Algebra Examples

$x - 4 y + 5 z = 0$ , $x + z - 3 y = 0$

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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Subtract $x$ from both sides of the equation.

$- 4 y + 5 z = - x$

$x + z - 3 y = 0$

Subtract $5 z$ from both sides of the equation.

$- 4 y = - x - 5 z$

$x + z - 3 y = 0$

$- 4 y = - x - 5 z$

$x + z - 3 y = 0$

Divide each term by $- 4$ and simplify.

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Divide each term in $- 4 y = - x - 5 z$ by $- 4$ .

$- \frac{4 y}{- 4} = - \frac{x}{- 4} - \frac{5 z}{- 4}$

$x + z - 3 y = 0$

Simplify the left side of the $e q u a t i o n$ by cancelling the common factors.

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Reduce the expression by cancelling the common factors.

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Factor $4$ out of $4 y$ .

$- \frac{4 (y)}{- 4} = - \frac{x}{- 4} - \frac{5 z}{- 4}$

$x + z - 3 y = 0$

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

$- (- 1 \cdot y) = - \frac{x}{- 4} - \frac{5 z}{- 4}$

$x + z - 3 y = 0$

$- (- 1 \cdot y) = - \frac{x}{- 4} - \frac{5 z}{- 4}$

$x + z - 3 y = 0$

Simplify the expression.

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Multiply $- 1$ by $y$ .

$- (- 1 y) = - \frac{x}{- 4} - \frac{5 z}{- 4}$

$x + z - 3 y = 0$

Rewrite $- 1 y$ as $- y$ .

$y = - \frac{x}{- 4} - \frac{5 z}{- 4}$

$x + z - 3 y = 0$

$y = - \frac{x}{- 4} - \frac{5 z}{- 4}$

$x + z - 3 y = 0$

$y = - \frac{x}{- 4} - \frac{5 z}{- 4}$

$x + z - 3 y = 0$

Simplify each term.

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Move the negative in front of the fraction.

$y = \frac{x}{4} - \frac{5 z}{- 4}$

$x + z - 3 y = 0$

Simplify $- - \frac{x}{4}$ .

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Multiply $- 1$ by $- 1$ .

$y = 1 (\frac{x}{4}) - \frac{5 z}{- 4}$

$x + z - 3 y = 0$

Multiply $\frac{x}{4}$ by $1$ .

$y = \frac{x}{4} - \frac{5 z}{- 4}$

$x + z - 3 y = 0$

$y = \frac{x}{4} - \frac{5 z}{- 4}$

$x + z - 3 y = 0$

Move the negative in front of the fraction.

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z - 3 y = 0$

Simplify $- - \frac{5 z}{4}$ .

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Multiply $- 1$ by $- 1$ .

$y = \frac{x}{4} + 1 (\frac{5 z}{4})$

$x + z - 3 y = 0$

Multiply $\frac{5 z}{4}$ by $1$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z - 3 y = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z - 3 y = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z - 3 y = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z - 3 y = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z - 3 y = 0$

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.

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z + (- 3 \frac{x}{4} - 3 \frac{5 z}{4}) = 0$

Simplify $- 3 \frac{x}{4}$ .

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Write $- 3$ as a fraction with denominator $1$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z + (\frac{- 3}{1} \cdot \frac{x}{4} - 3 \frac{5 z}{4}) = 0$

Multiply $\frac{- 3}{1}$ and $\frac{x}{4}$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z + (\frac{- 3 x}{4} - 3 \frac{5 z}{4}) = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z + (\frac{- 3 x}{4} - 3 \frac{5 z}{4}) = 0$

Simplify $- 3 \frac{5 z}{4}$ .

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Write $- 3$ as a fraction with denominator $1$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z + (\frac{- 3 x}{4} + \frac{- 3}{1} \cdot \frac{5 z}{4}) = 0$

Multiply $\frac{- 3}{1}$ and $\frac{5 z}{4}$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z + (\frac{- 3 x}{4} + \frac{- 3 (5 z)}{4}) = 0$

Multiply $5$ by $- 3$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z + (\frac{- 3 x}{4} + \frac{- 15 z}{4}) = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z + (\frac{- 3 x}{4} + \frac{- 15 z}{4}) = 0$

Simplify each term.

