - 7$

Step 1.2

Add $z$ to both sides of the equation.

$x = 3 - 2 y + z$

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

$6 x - 3 y - z = - 7$

$x = 3 - 2 y + z$

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

$6 x - 3 y - z = - 7$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $- 2 x - y + z = - 1$ with $3 - 2 y + z$ .

$- 2 (3 - 2 y + z) - y + z = - 1$

$x = 3 - 2 y + z$

$6 x - 3 y - z = - 7$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

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

$x = 3 - 2 y + z$

$6 x - 3 y - z = - 7$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 2$ by $3$ .

$- 6 - 2 (- 2 y) - 2 z - y + z = - 1$

$x = 3 - 2 y + z$

$6 x - 3 y - z = - 7$

Step 2.2.1.1.2.2

Multiply $- 2$ by $- 2$ .

$- 6 + 4 y - 2 z - y + z = - 1$

$x = 3 - 2 y + z$

$6 x - 3 y - z = - 7$

$- 6 + 4 y - 2 z - y + z = - 1$

$x = 3 - 2 y + z$

$6 x - 3 y - z = - 7$

$- 6 + 4 y - 2 z - y + z = - 1$

$x = 3 - 2 y + z$

$6 x - 3 y - z = - 7$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $y$ from $4 y$ .

$- 6 + 3 y - 2 z + z = - 1$

$x = 3 - 2 y + z$

$6 x - 3 y - z = - 7$

Step 2.2.1.2.2

Add $- 2 z$ and $z$ .

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$6 x - 3 y - z = - 7$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$6 x - 3 y - z = - 7$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$6 x - 3 y - z = - 7$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$6 x - 3 y - z = - 7$

Step 2.3

Replace all occurrences of $x$ in $6 x - 3 y - z = - 7$ with $3 - 2 y + z$ .

$6 (3 - 2 y + z) - 3 y - z = - 7$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $6 (3 - 2 y + z) - 3 y - z$ .

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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.

$6 \cdot 3 + 6 (- 2 y) + 6 z - 3 y - z = - 7$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $6$ by $3$ .

$18 + 6 (- 2 y) + 6 z - 3 y - z = - 7$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

Step 2.4.1.1.2.2

Multiply $- 2$ by $6$ .

$18 - 12 y + 6 z - 3 y - z = - 7$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$18 - 12 y + 6 z - 3 y - z = - 7$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$18 - 12 y + 6 z - 3 y - z = - 7$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $3 y$ from $- 12 y$ .

$18 - 15 y + 6 z - z = - 7$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

Step 2.4.1.2.2

Subtract $z$ from $6 z$ .

$18 - 15 y + 5 z = - 7$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$18 - 15 y + 5 z = - 7$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$18 - 15 y + 5 z = - 7$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$18 - 15 y + 5 z = - 7$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$18 - 15 y + 5 z = - 7$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

Step 3

Solve for $y$ in $18 - 15 y + 5 z = - 7$ .

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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 $18$ from both sides of the equation.

$- 15 y + 5 z = - 7 - 18$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

Step 3.1.2

Subtract $5 z$ from both sides of the equation.

$- 15 y = - 7 - 18 - 5 z$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

Step 3.1.3

Subtract $18$ from $- 7$ .

$- 15 y = - 25 - 5 z$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$- 15 y = - 25 - 5 z$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

Step 3.2

Divide each term in $- 15 y = - 25 - 5 z$ by $- 15$ and simplify.

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Step 3.2.1

Divide each term in $- 15 y = - 25 - 5 z$ by $- 15$ .

$\frac{- 15 y}{- 15} = \frac{- 25}{- 15} + \frac{- 5 z}{- 15}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 15$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 15 y}{- 15} = \frac{- 25}{- 15} + \frac{- 5 z}{- 15}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 25}{- 15} + \frac{- 5 z}{- 15}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$y = \frac{- 25}{- 15} + \frac{- 5 z}{- 15}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$y = \frac{- 25}{- 15} + \frac{- 5 z}{- 15}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

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

Cancel the common factor of $- 25$ and $- 15$ .

