Solve in Terms of the Arbitrary Variable x

$x + y - z = 3$ , $2 x - 8 y + 13 z = 1$

Step 1

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

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Step 1.1

Subtract $y$ from both sides of the equation.

$x - z = 3 - y$

$2 x - 8 y + 13 z = 1$

Step 1.2

Add $z$ to both sides of the equation.

$x = 3 - y + z$

$2 x - 8 y + 13 z = 1$

$x = 3 - y + z$

$2 x - 8 y + 13 z = 1$

Step 2

Solve the equation for $y$ .

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Step 2.1

Simplify $2 (3 - y + z) - 8 y + 13 z$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

$2 \cdot 3 + 2 (- y) + 2 z - 8 y + 13 z = 1$

$x = 3 - y + z$

Step 2.1.1.2

Simplify.

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Step 2.1.1.2.1

Multiply $2$ by $3$ .

$6 + 2 (- y) + 2 z - 8 y + 13 z = 1$

$x = 3 - y + z$

Step 2.1.1.2.2

Multiply $- 1$ by $2$ .

$6 - 2 y + 2 z - 8 y + 13 z = 1$

$x = 3 - y + z$

$6 - 2 y + 2 z - 8 y + 13 z = 1$

$x = 3 - y + z$

$6 - 2 y + 2 z - 8 y + 13 z = 1$

$x = 3 - y + z$

Step 2.1.2

Simplify by adding terms.

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Step 2.1.2.1

Subtract $8 y$ from $- 2 y$ .

$6 - 10 y + 2 z + 13 z = 1$

$x = 3 - y + z$

Step 2.1.2.2

Add $2 z$ and $13 z$ .

$6 - 10 y + 15 z = 1$

$x = 3 - y + z$

$6 - 10 y + 15 z = 1$

$x = 3 - y + z$

$6 - 10 y + 15 z = 1$

$x = 3 - y + z$

Step 2.2

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

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Step 2.2.1

Subtract $6$ from both sides of the equation.

$- 10 y + 15 z = 1 - 6$

$x = 3 - y + z$

Step 2.2.2

Subtract $15 z$ from both sides of the equation.

$- 10 y = 1 - 6 - 15 z$

$x = 3 - y + z$

Step 2.2.3

Subtract $6$ from $1$ .

$- 10 y = - 5 - 15 z$

$x = 3 - y + z$

$- 10 y = - 5 - 15 z$

$x = 3 - y + z$

Step 2.3

Divide each term in $- 10 y = - 5 - 15 z$ by $- 10$ and simplify.

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Step 2.3.1

Divide each term in $- 10 y = - 5 - 15 z$ by $- 10$ .

$\frac{- 10 y}{- 10} = \frac{- 5}{- 10} + \frac{- 15 z}{- 10}$

$x = 3 - y + z$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 10$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 10 y}{- 10} = \frac{- 5}{- 10} + \frac{- 15 z}{- 10}$

$x = 3 - y + z$

Step 2.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 5}{- 10} + \frac{- 15 z}{- 10}$

$x = 3 - y + z$

$y = \frac{- 5}{- 10} + \frac{- 15 z}{- 10}$

$x = 3 - y + z$

$y = \frac{- 5}{- 10} + \frac{- 15 z}{- 10}$

$x = 3 - y + z$

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

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

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Step 2.3.3.1.1.1

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

$y = \frac{- 5 \cdot 1}{- 10} + \frac{- 15 z}{- 10}$

$x = 3 - y + z$

Step 2.3.3.1.1.2

Cancel the common factors.

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Step 2.3.3.1.1.2.1

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

$y = \frac{- 5 \cdot 1}{- 5 \cdot 2} + \frac{- 15 z}{- 10}$

$x = 3 - y + z$

Step 2.3.3.1.1.2.2

Cancel the common factor.

$y = \frac{- 5 \cdot 1}{- 5 \cdot 2} + \frac{- 15 z}{- 10}$

$x = 3 - y + z$

Step 2.3.3.1.1.2.3

Rewrite the expression.

$y = \frac{1}{2} + \frac{- 15 z}{- 10}$

$x = 3 - y + z$

$y = \frac{1}{2} + \frac{- 15 z}{- 10}$

$x = 3 - y + z$

$y = \frac{1}{2} + \frac{- 15 z}{- 10}$

$x = 3 - y + z$

Step 2.3.3.1.2

Cancel the common factor of $- 15$ and $- 10$ .

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Step 2.3.3.1.2.1

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

$y = \frac{1}{2} + \frac{- 5 (3 z)}{- 10}$

$x = 3 - y + z$

Step 2.3.3.1.2.2

Cancel the common factors.

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Step 2.3.3.1.2.2.1

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

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

$x = 3 - y + z$

Step 2.3.3.1.2.2.2

Cancel the common factor.

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

$x = 3 - y + z$

Step 2.3.3.1.2.2.3

Rewrite the expression.

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

$x = 3 - y + z$

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

$x = 3 - y + z$

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

$x = 3 - y + z$

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

$x = 3 - y + z$

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

$x = 3 - y + z$

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

$x = 3 - y + z$

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

$x = 3 - y + z$

Step 3

Simplify the right side.

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Step 3.1

Simplify $3 - (\frac{1}{2} + \frac{3 z}{2}) + z$ .

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Step 3.1.1

Apply the distributive property.

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

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

Step 3.1.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

Step 3.1.3

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

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

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

Step 3.1.4

Combine the numerators over the common denominator.

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

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

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $3$ by $2$ .

$x = \frac{6 - 1}{2} - \frac{3 z}{2} + z$

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

Step 3.1.5.2

Subtract $1$ from $6$ .

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

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

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

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

Step 3.1.6

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

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

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

Step 3.1.7

Simplify terms.

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Step 3.1.7.1

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

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

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

Step 3.1.7.2

Combine the numerators over the common denominator.

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

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

Step 3.1.7.3

Combine the numerators over the common denominator.

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

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

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

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

Step 3.1.8

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

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

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

Step 3.1.9

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

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

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

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

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

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

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

Step 4

Simplify the right side.

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Step 4.1

Reorder $\frac{1}{2}$ and $\frac{3 z}{2}$ .

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

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

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

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

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