Algebra Examples

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

Solve the equation for $y$ .

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

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Since $6 x$ does not contain the variable to solve for, move it to the right side of the equation by subtracting $6 x$ from both sides.

$- y + 4 z = - 6 x$

$x - 7 y + z = 0$

Since $4 z$ does not contain the variable to solve for, move it to the right side of the equation by subtracting $4 z$ from both sides.

$- y = - 6 x - 4 z$

$x - 7 y + z = 0$

$- y = - 6 x - 4 z$

$x - 7 y + z = 0$

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

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

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

$x - 7 y + z = 0$

Simplify $- y \cdot - 1$ .

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Multiply $- 1$ by $- 1$ to get $1$ .

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

$x - 7 y + z = 0$

Multiply $y$ by $1$ to get $y$ .

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

$x - 7 y + z = 0$

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

$x - 7 y + z = 0$

Simplify each term.

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Multiply $- 1$ by $- 6$ to get $6$ .

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

$x - 7 y + z = 0$

Multiply $- 1$ by $- 4$ to get $4$ .

$y = 6 x + 4 z$

$x - 7 y + z = 0$

$y = 6 x + 4 z$

$x - 7 y + z = 0$

$y = 6 x + 4 z$

$x - 7 y + z = 0$

$y = 6 x + 4 z$

$x - 7 y + z = 0$

Solve the equation for $z$ .

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Simplify the left side.

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Simplify each term.

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Apply the distributive property.

$y = 6 x + 4 z$

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

Multiply $6$ by $- 7$ to get $- 42$ .

$y = 6 x + 4 z$

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

Multiply $4$ by $- 7$ to get $- 28$ .

$y = 6 x + 4 z$

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

$y = 6 x + 4 z$

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

Simplify by adding terms.

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Remove unnecessary parentheses.

$y = 6 x + 4 z$

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

Subtract $42 x$ from $x$ to get $- 41 x$ .

$y = 6 x + 4 z$

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

Add $- 28 z$ and $z$ to get $- 27 z$ .

$y = 6 x + 4 z$

$- 41 x - 27 z = 0$

$y = 6 x + 4 z$

$- 41 x - 27 z = 0$

$y = 6 x + 4 z$

$- 41 x - 27 z = 0$

Since $- 41 x$ does not contain the variable to solve for, move it to the right side of the equation by adding $41 x$ to both sides.

$y = 6 x + 4 z$

$- 27 z = 41 x$

Divide each term by $- 27$ and simplify.

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

$y = 6 x + 4 z$

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

Simplify the left side of the equation by cancelling the common factors.

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Reduce the expression by cancelling the common factors.

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Factor $27$ out of $- 27$ .

$y = 6 x + 4 z$

$- \frac{27 (z)}{27 (- 1)} = \frac{41 x}{- 27}$

Cancel the common factor.

$y = 6 x + 4 z$

$- \frac{27 z}{27 \cdot - 1} = \frac{41 x}{- 27}$

Rewrite the expression.

$y = 6 x + 4 z$

$- \frac{z}{- 1} = \frac{41 x}{- 27}$

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

$y = 6 x + 4 z$

$- (- 1 \cdot z) = \frac{41 x}{- 27}$

$y = 6 x + 4 z$

$- (- 1 \cdot z) = \frac{41 x}{- 27}$

Simplify the expression.

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Multiply $- 1$ by $z$ to get $- 1 z$ .

$y = 6 x + 4 z$

$- (- 1 z) = \frac{41 x}{- 27}$

Rewrite $- 1 z$ as $- z$ .

$y = 6 x + 4 z$

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

$y = 6 x + 4 z$

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

$y = 6 x + 4 z$

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

Move the negative in front of the fraction.

$y = 6 x + 4 z$

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

$y = 6 x + 4 z$

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

$y = 6 x + 4 z$

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

Simplify the right side.

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Simplify each term.

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

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Multiply $- 1$ by $4$ to get $- 4$ .

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

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

Write $- 4$ as a fraction with denominator $1$ .

$y = 6 x + \frac{- 4}{1} \cdot \frac{41 x}{27}$

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

Multiply $\frac{- 4}{1}$ and $\frac{41 x}{27}$ to get $\frac{- 4 (41 x)}{27}$ .

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

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

Multiply $41$ by $- 4$ to get $- 164$ .

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

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

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

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

Move the negative in front of the fraction.

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

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

Multiply $164$ by $x$ to get $164 x$ .

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

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

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

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

To write $\frac{6 x}{1}$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$y = \frac{6 x}{1} \cdot \frac{27}{27} - \frac{164 x}{27}$

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

Write each expression with a common denominator of $27$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$y = \frac{6 x \cdot 27}{1 \cdot 27} - \frac{164 x}{27}$

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

Multiply $27$ by $1$ to get $27$ .

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

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

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

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

Combine the numerators over the common denominator.

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

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

Simplify the numerator.

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

$y = \frac{x (6 \cdot 27 - 1 \cdot 164)}{27}$

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

Multiply $6$ by $27$ to get $162$ .

$y = \frac{x (162 - 1 \cdot 164)}{27}$

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

Multiply $- 1$ by $164$ to get $- 1 (164)$ .

$y = \frac{x (162 - 1 \cdot 164)}{27}$

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

Multiply $- 1$ by $164$ to get $- 164$ .

$y = \frac{x (162 - 164)}{27}$

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

Subtract $164$ from $162$ to get $- 2$ .

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

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

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

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

Simplify the expression.

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Move $- 2$ to the left of the expression $x \cdot - 2$ .

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

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

Move the negative in front of the fraction.

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

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

Multiply $2$ by $x$ to get $2 \cdot x$ .

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

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

Multiply $2$ by $x$ to get $2 x$ .

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

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

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

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

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

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

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