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Move the negative in front of the fraction.

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z + (- \frac{(3) x}{4} + \frac{- 15 z}{4}) = 0$

Multiply $3$ by $x$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z + (- \frac{3 x}{4} + \frac{- 15 z}{4}) = 0$

Move the negative in front of the fraction.

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z + (- \frac{3 x}{4} - \frac{(15) z}{4}) = 0$

Multiply $15$ by $z$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z + (- \frac{3 x}{4} - \frac{15 z}{4}) = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z + (- \frac{3 x}{4} - \frac{15 z}{4}) = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z + (- \frac{3 x}{4} - \frac{15 z}{4}) = 0$

Remove unnecessary parentheses.

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z - \frac{3 x}{4} - \frac{15 z}{4} = 0$

To write $\frac{x}{1}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$z + (\frac{x}{1} \cdot \frac{4}{4} - \frac{3 x}{4}) - \frac{15 z}{4} = 0$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$y = \frac{x}{4} + \frac{5 z}{4}$

$z + (\frac{x \cdot 4}{1 \cdot 4} - \frac{3 x}{4}) - \frac{15 z}{4} = 0$

Multiply $4$ by $1$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$z + (\frac{x \cdot 4}{4} - \frac{3 x}{4}) - \frac{15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$z + (\frac{x \cdot 4}{4} - \frac{3 x}{4}) - \frac{15 z}{4} = 0$

Combine the numerators over the common denominator.

$y = \frac{x}{4} + \frac{5 z}{4}$

$z + \frac{x \cdot 4 - (3 x)}{4} - \frac{15 z}{4} = 0$

Simplify each term.

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Simplify the numerator.

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Factor $x$ out of $x \cdot 4 - (3 x)$ .

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Factor $x$ out of $x \cdot 4$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$z + \frac{x (4) - (3 x)}{4} - \frac{15 z}{4} = 0$

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

$y = \frac{x}{4} + \frac{5 z}{4}$

$z + \frac{x (4) + x (- 1 \cdot 3)}{4} - \frac{15 z}{4} = 0$

Factor $x$ out of $x (4) + x (- 1 \cdot (3))$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$z + \frac{x (4 - 1 \cdot 3)}{4} - \frac{15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$z + \frac{x (4 - 1 \cdot 3)}{4} - \frac{15 z}{4} = 0$

Multiply $- 1$ by $3$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$z + \frac{x (4 - 1 \cdot 3)}{4} - \frac{15 z}{4} = 0$

Multiply $- 1$ by $3$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$z + \frac{x (4 - 3)}{4} - \frac{15 z}{4} = 0$

Subtract $3$ from $4$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$z + \frac{x \cdot 1}{4} - \frac{15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$z + \frac{x \cdot 1}{4} - \frac{15 z}{4} = 0$

Multiply $x$ by $1$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$z + \frac{x}{4} - \frac{15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$z + \frac{x}{4} - \frac{15 z}{4} = 0$

To write $\frac{z}{1}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$\frac{z}{1} \cdot \frac{4}{4} - \frac{15 z}{4} + \frac{x}{4} = 0$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$y = \frac{x}{4} + \frac{5 z}{4}$

$\frac{z \cdot 4}{1 \cdot 4} - \frac{15 z}{4} + \frac{x}{4} = 0$

Multiply $4$ by $1$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$\frac{z \cdot 4}{4} - \frac{15 z}{4} + \frac{x}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$\frac{z \cdot 4}{4} - \frac{15 z}{4} + \frac{x}{4} = 0$

Combine the numerators over the common denominator.

$y = \frac{x}{4} + \frac{5 z}{4}$

$\frac{z \cdot 4 - (15 z)}{4} + \frac{x}{4} = 0$

Simplify each term.

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Simplify the numerator.

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Factor $z$ out of $z \cdot 4 - (15 z)$ .