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Step 3.2.3.1.1.1

Factor $- 5$ out of $- 25$ .

$y = \frac{- 5 \cdot 5}{- 15} + \frac{- 5 z}{- 15}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

Step 3.2.3.1.1.2

Cancel the common factors.

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Step 3.2.3.1.1.2.1

Factor $- 5$ out of $- 15$ .

$y = \frac{- 5 \cdot 5}{- 5 \cdot 3} + \frac{- 5 z}{- 15}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

Step 3.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{- 5 \cdot 5}{- 5 \cdot 3} + \frac{- 5 z}{- 15}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

Step 3.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{5}{3} + \frac{- 5 z}{- 15}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$y = \frac{5}{3} + \frac{- 5 z}{- 15}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$y = \frac{5}{3} + \frac{- 5 z}{- 15}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

Step 3.2.3.1.2

Cancel the common factor of $- 5$ and $- 15$ .

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Step 3.2.3.1.2.1

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

$y = \frac{5}{3} + \frac{- 5 z}{- 15}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

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 $- 15$ .

$y = \frac{5}{3} + \frac{- 5 z}{- 5 \cdot 3}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

Step 3.2.3.1.2.2.2

Cancel the common factor.

$y = \frac{5}{3} + \frac{- 5 z}{- 5 \cdot 3}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

Step 3.2.3.1.2.2.3

Rewrite the expression.

$y = \frac{5}{3} + \frac{z}{3}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$y = \frac{5}{3} + \frac{z}{3}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$y = \frac{5}{3} + \frac{z}{3}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$y = \frac{5}{3} + \frac{z}{3}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$y = \frac{5}{3} + \frac{z}{3}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$y = \frac{5}{3} + \frac{z}{3}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

$y = \frac{5}{3} + \frac{z}{3}$

$- 6 + 3 y - z = - 1$

$x = 3 - 2 y + z$

Step 4

Replace all occurrences of $y$ with $\frac{5}{3} + \frac{z}{3}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $- 6 + 3 y - z = - 1$ with $\frac{5}{3} + \frac{z}{3}$ .

$- 6 + 3 (\frac{5}{3} + \frac{z}{3}) - z = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

$x = 3 - 2 y + z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 6 + 3 (\frac{5}{3} + \frac{z}{3}) - z$ .

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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.

$- 6 + 3 (\frac{5}{3}) + 3 (\frac{z}{3}) - z = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

$x = 3 - 2 y + z$

Step 4.2.1.1.2

Cancel the common factor of $3$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

$- 6 + 3 (\frac{5}{3}) + 3 (\frac{z}{3}) - z = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

$x = 3 - 2 y + z$

Step 4.2.1.1.2.2

Rewrite the expression.

$- 6 + 5 + 3 (\frac{z}{3}) - z = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

$x = 3 - 2 y + z$

$- 6 + 5 + 3 (\frac{z}{3}) - z = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

$x = 3 - 2 y + z$

Step 4.2.1.1.3

Cancel the common factor of $3$ .

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Step 4.2.1.1.3.1

Cancel the common factor.

$- 6 + 5 + 3 (\frac{z}{3}) - z = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

$x = 3 - 2 y + z$

Step 4.2.1.1.3.2

Rewrite the expression.

$- 6 + 5 + z - z = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

$x = 3 - 2 y + z$

$- 6 + 5 + z - z = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

$x = 3 - 2 y + z$

$- 6 + 5 + z - z = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

$x = 3 - 2 y + z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Combine the opposite terms in $- 6 + 5 + z - z$ .

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Step 4.2.1.2.1.1

Subtract $z$ from $z$ .

$- 6 + 5 + 0 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

$x = 3 - 2 y + z$

Step 4.2.1.2.1.2

Add $- 6 + 5$ and $0$ .

