Algebra Examples

$- 3 x - 4 y + z = - 1$ $2 x + y - z = - 8$ $x + 8 y - z = 23$

Step 1

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

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Step 1.1

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

$- 4 y + z = - 1 + 3 x$

$2 x + y - z = - 8$

$x + 8 y - z = 23$

Step 1.2

Add $4 y$ to both sides of the equation.

$z = - 1 + 3 x + 4 y$

$2 x + y - z = - 8$

$x + 8 y - z = 23$

$z = - 1 + 3 x + 4 y$

$2 x + y - z = - 8$

$x + 8 y - z = 23$

Step 2

Replace all occurrences of $z$ with $- 1 + 3 x + 4 y$ in each equation.

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Step 2.1

Replace all occurrences of $z$ in $2 x + y - z = - 8$ with $- 1 + 3 x + 4 y$ .

$2 x + y - (- 1 + 3 x + 4 y) = - 8$

$z = - 1 + 3 x + 4 y$

$x + 8 y - z = 23$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 x + y - (- 1 + 3 x + 4 y)$ .

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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.

$2 x + y + 1 - (3 x) - (4 y) = - 8$

$z = - 1 + 3 x + 4 y$

$x + 8 y - z = 23$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$2 x + y + 1 - (3 x) - (4 y) = - 8$

$z = - 1 + 3 x + 4 y$

$x + 8 y - z = 23$

Step 2.2.1.1.2.2

Multiply $3$ by $- 1$ .

$2 x + y + 1 - 3 x - (4 y) = - 8$

$z = - 1 + 3 x + 4 y$

$x + 8 y - z = 23$

Step 2.2.1.1.2.3

Multiply $4$ by $- 1$ .

$2 x + y + 1 - 3 x - 4 y = - 8$

$z = - 1 + 3 x + 4 y$

$x + 8 y - z = 23$

$2 x + y + 1 - 3 x - 4 y = - 8$

$z = - 1 + 3 x + 4 y$

$x + 8 y - z = 23$

$2 x + y + 1 - 3 x - 4 y = - 8$

$z = - 1 + 3 x + 4 y$

$x + 8 y - z = 23$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $3 x$ from $2 x$ .

$- x + y + 1 - 4 y = - 8$

$z = - 1 + 3 x + 4 y$

$x + 8 y - z = 23$

Step 2.2.1.2.2

Subtract $4 y$ from $y$ .

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

$x + 8 y - z = 23$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

$x + 8 y - z = 23$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

$x + 8 y - z = 23$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

$x + 8 y - z = 23$

Step 2.3

Replace all occurrences of $z$ in $x + 8 y - z = 23$ with $- 1 + 3 x + 4 y$ .

$x + 8 y - (- 1 + 3 x + 4 y) = 23$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $x + 8 y - (- 1 + 3 x + 4 y)$ .

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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.

$x + 8 y + 1 - (3 x) - (4 y) = 23$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$x + 8 y + 1 - (3 x) - (4 y) = 23$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

Step 2.4.1.1.2.2

Multiply $3$ by $- 1$ .

$x + 8 y + 1 - 3 x - (4 y) = 23$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

Step 2.4.1.1.2.3

Multiply $4$ by $- 1$ .

$x + 8 y + 1 - 3 x - 4 y = 23$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

$x + 8 y + 1 - 3 x - 4 y = 23$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

$x + 8 y + 1 - 3 x - 4 y = 23$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $3 x$ from $x$ .

$- 2 x + 8 y + 1 - 4 y = 23$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

Step 2.4.1.2.2

Subtract $4 y$ from $8 y$ .

$- 2 x + 4 y + 1 = 23$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

$- 2 x + 4 y + 1 = 23$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

$- 2 x + 4 y + 1 = 23$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

$- 2 x + 4 y + 1 = 23$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

$- 2 x + 4 y + 1 = 23$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

Step 3

Solve for $x$ in $- 2 x + 4 y + 1 = 23$ .

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Step 3.1

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

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Step 3.1.1

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

$- 2 x + 1 = 23 - 4 y$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

Step 3.1.2

Subtract $1$ from both sides of the equation.

$- 2 x = 23 - 4 y - 1$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

Step 3.1.3

Subtract $1$ from $23$ .