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Factor $z$ out of $z \cdot 4$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$\frac{z (4) - (15 z)}{4} + \frac{x}{4} = 0$

Factor $z$ out of $- (15 z)$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$\frac{z (4) + z (- 1 \cdot 15)}{4} + \frac{x}{4} = 0$

Factor $z$ out of $z (4) + z (- 1 \cdot (15))$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$\frac{z (4 - 1 \cdot 15)}{4} + \frac{x}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$\frac{z (4 - 1 \cdot 15)}{4} + \frac{x}{4} = 0$

Multiply $- 1$ by $15$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$\frac{z (4 - 1 \cdot 15)}{4} + \frac{x}{4} = 0$

Multiply $- 1$ by $15$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$\frac{z (4 - 15)}{4} + \frac{x}{4} = 0$

Subtract $15$ from $4$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$\frac{z \cdot - 11}{4} + \frac{x}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$\frac{z \cdot - 11}{4} + \frac{x}{4} = 0$

Move $- 11$ to the left of the expression $z \cdot - 11$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$\frac{- 11 \cdot z}{4} + \frac{x}{4} = 0$

Move the negative in front of the fraction.

$y = \frac{x}{4} + \frac{5 z}{4}$

$- \frac{(11) \cdot z}{4} + \frac{x}{4} = 0$

Multiply $11$ by $z$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$- \frac{11 \cdot z}{4} + \frac{x}{4} = 0$

Multiply $11$ by $z$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$- \frac{11 z}{4} + \frac{x}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$- \frac{11 z}{4} + \frac{x}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$- \frac{11 z}{4} + \frac{x}{4} = 0$

Subtract $\frac{x}{4}$ from both sides of the equation.

$y = \frac{x}{4} + \frac{5 z}{4}$

$- \frac{11 z}{4} = - \frac{x}{4}$

Solve for $z$ .

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Multiply both sides of the equation by $4$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$- 11 z = - \frac{x}{4} \cdot 4$

Simplify the right side.

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Cancel the common factor of $4$ .

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Rewrite.

$y = \frac{x}{4} + \frac{5 z}{4}$

$- 11 z = \frac{- x}{4} \cdot 4$

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

$y = \frac{x}{4} + \frac{5 z}{4}$

$- 11 z = \frac{- x}{4} \cdot \frac{4}{1}$

Factor out the greatest common factor $4$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$- 11 z = \frac{- x}{4 \cdot 1} \cdot \frac{4 \cdot 1}{1}$

Cancel the common factor.

$y = \frac{x}{4} + \frac{5 z}{4}$

$- 11 z = \frac{- x}{4 \cdot 1} \cdot \frac{4 \cdot 1}{1}$

Rewrite the expression.

$y = \frac{x}{4} + \frac{5 z}{4}$

$- 11 z = \frac{- x}{1} \cdot \frac{1}{1}$

$y = \frac{x}{4} + \frac{5 z}{4}$

$- 11 z = \frac{- x}{1} \cdot \frac{1}{1}$

Simplify.

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Multiply $\frac{- x}{1}$ and $\frac{1}{1}$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$- 11 z = \frac{- x}{1}$

Divide $- x$ by $1$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$- 11 z = - x$

$y = \frac{x}{4} + \frac{5 z}{4}$

$- 11 z = - x$

$y = \frac{x}{4} + \frac{5 z}{4}$

$- 11 z = - x$

Divide each term by $- 11$ and simplify.

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Divide each term in $- 11 z = - x$ by $- 11$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$- \frac{11 z}{- 11} = - \frac{x}{- 11}$

Simplify the left side of the $e q u a t i o n$ by cancelling the common factors.

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Reduce the expression by cancelling the common factors.

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Factor $11$ out of $11 z$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$- \frac{11 (z)}{- 11} = - \frac{x}{- 11}$

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

$y = \frac{x}{4} + \frac{5 z}{4}$

$- (- 1 \cdot z) = - \frac{x}{- 11}$

$y = \frac{x}{4} + \frac{5 z}{4}$

$- (- 1 \cdot z) = - \frac{x}{- 11}$

Simplify the expression.

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Multiply $- 1$ by $z$ .

$y = \frac{x}{4} + \frac{5 z}{4}$

$- (- 1 z) = - \frac{x}{- 11}$

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

$y = \frac{x}{4} + \frac{5 z}{4}$

$z = - \frac{x}{- 11}$

$y = \frac{x}{4} + \frac{5 z}{4}$

$z = - \frac{x}{- 11}$

$y = \frac{x}{4} + \frac{5 z}{4}$

$z = - \frac{x}{- 11}$

Move the negative in front of the fraction.

