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 $y$ .

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Step 1.1

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

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Step 1.1.1

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

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

$- x + y - z = 1$

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

Step 1.1.2

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

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

$- x + y - z = 1$

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

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

$- x + y - z = 1$

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

Step 1.2

Divide each term in $- y = - 5 - 2 x - 3 z$ by $- 1$ and simplify.

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Step 1.2.1

Divide each term in $- y = - 5 - 2 x - 3 z$ by $- 1$ .

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

$- x + y - z = 1$

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Dividing two negative values results in a positive value.

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

$- x + y - z = 1$

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

Step 1.2.2.2

Divide $y$ by $1$ .

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

$- x + y - z = 1$

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

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

$- x + y - z = 1$

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

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

Divide $- 5$ by $- 1$ .

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

$- x + y - z = 1$

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

Step 1.2.3.1.2

Move the negative one from the denominator of $\frac{- 2 x}{- 1}$ .

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

$- x + y - z = 1$

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

Step 1.2.3.1.3

Rewrite $- 1 \cdot (- 2 x)$ as $- (- 2 x)$ .

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

$- x + y - z = 1$

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

Step 1.2.3.1.4

Multiply $- 2$ by $- 1$ .

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

$- x + y - z = 1$

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

Step 1.2.3.1.5

Move the negative one from the denominator of $\frac{- 3 z}{- 1}$ .

$y = 5 + 2 x - 1 \cdot (- 3 z)$

$- x + y - z = 1$

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

Step 1.2.3.1.6

Rewrite $- 1 \cdot (- 3 z)$ as $- (- 3 z)$ .

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

$- x + y - z = 1$

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

Step 1.2.3.1.7

Multiply $- 3$ by $- 1$ .

$y = 5 + 2 x + 3 z$

$- x + y - z = 1$

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

$y = 5 + 2 x + 3 z$

$- x + y - z = 1$

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

$y = 5 + 2 x + 3 z$

$- x + y - z = 1$

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

$y = 5 + 2 x + 3 z$

$- x + y - z = 1$

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

$y = 5 + 2 x + 3 z$

$- x + y - z = 1$

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

Step 2

Solve the equation for $z$ .

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Step 2.1

Simplify $- x + 5 + 2 x + 3 z - z$ .

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Step 2.1.1

Add $- x$ and $2 x$ .

$x + 5 + 3 z - z = 1$

$y = 5 + 2 x + 3 z$

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

Step 2.1.2

Subtract $z$ from $3 z$ .

$x + 5 + 2 z = 1$

$y = 5 + 2 x + 3 z$

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

$x + 5 + 2 z = 1$

$y = 5 + 2 x + 3 z$

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

Step 2.2

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

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Step 2.2.1

Subtract $x$ from both sides of the equation.

$5 + 2 z = 1 - x$

$y = 5 + 2 x + 3 z$

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

Step 2.2.2

Subtract $5$ from both sides of the equation.

$2 z = 1 - x - 5$

$y = 5 + 2 x + 3 z$

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

Step 2.2.3

Subtract $5$ from $1$ .

$2 z = - x - 4$

$y = 5 + 2 x + 3 z$

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

$2 z = - x - 4$

$y = 5 + 2 x + 3 z$

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

Step 2.3

Divide each term in $2 z = - x - 4$ by $2$ and simplify.

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Step 2.3.1

Divide each term in $2 z = - x - 4$ by $2$ .

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

$y = 5 + 2 x + 3 z$

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1

Cancel the common factor.

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

$y = 5 + 2 x + 3 z$

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

Step 2.3.2.1.2

Divide $z$ by $1$ .

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

$y = 5 + 2 x + 3 z$

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

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

$y = 5 + 2 x + 3 z$

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

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

$y = 5 + 2 x + 3 z$

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

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Move the negative in front of the fraction.

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

$y = 5 + 2 x + 3 z$

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

Step 2.3.3.1.2

Divide $- 4$ by $2$ .

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

$y = 5 + 2 x + 3 z$

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

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

$y = 5 + 2 x + 3 z$

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

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

$y = 5 + 2 x + 3 z$

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

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

$y = 5 + 2 x + 3 z$

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

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

$y = 5 + 2 x + 3 z$

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

Step 3

Simplify the right side.

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Step 3.1

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

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Apply the distributive property.

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

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

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

Step 3.1.1.2

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

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Step 3.1.1.2.1

Multiply $- 1$ by $3$ .

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

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

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

Step 3.1.1.2.2

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

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

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

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

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

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

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

Step 3.1.1.3

Multiply $3$ by $- 2$ .

$y = 5 + 2 x + \frac{- 3 x}{2} - 6$

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

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

Step 3.1.1.4

Move the negative in front of the fraction.

$y = 5 + 2 x - \frac{3 x}{2} - 6$

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

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

$y = 5 + 2 x - \frac{3 x}{2} - 6$

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

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

Step 3.1.2

Subtract $6$ from $5$ .

$y = 2 x - \frac{3 x}{2} - 1$

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

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

Step 3.1.3

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

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

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

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

Step 3.1.4

Simplify terms.

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Step 3.1.4.1

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

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

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

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

Step 3.1.4.2

Combine the numerators over the common denominator.

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

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

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

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

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

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

Step 3.1.5

Simplify each term.

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Step 3.1.5.1

Simplify the numerator.

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Step 3.1.5.1.1

Factor $x$ out of $2 x \cdot 2 - 3 x$ .

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Step 3.1.5.1.1.1

Factor $x$ out of $2 x \cdot 2$ .

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

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

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

Step 3.1.5.1.1.2

Factor $x$ out of $- 3 x$ .

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

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

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

Step 3.1.5.1.1.3

Factor $x$ out of $x (2 \cdot 2) + x \cdot - 3$ .

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

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

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

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

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

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

Step 3.1.5.1.2

Multiply $2$ by $2$ .

$y = \frac{x (4 - 3)}{2} - 1$

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

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

Step 3.1.5.1.3

Subtract $3$ from $4$ .

$y = \frac{x \cdot 1}{2} - 1$

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

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

$y = \frac{x \cdot 1}{2} - 1$

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

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

Step 3.1.5.2

Multiply $x$ by $1$ .

$y = \frac{x}{2} - 1$

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

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

$y = \frac{x}{2} - 1$

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

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

$y = \frac{x}{2} - 1$

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

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

$y = \frac{x}{2} - 1$

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

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