Algebra Examples

$6 x - y + 4 z = 0$ , $x - 7 y + z = 0$

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 $6 x$ from both sides of the equation.

$- y + 4 z = - 6 x$

$x - 7 y + z = 0$

Step 1.1.2

Subtract $4 z$ from both sides of the equation.

$- y = - 6 x - 4 z$

$x - 7 y + z = 0$

$- y = - 6 x - 4 z$

$x - 7 y + z = 0$

Step 1.2

Divide each term in $- y = - 6 x - 4 z$ by $- 1$ and simplify.

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Step 1.2.1

Divide each term in $- y = - 6 x - 4 z$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 6 x}{- 1} + \frac{- 4 z}{- 1}$

$x - 7 y + z = 0$

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.

$\frac{y}{1} = \frac{- 6 x}{- 1} + \frac{- 4 z}{- 1}$

$x - 7 y + z = 0$

Step 1.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 6 x}{- 1} + \frac{- 4 z}{- 1}$

$x - 7 y + z = 0$

$y = \frac{- 6 x}{- 1} + \frac{- 4 z}{- 1}$

$x - 7 y + z = 0$

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 one from the denominator of $\frac{- 6 x}{- 1}$ .

$y = - 1 \cdot (- 6 x) + \frac{- 4 z}{- 1}$

$x - 7 y + z = 0$

Step 1.2.3.1.2

Rewrite $- 1 \cdot (- 6 x)$ as $- (- 6 x)$ .

$y = - (- 6 x) + \frac{- 4 z}{- 1}$

$x - 7 y + z = 0$

Step 1.2.3.1.3

Multiply $- 6$ by $- 1$ .

$y = 6 x + \frac{- 4 z}{- 1}$

$x - 7 y + z = 0$

Step 1.2.3.1.4

Move the negative one from the denominator of $\frac{- 4 z}{- 1}$ .

$y = 6 x - 1 \cdot (- 4 z)$

$x - 7 y + z = 0$

Step 1.2.3.1.5

Rewrite $- 1 \cdot (- 4 z)$ as $- (- 4 z)$ .

$y = 6 x - (- 4 z)$

$x - 7 y + z = 0$

Step 1.2.3.1.6

Multiply $- 4$ by $- 1$ .

$y = 6 x + 4 z$

$x - 7 y + z = 0$

$y = 6 x + 4 z$

$x - 7 y + z = 0$

$y = 6 x + 4 z$

$x - 7 y + z = 0$

$y = 6 x + 4 z$

$x - 7 y + z = 0$

$y = 6 x + 4 z$

$x - 7 y + z = 0$

Step 2

Solve the equation for $z$ .

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Step 2.1

Simplify $x - 7 (6 x + 4 z) + z$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

$x - 7 (6 x) - 7 (4 z) + z = 0$

$y = 6 x + 4 z$

Step 2.1.1.2

Multiply $6$ by $- 7$ .

$x - 42 x - 7 (4 z) + z = 0$

$y = 6 x + 4 z$

Step 2.1.1.3

Multiply $4$ by $- 7$ .

$x - 42 x - 28 z + z = 0$

$y = 6 x + 4 z$

$x - 42 x - 28 z + z = 0$

$y = 6 x + 4 z$

Step 2.1.2

Simplify by adding terms.

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Step 2.1.2.1

Subtract $42 x$ from $x$ .

$- 41 x - 28 z + z = 0$

$y = 6 x + 4 z$

Step 2.1.2.2

Add $- 28 z$ and $z$ .

$- 41 x - 27 z = 0$

$y = 6 x + 4 z$

$- 41 x - 27 z = 0$

$y = 6 x + 4 z$

$- 41 x - 27 z = 0$

$y = 6 x + 4 z$

Step 2.2

Add $41 x$ to both sides of the equation.

$- 27 z = 41 x$

$y = 6 x + 4 z$

Step 2.3

Divide each term in $- 27 z = 41 x$ by $- 27$ and simplify.

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Step 2.3.1

Divide each term in $- 27 z = 41 x$ by $- 27$ .

$\frac{- 27 z}{- 27} = \frac{41 x}{- 27}$

$y = 6 x + 4 z$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 27$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 27 z}{- 27} = \frac{41 x}{- 27}$

$y = 6 x + 4 z$

Step 2.3.2.1.2

Divide $z$ by $1$ .

