Algebra Examples

$2 x - y + 3 z = - 5$ $- x + y - z = 1$ $4 x - 3 y + 5 z = - 5$

Step 1

Solve the equation for $x$ .

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Step 1.1

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

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Step 1.1.1

Add $y$ to both sides of the equation.

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

$- x + y - z = 1$

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

Step 1.1.2

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

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

$- x + y - z = 1$

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

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

$- x + y - z = 1$

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

Step 1.2

Divide each term in $2 x = - 5 + y - 3 z$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 x = - 5 + y - 3 z$ by $2$ .

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

$- x + y - z = 1$

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

$- x + y - z = 1$

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

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$- x + y - z = 1$

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

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

$- x + y - z = 1$

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

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

$- x + y - z = 1$

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

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Move the negative in front of the fraction.

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

$- x + y - z = 1$

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

Step 1.2.3.1.2

Move the negative in front of the fraction.

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

$- x + y - z = 1$

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

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

$- x + y - z = 1$

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

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

$- x + y - z = 1$

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

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

$- x + y - z = 1$

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

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

$- x + y - z = 1$

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

Step 2

Solve the equation for $y$ .

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Step 2.1

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

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

$\frac{5}{2} - \frac{y}{2} + \frac{3 z}{2} + y - z = 1$

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

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

Step 2.1.1.2

Simplify.

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Step 2.1.1.2.1

Multiply $- - \frac{5}{2}$ .

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Step 2.1.1.2.1.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{5}{2}) - \frac{y}{2} + \frac{3 z}{2} + y - z = 1$

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

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

Step 2.1.1.2.1.2

Multiply $\frac{5}{2}$ by $1$ .

$\frac{5}{2} - \frac{y}{2} + \frac{3 z}{2} + y - z = 1$

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

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

$\frac{5}{2} - \frac{y}{2} + \frac{3 z}{2} + y - z = 1$

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

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

Step 2.1.1.2.2

Multiply $- - \frac{3 z}{2}$ .

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Step 2.1.1.2.2.1

Multiply $- 1$ by $- 1$ .

$\frac{5}{2} - \frac{y}{2} + 1 (\frac{3 z}{2}) + y - z = 1$

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

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

Step 2.1.1.2.2.2

Multiply $\frac{3 z}{2}$ by $1$ .

$\frac{5}{2} - \frac{y}{2} + \frac{3 z}{2} + y - z = 1$

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

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

$\frac{5}{2} - \frac{y}{2} + \frac{3 z}{2} + y - z = 1$

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

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

$\frac{5}{2} - \frac{y}{2} + \frac{3 z}{2} + y - z = 1$

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

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

$\frac{5}{2} - \frac{y}{2} + \frac{3 z}{2} + y - z = 1$

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

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

Step 2.1.2

To write $y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5}{2} + \frac{3 z}{2} - \frac{y}{2} + y \cdot \frac{2}{2} - z = 1$

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

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

Step 2.1.3

Simplify terms.

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Step 2.1.3.1

Combine $y$ and $\frac{2}{2}$ .

$\frac{5}{2} + \frac{3 z}{2} - \frac{y}{2} + \frac{y \cdot 2}{2} - z = 1$

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

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

Step 2.1.3.2

Combine the numerators over the common denominator.

$\frac{5}{2} + \frac{3 z}{2} + \frac{- y + y \cdot 2}{2} - z = 1$

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

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

Step 2.1.3.3

Combine the numerators over the common denominator.

$- z + \frac{5 + 3 z - y + y \cdot 2}{2} = 1$

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

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

$- z + \frac{5 + 3 z - y + y \cdot 2}{2} = 1$

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

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

Step 2.1.4

Move $2$ to the left of $y$ .

$- z + \frac{5 + 3 z - y + 2 y}{2} = 1$

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

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

Step 2.1.5

Add $- y$ and $2 y$ .

