Algebra Examples

$5 x + 4 y + 3 z = 0$ $- 5 x - 6 y + 10 z = 10$ $5 x + 3 y + 3 z = 5$

Step 1

Solve for $x$ in $5 x + 4 y + 3 z = 0$ .

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Step 1.1

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

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Step 1.1.1

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

$5 x + 3 z = - 4 y$

$- 5 x - 6 y + 10 z = 10$

$5 x + 3 y + 3 z = 5$

Step 1.1.2

Subtract $3 z$ from both sides of the equation.

$5 x = - 4 y - 3 z$

$- 5 x - 6 y + 10 z = 10$

$5 x + 3 y + 3 z = 5$

$5 x = - 4 y - 3 z$

$- 5 x - 6 y + 10 z = 10$

$5 x + 3 y + 3 z = 5$

Step 1.2

Divide each term in $5 x = - 4 y - 3 z$ by $5$ and simplify.

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Step 1.2.1

Divide each term in $5 x = - 4 y - 3 z$ by $5$ .

$\frac{5 x}{5} = \frac{- 4 y}{5} + \frac{- 3 z}{5}$

$- 5 x - 6 y + 10 z = 10$

$5 x + 3 y + 3 z = 5$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $5$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 4 y}{5} + \frac{- 3 z}{5}$

$- 5 x - 6 y + 10 z = 10$

$5 x + 3 y + 3 z = 5$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4 y}{5} + \frac{- 3 z}{5}$

$- 5 x - 6 y + 10 z = 10$

$5 x + 3 y + 3 z = 5$

$x = \frac{- 4 y}{5} + \frac{- 3 z}{5}$

$- 5 x - 6 y + 10 z = 10$

$5 x + 3 y + 3 z = 5$

$x = \frac{- 4 y}{5} + \frac{- 3 z}{5}$

$- 5 x - 6 y + 10 z = 10$

$5 x + 3 y + 3 z = 5$

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 in front of the fraction.

$x = - \frac{4 y}{5} + \frac{- 3 z}{5}$

$- 5 x - 6 y + 10 z = 10$

$5 x + 3 y + 3 z = 5$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$- 5 x - 6 y + 10 z = 10$

$5 x + 3 y + 3 z = 5$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$- 5 x - 6 y + 10 z = 10$

$5 x + 3 y + 3 z = 5$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$- 5 x - 6 y + 10 z = 10$

$5 x + 3 y + 3 z = 5$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$- 5 x - 6 y + 10 z = 10$

$5 x + 3 y + 3 z = 5$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$- 5 x - 6 y + 10 z = 10$

$5 x + 3 y + 3 z = 5$

Step 2

Replace all occurrences of $x$ with $- \frac{4 y}{5} - \frac{3 z}{5}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- 5 x - 6 y + 10 z = 10$ with $- \frac{4 y}{5} - \frac{3 z}{5}$ .

$- 5 (- \frac{4 y}{5} - \frac{3 z}{5}) - 6 y + 10 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$5 x + 3 y + 3 z = 5$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 5 (- \frac{4 y}{5} - \frac{3 z}{5}) - 6 y + 10 z$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$- 5 (- \frac{4 y}{5}) - 5 (- \frac{3 z}{5}) - 6 y + 10 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$5 x + 3 y + 3 z = 5$

Step 2.2.1.1.2

Cancel the common factor of $5$ .

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Step 2.2.1.1.2.1

Move the leading negative in $- \frac{4 y}{5}$ into the numerator.

$- 5 \frac{- 4 y}{5} - 5 (- \frac{3 z}{5}) - 6 y + 10 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$5 x + 3 y + 3 z = 5$

Step 2.2.1.1.2.2

Factor $5$ out of $- 5$ .

$5 (- 1) (\frac{- 4 y}{5}) - 5 (- \frac{3 z}{5}) - 6 y + 10 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$5 x + 3 y + 3 z = 5$

Step 2.2.1.1.2.3

Cancel the common factor.

$5 \cdot (- 1 \frac{- 4 y}{5}) - 5 (- \frac{3 z}{5}) - 6 y + 10 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$5 x + 3 y + 3 z = 5$

Step 2.2.1.1.2.4

Rewrite the expression.

$- 1 (- 4 y) - 5 (- \frac{3 z}{5}) - 6 y + 10 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$5 x + 3 y + 3 z = 5$

$- 1 (- 4 y) - 5 (- \frac{3 z}{5}) - 6 y + 10 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$5 x + 3 y + 3 z = 5$

Step 2.2.1.1.3

Multiply $- 4$ by $- 1$ .

$4 y - 5 (- \frac{3 z}{5}) - 6 y + 10 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$5 x + 3 y + 3 z = 5$

Step 2.2.1.1.4

Cancel the common factor of $5$ .

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Step 2.2.1.1.4.1

Move the leading negative in $- \frac{3 z}{5}$ into the numerator.