$z = \frac{41 x}{- 27}$

$y = 6 x + 4 z$

$z = \frac{41 x}{- 27}$

$y = 6 x + 4 z$

$z = \frac{41 x}{- 27}$

$y = 6 x + 4 z$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Move the negative in front of the fraction.

$z = - \frac{41 x}{27}$

$y = 6 x + 4 z$

$z = - \frac{41 x}{27}$

$y = 6 x + 4 z$

$z = - \frac{41 x}{27}$

$y = 6 x + 4 z$

$z = - \frac{41 x}{27}$

$y = 6 x + 4 z$

Step 3

Simplify the right side.

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Step 3.1

Simplify $6 x + 4 (- \frac{41 x}{27})$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Multiply $4 (- \frac{41 x}{27})$ .

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Step 3.1.1.1.1

Multiply $- 1$ by $4$ .

$y = 6 x - 4 \frac{41 x}{27}$

$z = - \frac{41 x}{27}$

Step 3.1.1.1.2

Combine $- 4$ and $\frac{41 x}{27}$ .

$y = 6 x + \frac{- 4 (41 x)}{27}$

$z = - \frac{41 x}{27}$

Step 3.1.1.1.3

Multiply $41$ by $- 4$ .

$y = 6 x + \frac{- 164 x}{27}$

$z = - \frac{41 x}{27}$

$y = 6 x + \frac{- 164 x}{27}$

$z = - \frac{41 x}{27}$

Step 3.1.1.2

Move the negative in front of the fraction.

$y = 6 x - \frac{164 x}{27}$

$z = - \frac{41 x}{27}$

$y = 6 x - \frac{164 x}{27}$

$z = - \frac{41 x}{27}$

Step 3.1.2

To write $6 x$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$y = 6 x \cdot \frac{27}{27} - \frac{164 x}{27}$

$z = - \frac{41 x}{27}$

Step 3.1.3

Simplify terms.

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Step 3.1.3.1

Combine $6 x$ and $\frac{27}{27}$ .

$y = \frac{6 x \cdot 27}{27} - \frac{164 x}{27}$

$z = - \frac{41 x}{27}$

Step 3.1.3.2

Combine the numerators over the common denominator.

$y = \frac{6 x \cdot 27 - 164 x}{27}$

$z = - \frac{41 x}{27}$

$y = \frac{6 x \cdot 27 - 164 x}{27}$

$z = - \frac{41 x}{27}$

Step 3.1.4

Simplify the numerator.

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Step 3.1.4.1

Factor $2 x$ out of $6 x \cdot 27 - 164 x$ .

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Step 3.1.4.1.1

Factor $2 x$ out of $6 x \cdot 27$ .

$y = \frac{2 x (3 \cdot 27) - 164 x}{27}$

$z = - \frac{41 x}{27}$

Step 3.1.4.1.2

Factor $2 x$ out of $- 164 x$ .

$y = \frac{2 x (3 \cdot 27) + 2 x (- 82)}{27}$

$z = - \frac{41 x}{27}$

Step 3.1.4.1.3

Factor $2 x$ out of $2 x (3 \cdot 27) + 2 x (- 82)$ .

$y = \frac{2 x (3 \cdot 27 - 82)}{27}$

$z = - \frac{41 x}{27}$

$y = \frac{2 x (3 \cdot 27 - 82)}{27}$

$z = - \frac{41 x}{27}$

Step 3.1.4.2

Multiply $3$ by $27$ .

$y = \frac{2 x (81 - 82)}{27}$

$z = - \frac{41 x}{27}$

Step 3.1.4.3

Subtract $82$ from $81$ .

$y = \frac{2 x \cdot - 1}{27}$

$z = - \frac{41 x}{27}$

Step 3.1.4.4

Multiply $- 1$ by $2$ .

$y = \frac{- 2 x}{27}$

$z = - \frac{41 x}{27}$

$y = \frac{- 2 x}{27}$

$z = - \frac{41 x}{27}$

Step 3.1.5

Move the negative in front of the fraction.

$y = - \frac{2 x}{27}$

$z = - \frac{41 x}{27}$

$y = - \frac{2 x}{27}$

$z = - \frac{41 x}{27}$

$y = - \frac{2 x}{27}$

$z = - \frac{41 x}{27}$