Algebra Examples

$x - y + z = - 9$ , $4 x + y + z = 10$

Step 1

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

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Step 1.1

Add $y$ to both sides of the equation.

$x + z = - 9 + y$

$4 x + y + z = 10$

Step 1.2

Subtract $z$ from both sides of the equation.

$x = - 9 + y - z$

$4 x + y + z = 10$

$x = - 9 + y - z$

$4 x + y + z = 10$

Step 2

Solve the equation for $y$ .

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Step 2.1

Simplify $4 (- 9 + y - z) + y + z$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

$4 \cdot - 9 + 4 y + 4 (- z) + y + z = 10$

$x = - 9 + y - z$

Step 2.1.1.2

Simplify.

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Step 2.1.1.2.1

Multiply $4$ by $- 9$ .

$- 36 + 4 y + 4 (- z) + y + z = 10$

$x = - 9 + y - z$

Step 2.1.1.2.2

Multiply $- 1$ by $4$ .

$- 36 + 4 y - 4 z + y + z = 10$

$x = - 9 + y - z$

$- 36 + 4 y - 4 z + y + z = 10$

$x = - 9 + y - z$

$- 36 + 4 y - 4 z + y + z = 10$

$x = - 9 + y - z$

Step 2.1.2

Simplify by adding terms.

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Step 2.1.2.1

Add $4 y$ and $y$ .

$- 36 + 5 y - 4 z + z = 10$

$x = - 9 + y - z$

Step 2.1.2.2

Add $- 4 z$ and $z$ .

$- 36 + 5 y - 3 z = 10$

$x = - 9 + y - z$

$- 36 + 5 y - 3 z = 10$

$x = - 9 + y - z$

$- 36 + 5 y - 3 z = 10$

$x = - 9 + y - z$

Step 2.2

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

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Step 2.2.1

Add $36$ to both sides of the equation.

$5 y - 3 z = 10 + 36$

$x = - 9 + y - z$

Step 2.2.2

Add $3 z$ to both sides of the equation.

$5 y = 10 + 36 + 3 z$

$x = - 9 + y - z$

Step 2.2.3

Add $10$ and $36$ .

$5 y = 46 + 3 z$

$x = - 9 + y - z$

$5 y = 46 + 3 z$

$x = - 9 + y - z$

Step 2.3

Divide each term in $5 y = 46 + 3 z$ by $5$ and simplify.

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Step 2.3.1

Divide each term in $5 y = 46 + 3 z$ by $5$ .

$\frac{5 y}{5} = \frac{46}{5} + \frac{3 z}{5}$

$x = - 9 + y - z$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $5$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{46}{5} + \frac{3 z}{5}$

$x = - 9 + y - z$

Step 2.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{46}{5} + \frac{3 z}{5}$

$x = - 9 + y - z$

$y = \frac{46}{5} + \frac{3 z}{5}$

$x = - 9 + y - z$

$y = \frac{46}{5} + \frac{3 z}{5}$

$x = - 9 + y - z$

$y = \frac{46}{5} + \frac{3 z}{5}$

$x = - 9 + y - z$

$y = \frac{46}{5} + \frac{3 z}{5}$

$x = - 9 + y - z$

Step 3

Substitute the calculated values for $y$ into the equation for $x$ .

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Step 3.1

Simplify the left side.

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Step 3.1.1

Remove parentheses.

$x = - 9 + \frac{46}{5} + \frac{3 z}{5} - z$

$y = \frac{46}{5} + \frac{3 z}{5}$

$x = - 9 + \frac{46}{5} + \frac{3 z}{5} - z$

$y = \frac{46}{5} + \frac{3 z}{5}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $- 9 + \frac{46}{5} + \frac{3 z}{5} - z$ .

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Step 3.2.1.1

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x = - 9 \cdot \frac{5}{5} + \frac{46}{5} + \frac{3 z}{5} - z$

$y = \frac{46}{5} + \frac{3 z}{5}$

Step 3.2.1.2

Combine $- 9$ and $\frac{5}{5}$ .

$x = \frac{- 9 \cdot 5}{5} + \frac{46}{5} + \frac{3 z}{5} - z$

$y = \frac{46}{5} + \frac{3 z}{5}$

Step 3.2.1.3

Combine the numerators over the common denominator.

$x = \frac{- 9 \cdot 5 + 46}{5} + \frac{3 z}{5} - z$

$y = \frac{46}{5} + \frac{3 z}{5}$

Step 3.2.1.4

Simplify the numerator.

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Step 3.2.1.4.1

Multiply $- 9$ by $5$ .

$x = \frac{- 45 + 46}{5} + \frac{3 z}{5} - z$

$y = \frac{46}{5} + \frac{3 z}{5}$

Step 3.2.1.4.2

Add $- 45$ and $46$ .

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

$y = \frac{46}{5} + \frac{3 z}{5}$

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

$y = \frac{46}{5} + \frac{3 z}{5}$

Step 3.2.1.5

To write $- z$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

$y = \frac{46}{5} + \frac{3 z}{5}$

Step 3.2.1.6

Combine $- z$ and $\frac{5}{5}$ .

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

$y = \frac{46}{5} + \frac{3 z}{5}$

Step 3.2.1.7

Combine the numerators over the common denominator.

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

$y = \frac{46}{5} + \frac{3 z}{5}$

Step 3.2.1.8

Combine the numerators over the common denominator.

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

$y = \frac{46}{5} + \frac{3 z}{5}$

Step 3.2.1.9

Multiply $5$ by $- 1$ .

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

$y = \frac{46}{5} + \frac{3 z}{5}$

Step 3.2.1.10

Subtract $5 z$ from $3 z$ .

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

$y = \frac{46}{5} + \frac{3 z}{5}$

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

$y = \frac{46}{5} + \frac{3 z}{5}$

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

$y = \frac{46}{5} + \frac{3 z}{5}$

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

$y = \frac{46}{5} + \frac{3 z}{5}$

Step 4

Simplify the right side.

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Step 4.1

Reorder $\frac{46}{5}$ and $\frac{3 z}{5}$ .

$y = \frac{3 z}{5} + \frac{46}{5}$

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

$y = \frac{3 z}{5} + \frac{46}{5}$

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