$4 y - 5 \frac{- 3 z}{5} - 6 y + 10 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$5 x + 3 y + 3 z = 5$

Step 2.2.1.1.4.2

Factor $5$ out of $- 5$ .

$4 y + 5 (- 1) (\frac{- 3 z}{5}) - 6 y + 10 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$5 x + 3 y + 3 z = 5$

Step 2.2.1.1.4.3

Cancel the common factor.

$4 y + 5 \cdot (- 1 \frac{- 3 z}{5}) - 6 y + 10 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$5 x + 3 y + 3 z = 5$

Step 2.2.1.1.4.4

Rewrite the expression.

$4 y - 1 (- 3 z) - 6 y + 10 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$5 x + 3 y + 3 z = 5$

$4 y - 1 (- 3 z) - 6 y + 10 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$5 x + 3 y + 3 z = 5$

Step 2.2.1.1.5

Multiply $- 3$ by $- 1$ .

$4 y + 3 z - 6 y + 10 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$5 x + 3 y + 3 z = 5$

$4 y + 3 z - 6 y + 10 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$5 x + 3 y + 3 z = 5$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $6 y$ from $4 y$ .

$- 2 y + 3 z + 10 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$5 x + 3 y + 3 z = 5$

Step 2.2.1.2.2

Add $3 z$ and $10 z$ .

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$5 x + 3 y + 3 z = 5$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$5 x + 3 y + 3 z = 5$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$5 x + 3 y + 3 z = 5$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$5 x + 3 y + 3 z = 5$

Step 2.3

Replace all occurrences of $x$ in $5 x + 3 y + 3 z = 5$ with $- \frac{4 y}{5} - \frac{3 z}{5}$ .

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

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $5 (- \frac{4 y}{5} - \frac{3 z}{5}) + 3 y + 3 z$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Apply the distributive property.

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

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

Step 2.4.1.1.2

Cancel the common factor of $5$ .

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Step 2.4.1.1.2.1

Move the leading negative in $- \frac{4 y}{5}$ into the numerator.

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

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

Step 2.4.1.1.2.2

Cancel the common factor.

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

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

Step 2.4.1.1.2.3

Rewrite the expression.

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

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

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

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

Step 2.4.1.1.3

Cancel the common factor of $5$ .

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Step 2.4.1.1.3.1

Move the leading negative in $- \frac{3 z}{5}$ into the numerator.

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

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

Step 2.4.1.1.3.2

Cancel the common factor.

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

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

Step 2.4.1.1.3.3

Rewrite the expression.

$- 4 y - 3 z + 3 y + 3 z = 5$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$- 4 y - 3 z + 3 y + 3 z = 5$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$- 4 y - 3 z + 3 y + 3 z = 5$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Combine the opposite terms in $- 4 y - 3 z + 3 y + 3 z$ .

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Step 2.4.1.2.1.1

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

$- 4 y + 3 y + 0 = 5$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

Step 2.4.1.2.1.2

Add $- 4 y + 3 y$ and $0$ .

$- 4 y + 3 y = 5$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$- 4 y + 3 y = 5$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

Step 2.4.1.2.2

Add $- 4 y$ and $3 y$ .

$- y = 5$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$- y = 5$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$- y = 5$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$- y = 5$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$- y = 5$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

Step 3

Divide each term in $- y = 5$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- y = 5$ by $- 1$ .

$\frac{- y}{- 1} = \frac{5}{- 1}$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{5}{- 1}$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

Step 3.2.2

Divide $y$ by $1$ .

$y = \frac{5}{- 1}$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$y = \frac{5}{- 1}$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $5$ by $- 1$ .

$y = - 5$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$y = - 5$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$y = - 5$

$- 2 y + 13 z = 10$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

Step 4

Replace all occurrences of $y$ with $- 5$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $- 2 y + 13 z = 10$ with $- 5$ .

$- 2 \cdot - 5 + 13 z = 10$

$y = - 5$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Multiply $- 2$ by $- 5$ .

$10 + 13 z = 10$

$y = - 5$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

$10 + 13 z = 10$

$y = - 5$

$x = - \frac{4 y}{5} - \frac{3 z}{5}$

Step 4.3

Replace all occurrences of $y$ in $x = - \frac{4 y}{5} - \frac{3 z}{5}$ with $- 5$ .

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

$10 + 13 z = 10$

$y = - 5$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify each term.

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Step 4.4.1.1

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

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Step 4.4.1.1.1

Factor $5$ out of $4 (- 5)$ .

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

$10 + 13 z = 10$

$y = - 5$

Step 4.4.1.1.2

Cancel the common factors.

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Step 4.4.1.1.2.1

Factor $5$ out of $5$ .

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

$10 + 13 z = 10$

$y = - 5$

Step 4.4.1.1.2.2

Cancel the common factor.

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

$10 + 13 z = 10$

$y = - 5$

Step 4.4.1.1.2.3

Rewrite the expression.