$- 2 x = - 4 y + 22$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

$- 2 x = - 4 y + 22$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

Step 3.2

Divide each term in $- 2 x = - 4 y + 22$ by $- 2$ and simplify.

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Step 3.2.1

Divide each term in $- 2 x = - 4 y + 22$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 4 y}{- 2} + \frac{22}{- 2}$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 4 y}{- 2} + \frac{22}{- 2}$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4 y}{- 2} + \frac{22}{- 2}$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

$x = \frac{- 4 y}{- 2} + \frac{22}{- 2}$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

$x = \frac{- 4 y}{- 2} + \frac{22}{- 2}$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Cancel the common factor of $- 4$ and $- 2$ .

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Step 3.2.3.1.1.1

Factor $- 2$ out of $- 4 y$ .

$x = \frac{- 2 (2 y)}{- 2} + \frac{22}{- 2}$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

Step 3.2.3.1.1.2

Cancel the common factors.

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Step 3.2.3.1.1.2.1

Factor $- 2$ out of $- 2$ .

$x = \frac{- 2 (2 y)}{- 2 \cdot 1} + \frac{22}{- 2}$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

Step 3.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{- 2 (2 y)}{- 2 \cdot 1} + \frac{22}{- 2}$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

Step 3.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{2 y}{1} + \frac{22}{- 2}$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

Step 3.2.3.1.1.2.4

Divide $2 y$ by $1$ .

$x = 2 y + \frac{22}{- 2}$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

$x = 2 y + \frac{22}{- 2}$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

$x = 2 y + \frac{22}{- 2}$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

Step 3.2.3.1.2

Divide $22$ by $- 2$ .

$x = 2 y - 11$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

$x = 2 y - 11$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

$x = 2 y - 11$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

$x = 2 y - 11$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

$x = 2 y - 11$

$- x - 3 y + 1 = - 8$

$z = - 1 + 3 x + 4 y$

Step 4

Replace all occurrences of $x$ with $2 y - 11$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $- x - 3 y + 1 = - 8$ with $2 y - 11$ .

$- (2 y - 11) - 3 y + 1 = - 8$

$x = 2 y - 11$

$z = - 1 + 3 x + 4 y$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- (2 y - 11) - 3 y + 1$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$- (2 y) + 11 - 3 y + 1 = - 8$

$x = 2 y - 11$

$z = - 1 + 3 x + 4 y$

Step 4.2.1.1.2

Multiply $2$ by $- 1$ .

$- 2 y + 11 - 3 y + 1 = - 8$

$x = 2 y - 11$

$z = - 1 + 3 x + 4 y$

Step 4.2.1.1.3

Multiply $- 1$ by $- 11$ .

$- 2 y + 11 - 3 y + 1 = - 8$

$x = 2 y - 11$

$z = - 1 + 3 x + 4 y$

$- 2 y + 11 - 3 y + 1 = - 8$

$x = 2 y - 11$

$z = - 1 + 3 x + 4 y$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Subtract $3 y$ from $- 2 y$ .

$- 5 y + 11 + 1 = - 8$

$x = 2 y - 11$

$z = - 1 + 3 x + 4 y$

Step 4.2.1.2.2

Add $11$ and $1$ .

$- 5 y + 12 = - 8$

$x = 2 y - 11$

$z = - 1 + 3 x + 4 y$

$- 5 y + 12 = - 8$

$x = 2 y - 11$

$z = - 1 + 3 x + 4 y$

$- 5 y + 12 = - 8$

$x = 2 y - 11$

$z = - 1 + 3 x + 4 y$

$- 5 y + 12 = - 8$

$x = 2 y - 11$

$z = - 1 + 3 x + 4 y$

Step 4.3

Replace all occurrences of $x$ in $z = - 1 + 3 x + 4 y$ with $2 y - 11$ .

$z = - 1 + 3 (2 y - 11) + 4 y$

$- 5 y + 12 = - 8$

$x = 2 y - 11$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $- 1 + 3 (2 y - 11) + 4 y$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$z = - 1 + 3 (2 y) + 3 \cdot - 11 + 4 y$

$- 5 y + 12 = - 8$

$x = 2 y - 11$

Step 4.4.1.1.2

Multiply $2$ by $3$ .