$- z + \frac{5 + 3 z + y}{2} = 1$

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

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

Step 2.1.6

To write $- z$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- z \cdot \frac{2}{2} + \frac{5 + 3 z + y}{2} = 1$

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

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

Step 2.1.7

Simplify terms.

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Step 2.1.7.1

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

$\frac{- z \cdot 2}{2} + \frac{5 + 3 z + y}{2} = 1$

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

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

Step 2.1.7.2

Combine the numerators over the common denominator.

$\frac{- z \cdot 2 + 5 + 3 z + y}{2} = 1$

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

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

$\frac{- z \cdot 2 + 5 + 3 z + y}{2} = 1$

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

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

Step 2.1.8

Simplify the numerator.

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Step 2.1.8.1

Multiply $2$ by $- 1$ .

$\frac{- 2 z + 5 + 3 z + y}{2} = 1$

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

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

Step 2.1.8.2

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

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

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

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

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

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

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

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

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

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

Step 2.2

Multiply both sides of the equation by $2$ .

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

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

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

Step 2.3

Simplify both sides of the equation.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Cancel the common factor of $2$ .

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Step 2.3.1.1.1

Cancel the common factor.

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

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

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

Step 2.3.1.1.2

Rewrite the expression.

$z + 5 + y = 2 \cdot 1$

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

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

$z + 5 + y = 2 \cdot 1$

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

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

$z + 5 + y = 2 \cdot 1$

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

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

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $2$ by $1$ .

$z + 5 + y = 2$

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

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

$z + 5 + y = 2$

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

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

$z + 5 + y = 2$

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

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

Step 2.4

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

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Step 2.4.1

Subtract $z$ from both sides of the equation.

$5 + y = 2 - z$

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

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

Step 2.4.2

Subtract $5$ from both sides of the equation.

$y = 2 - z - 5$

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

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

Step 2.4.3

Subtract $5$ from $2$ .

$y = - z - 3$

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

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

$y = - z - 3$

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

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

$y = - z - 3$

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

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

Step 3

Simplify the right side.

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Step 3.1

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

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Step 3.1.1

Combine the numerators over the common denominator.

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

$y = - z - 3$

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

Step 3.1.2

Subtract $3$ from $- 5$ .

$x = \frac{- z - 8 - 3 z}{2}$

$y = - z - 3$

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

Step 3.1.3

Subtract $3 z$ from $- z$ .

$x = \frac{- 4 z - 8}{2}$

$y = - z - 3$

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

Step 3.1.4

Cancel the common factor of $- 4 z - 8$ and $2$ .

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Step 3.1.4.1

Factor $2$ out of $- 4 z$ .

$x = \frac{2 (- 2 z) - 8}{2}$

$y = - z - 3$

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

Step 3.1.4.2

Factor $2$ out of $- 8$ .

$x = \frac{2 (- 2 z) + 2 \cdot - 4}{2}$

$y = - z - 3$

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

Step 3.1.4.3

Factor $2$ out of $2 (- 2 z) + 2 (- 4)$ .

$x = \frac{2 (- 2 z - 4)}{2}$

$y = - z - 3$

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

Step 3.1.4.4

Cancel the common factors.

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Step 3.1.4.4.1

Factor $2$ out of $2$ .

$x = \frac{2 (- 2 z - 4)}{2 (1)}$

$y = - z - 3$

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

Step 3.1.4.4.2

Cancel the common factor.

$x = \frac{2 (- 2 z - 4)}{2 \cdot 1}$

$y = - z - 3$

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

Step 3.1.4.4.3

Rewrite the expression.

$x = \frac{- 2 z - 4}{1}$

$y = - z - 3$

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

Step 3.1.4.4.4

Divide $- 2 z - 4$ by $1$ .

$x = - 2 z - 4$

$y = - z - 3$

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

$x = - 2 z - 4$

$y = - z - 3$

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

$x = - 2 z - 4$

$y = - z - 3$

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

$x = - 2 z - 4$

$y = - z - 3$

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

$x = - 2 z - 4$

$y = - z - 3$

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