$y = \frac{x}{4} + \frac{5 z}{4}$

$z = \frac{x}{11}$

$y = \frac{x}{4} + \frac{5 z}{4}$

$z = \frac{x}{11}$

$y = \frac{x}{4} + \frac{5 z}{4}$

$z = \frac{x}{11}$

$y = \frac{x}{4} + \frac{5 z}{4}$

$z = \frac{x}{11}$

Simplify the right side.

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Simplify each term.

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Multiply $5$ by $\frac{x}{11}$ .

$y = \frac{x}{4} + \frac{5 (\frac{x}{11})}{4}$

$z = \frac{x}{11}$

Simplify the numerator.

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Write $5$ as a fraction with denominator $1$ .

$y = \frac{x}{4} + \frac{\frac{5}{1} \cdot \frac{x}{11}}{4}$

$z = \frac{x}{11}$

Multiply $\frac{5}{1}$ and $\frac{x}{11}$ .

$y = \frac{x}{4} + \frac{\frac{5 x}{11}}{4}$

$z = \frac{x}{11}$

$y = \frac{x}{4} + \frac{\frac{5 x}{11}}{4}$

$z = \frac{x}{11}$

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{x}{4} + \frac{5 x}{11} \cdot \frac{1}{4}$

$z = \frac{x}{11}$

Simplify $\frac{5 x}{11} \frac{1}{4}$ .

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Multiply $\frac{5 x}{11}$ and $\frac{1}{4}$ .

$y = \frac{x}{4} + \frac{5 x}{11 \cdot 4}$

$z = \frac{x}{11}$

Multiply $11$ by $4$ .

$y = \frac{x}{4} + \frac{5 x}{44}$

$z = \frac{x}{11}$

$y = \frac{x}{4} + \frac{5 x}{44}$

$z = \frac{x}{11}$

$y = \frac{x}{4} + \frac{5 x}{44}$

$z = \frac{x}{11}$

To write $\frac{x}{4}$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$y = \frac{x}{4} \cdot \frac{11}{11} + \frac{5 x}{44}$

$z = \frac{x}{11}$

Write each expression with a common denominator of $44$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$y = \frac{x \cdot 11}{4 \cdot 11} + \frac{5 x}{44}$

$z = \frac{x}{11}$

Multiply $4$ by $11$ .

$y = \frac{x \cdot 11}{44} + \frac{5 x}{44}$

$z = \frac{x}{11}$

$y = \frac{x \cdot 11}{44} + \frac{5 x}{44}$

$z = \frac{x}{11}$

Combine the numerators over the common denominator.

$y = \frac{x \cdot 11 + 5 x}{44}$

$z = \frac{x}{11}$

Simplify the numerator.

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Move $11$ to the left of the expression $x \cdot 11$ .

$y = \frac{11 \cdot x + 5 x}{44}$

$z = \frac{x}{11}$

Multiply $11$ by $x$ .

$y = \frac{11 x + 5 x}{44}$

$z = \frac{x}{11}$

Add $11 x$ and $5 x$ .

$y = \frac{16 x}{44}$

$z = \frac{x}{11}$

$y = \frac{16 x}{44}$

$z = \frac{x}{11}$

Reduce the expression by cancelling the common factors.

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Factor $4$ out of $16 x$ .

$y = \frac{4 (4 x)}{44}$

$z = \frac{x}{11}$

Cancel the common factors.

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Factor $4$ out of $44$ .

$y = \frac{4 (4 x)}{4 (11)}$

$z = \frac{x}{11}$

Cancel the common factor.

$y = \frac{4 (4 x)}{4 \cdot 11}$

$z = \frac{x}{11}$

Rewrite the expression.

$y = \frac{4 x}{11}$

$z = \frac{x}{11}$

$y = \frac{4 x}{11}$

$z = \frac{x}{11}$

$y = \frac{4 x}{11}$

$z = \frac{x}{11}$

$y = \frac{4 x}{11}$

$z = \frac{x}{11}$

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