$z = - 1 + 6 y + 3 \cdot - 11 + 4 y$

$- 5 y + 12 = - 8$

$x = 2 y - 11$

Step 4.4.1.1.3

Multiply $3$ by $- 11$ .

$z = - 1 + 6 y - 33 + 4 y$

$- 5 y + 12 = - 8$

$x = 2 y - 11$

$z = - 1 + 6 y - 33 + 4 y$

$- 5 y + 12 = - 8$

$x = 2 y - 11$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Subtract $33$ from $- 1$ .

$z = 6 y - 34 + 4 y$

$- 5 y + 12 = - 8$

$x = 2 y - 11$

Step 4.4.1.2.2

Add $6 y$ and $4 y$ .

$z = 10 y - 34$

$- 5 y + 12 = - 8$

$x = 2 y - 11$

$z = 10 y - 34$

$- 5 y + 12 = - 8$

$x = 2 y - 11$

$z = 10 y - 34$

$- 5 y + 12 = - 8$

$x = 2 y - 11$

$z = 10 y - 34$

$- 5 y + 12 = - 8$

$x = 2 y - 11$

$z = 10 y - 34$

$- 5 y + 12 = - 8$

$x = 2 y - 11$

Step 5

Solve for $y$ in $- 5 y + 12 = - 8$ .

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Step 5.1

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

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Step 5.1.1

Subtract $12$ from both sides of the equation.

$- 5 y = - 8 - 12$

$z = 10 y - 34$

$x = 2 y - 11$

Step 5.1.2

Subtract $12$ from $- 8$ .

$- 5 y = - 20$

$z = 10 y - 34$

$x = 2 y - 11$

$- 5 y = - 20$

$z = 10 y - 34$

$x = 2 y - 11$

Step 5.2

Divide each term in $- 5 y = - 20$ by $- 5$ and simplify.

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Step 5.2.1

Divide each term in $- 5 y = - 20$ by $- 5$ .

$\frac{- 5 y}{- 5} = \frac{- 20}{- 5}$

$z = 10 y - 34$

$x = 2 y - 11$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 5$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 5 y}{- 5} = \frac{- 20}{- 5}$

$z = 10 y - 34$

$x = 2 y - 11$

Step 5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 20}{- 5}$

$z = 10 y - 34$

$x = 2 y - 11$

$y = \frac{- 20}{- 5}$

$z = 10 y - 34$

$x = 2 y - 11$

$y = \frac{- 20}{- 5}$

$z = 10 y - 34$

$x = 2 y - 11$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $- 20$ by $- 5$ .

$y = 4$

$z = 10 y - 34$

$x = 2 y - 11$

$y = 4$

$z = 10 y - 34$

$x = 2 y - 11$

$y = 4$

$z = 10 y - 34$

$x = 2 y - 11$

$y = 4$

$z = 10 y - 34$

$x = 2 y - 11$

Step 6

Replace all occurrences of $y$ with $4$ in each equation.

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Step 6.1

Replace all occurrences of $y$ in $z = 10 y - 34$ with $4$ .

$z = 10 (4) - 34$

$y = 4$

$x = 2 y - 11$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $10 (4) - 34$ .

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Step 6.2.1.1

Multiply $10$ by $4$ .

$z = 40 - 34$

$y = 4$

$x = 2 y - 11$

Step 6.2.1.2

Subtract $34$ from $40$ .

$z = 6$

$y = 4$

$x = 2 y - 11$

$z = 6$

$y = 4$

$x = 2 y - 11$

$z = 6$

$y = 4$

$x = 2 y - 11$

Step 6.3

Replace all occurrences of $y$ in $x = 2 y - 11$ with $4$ .

$x = 2 (4) - 11$

$z = 6$

$y = 4$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $2 (4) - 11$ .

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Step 6.4.1.1

Multiply $2$ by $4$ .

$x = 8 - 11$

$z = 6$

$y = 4$

Step 6.4.1.2

Subtract $11$ from $8$ .

$x = - 3$

$z = 6$

$y = 4$

$x = - 3$

$z = 6$

$y = 4$

$x = - 3$

$z = 6$

$y = 4$

$x = - 3$

$z = 6$

$y = 4$

Step 7

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

$(- 3, 4, 6)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 3, 4, 6)$

Equation Form:

$x = - 3, y = 4, z = 6$