$- 6 + 5 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

$x = 3 - 2 y + z$

$- 6 + 5 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

$x = 3 - 2 y + z$

Step 4.2.1.2.2

Add $- 6$ and $5$ .

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

$x = 3 - 2 y + z$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

$x = 3 - 2 y + z$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

$x = 3 - 2 y + z$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

$x = 3 - 2 y + z$

Step 4.3

Replace all occurrences of $y$ in $x = 3 - 2 y + z$ with $\frac{5}{3} + \frac{z}{3}$ .

$x = 3 - 2 (\frac{5}{3} + \frac{z}{3}) + z$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $3 - 2 (\frac{5}{3} + \frac{z}{3}) + z$ .

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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.

$x = 3 - 2 (\frac{5}{3}) - 2 \frac{z}{3} + z$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.1.2

Multiply $- 2 (\frac{5}{3})$ .

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Step 4.4.1.1.2.1

Combine $- 2$ and $\frac{5}{3}$ .

$x = 3 + \frac{- 2 \cdot 5}{3} - 2 \frac{z}{3} + z$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.1.2.2

Multiply $- 2$ by $5$ .

$x = 3 + \frac{- 10}{3} - 2 \frac{z}{3} + z$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

$x = 3 + \frac{- 10}{3} - 2 \frac{z}{3} + z$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.1.3

Combine $- 2$ and $\frac{z}{3}$ .

$x = 3 + \frac{- 10}{3} + \frac{- 2 z}{3} + z$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.1.4

Simplify each term.

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Step 4.4.1.1.4.1

Move the negative in front of the fraction.

$x = 3 - \frac{10}{3} + \frac{- 2 z}{3} + z$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.1.4.2

Move the negative in front of the fraction.

$x = 3 - \frac{10}{3} - \frac{2 z}{3} + z$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

$x = 3 - \frac{10}{3} - \frac{2 z}{3} + z$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

$x = 3 - \frac{10}{3} - \frac{2 z}{3} + z$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = 3 \cdot \frac{3}{3} - \frac{10}{3} - \frac{2 z}{3} + z$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.3

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

$x = \frac{3 \cdot 3}{3} - \frac{10}{3} - \frac{2 z}{3} + z$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.4

Combine the numerators over the common denominator.

$x = \frac{3 \cdot 3 - 10}{3} - \frac{2 z}{3} + z$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.5

Simplify the numerator.

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Step 4.4.1.5.1

Multiply $3$ by $3$ .

$x = \frac{9 - 10}{3} - \frac{2 z}{3} + z$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.5.2

Subtract $10$ from $9$ .

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

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

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

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.6

Move the negative in front of the fraction.

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

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.7

To write $z$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = - \frac{1}{3} - \frac{2 z}{3} + z \cdot \frac{3}{3}$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.8

Simplify terms.

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Step 4.4.1.8.1

Combine $z$ and $\frac{3}{3}$ .

$x = - \frac{1}{3} - \frac{2 z}{3} + \frac{z \cdot 3}{3}$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.8.2

Combine the numerators over the common denominator.

$x = - \frac{1}{3} + \frac{- 2 z + z \cdot 3}{3}$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.8.3

Combine the numerators over the common denominator.

$x = \frac{- 1 - 2 z + z \cdot 3}{3}$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

$x = \frac{- 1 - 2 z + z \cdot 3}{3}$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.9

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

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

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.10

Simplify terms.

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Step 4.4.1.10.1

Add $- 2 z$ and $3 z$ .

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

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.10.2

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 + z}{3}$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.10.3

Factor $- 1$ out of $z$ .

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

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.10.4

Factor $- 1$ out of $- 1 (1) - 1 (- z)$ .

$x = \frac{- 1 (1 - z)}{3}$

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 4.4.1.10.5

Move the negative in front of the fraction.

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

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

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

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

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

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

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

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

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

$- 1 = - 1$

$y = \frac{5}{3} + \frac{z}{3}$

Step 5

Remove any equations from the system that are always true.

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

$y = \frac{5}{3} + \frac{z}{3}$