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

$10 + 13 z = 10$

$y = - 5$

Step 4.4.1.1.2.4

Divide $4 \cdot (- 1)$ by $1$ .

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

$10 + 13 z = 10$

$y = - 5$

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

$10 + 13 z = 10$

$y = - 5$

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

$10 + 13 z = 10$

$y = - 5$

Step 4.4.1.2

Multiply $- (4 \cdot (- 1))$ .

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Step 4.4.1.2.1

Multiply $4$ by $- 1$ .

$x = 4 - \frac{3 z}{5}$

$10 + 13 z = 10$

$y = - 5$

Step 4.4.1.2.2

Multiply $- 1$ by $- 4$ .

$x = 4 - \frac{3 z}{5}$

$10 + 13 z = 10$

$y = - 5$

$x = 4 - \frac{3 z}{5}$

$10 + 13 z = 10$

$y = - 5$

$x = 4 - \frac{3 z}{5}$

$10 + 13 z = 10$

$y = - 5$

$x = 4 - \frac{3 z}{5}$

$10 + 13 z = 10$

$y = - 5$

$x = 4 - \frac{3 z}{5}$

$10 + 13 z = 10$

$y = - 5$

Step 5

Solve for $z$ in $10 + 13 z = 10$ .

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Step 5.1

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

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Step 5.1.1

Subtract $10$ from both sides of the equation.

$13 z = 10 - 10$

$x = 4 - \frac{3 z}{5}$

$y = - 5$

Step 5.1.2

Subtract $10$ from $10$ .

$13 z = 0$

$x = 4 - \frac{3 z}{5}$

$y = - 5$

$13 z = 0$

$x = 4 - \frac{3 z}{5}$

$y = - 5$

Step 5.2

Divide each term in $13 z = 0$ by $13$ and simplify.

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Step 5.2.1

Divide each term in $13 z = 0$ by $13$ .

$\frac{13 z}{13} = \frac{0}{13}$

$x = 4 - \frac{3 z}{5}$

$y = - 5$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $13$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{13 z}{13} = \frac{0}{13}$

$x = 4 - \frac{3 z}{5}$

$y = - 5$

Step 5.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{0}{13}$

$x = 4 - \frac{3 z}{5}$

$y = - 5$

$z = \frac{0}{13}$

$x = 4 - \frac{3 z}{5}$

$y = - 5$

$z = \frac{0}{13}$

$x = 4 - \frac{3 z}{5}$

$y = - 5$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $0$ by $13$ .

$z = 0$

$x = 4 - \frac{3 z}{5}$

$y = - 5$

$z = 0$

$x = 4 - \frac{3 z}{5}$

$y = - 5$

$z = 0$

$x = 4 - \frac{3 z}{5}$

$y = - 5$

$z = 0$

$x = 4 - \frac{3 z}{5}$

$y = - 5$

Step 6

Replace all occurrences of $z$ with $0$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $x = 4 - \frac{3 z}{5}$ with $0$ .

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

$z = 0$

$y = - 5$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $4 - \frac{3 (0)}{5}$ .

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Step 6.2.1.1

Simplify each term.

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Step 6.2.1.1.1

Cancel the common factor of $0$ and $5$ .

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Step 6.2.1.1.1.1

Factor $5$ out of $3 (0)$ .

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

$z = 0$

$y = - 5$

Step 6.2.1.1.1.2

Cancel the common factors.

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Step 6.2.1.1.1.2.1

Factor $5$ out of $5$ .

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

$z = 0$

$y = - 5$

Step 6.2.1.1.1.2.2

Cancel the common factor.

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

$z = 0$

$y = - 5$

Step 6.2.1.1.1.2.3

Rewrite the expression.

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

$z = 0$

$y = - 5$

Step 6.2.1.1.1.2.4

Divide $3 \cdot (0)$ by $1$ .

$x = 4 - (3 \cdot (0))$

$z = 0$

$y = - 5$

$x = 4 - (3 \cdot (0))$

$z = 0$

$y = - 5$

$x = 4 - (3 \cdot (0))$

$z = 0$

$y = - 5$

Step 6.2.1.1.2

Multiply $- (3 \cdot (0))$ .

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Step 6.2.1.1.2.1

Multiply $3$ by $0$ .

$x = 4 - 0$

$z = 0$

$y = - 5$

Step 6.2.1.1.2.2

Multiply $- 1$ by $0$ .

$x = 4 + 0$

$z = 0$

$y = - 5$

$x = 4 + 0$

$z = 0$

$y = - 5$

$x = 4 + 0$

$z = 0$

$y = - 5$

Step 6.2.1.2

Add $4$ and $0$ .

$x = 4$

$z = 0$

$y = - 5$

$x = 4$

$z = 0$

$y = - 5$

$x = 4$

$z = 0$

$y = - 5$

$x = 4$

$z = 0$

$y = - 5$

Step 7

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(4, - 5, 0)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(4, - 5, 0)$

Equation Form:

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