Algebra Examples

$- x - 3 y + 3 z = - 5$ $x - 5 y - z = 7$ $x - 7 y + z = - 3$

Step 1

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

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Step 1.1

Add $5 y$ to both sides of the equation.

$x - z = 7 + 5 y$

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

$x - 7 y + z = - 3$

Step 1.2

Add $z$ to both sides of the equation.

$x = 7 + 5 y + z$

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

$x - 7 y + z = - 3$

$x = 7 + 5 y + z$

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

$x - 7 y + z = - 3$

Step 2

Replace all occurrences of $x$ with $7 + 5 y + z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- x - 3 y + 3 z = - 5$ with $7 + 5 y + z$ .

$- (7 + 5 y + z) - 3 y + 3 z = - 5$

$x = 7 + 5 y + z$

$x - 7 y + z = - 3$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- (7 + 5 y + z) - 3 y + 3 z$ .

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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.

$- 1 \cdot 7 - (5 y) - z - 3 y + 3 z = - 5$

$x = 7 + 5 y + z$

$x - 7 y + z = - 3$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 1$ by $7$ .

$- 7 - (5 y) - z - 3 y + 3 z = - 5$

$x = 7 + 5 y + z$

$x - 7 y + z = - 3$

Step 2.2.1.1.2.2

Multiply $5$ by $- 1$ .

$- 7 - 5 y - z - 3 y + 3 z = - 5$

$x = 7 + 5 y + z$

$x - 7 y + z = - 3$

$- 7 - 5 y - z - 3 y + 3 z = - 5$

$x = 7 + 5 y + z$

$x - 7 y + z = - 3$

$- 7 - 5 y - z - 3 y + 3 z = - 5$

$x = 7 + 5 y + z$

$x - 7 y + z = - 3$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $3 y$ from $- 5 y$ .

$- 7 - 8 y - z + 3 z = - 5$

$x = 7 + 5 y + z$

$x - 7 y + z = - 3$

Step 2.2.1.2.2

Add $- z$ and $3 z$ .

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

$x - 7 y + z = - 3$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

$x - 7 y + z = - 3$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

$x - 7 y + z = - 3$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

$x - 7 y + z = - 3$

Step 2.3

Replace all occurrences of $x$ in $x - 7 y + z = - 3$ with $7 + 5 y + z$ .

$(7 + 5 y + z) - 7 y + z = - 3$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $(7 + 5 y + z) - 7 y + z$ .

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Step 2.4.1.1

Subtract $7 y$ from $5 y$ .

$7 - 2 y + z + z = - 3$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

Step 2.4.1.2

Add $z$ and $z$ .

$7 - 2 y + 2 z = - 3$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

$7 - 2 y + 2 z = - 3$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

$7 - 2 y + 2 z = - 3$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

$7 - 2 y + 2 z = - 3$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

Step 3

Solve for $y$ in $7 - 2 y + 2 z = - 3$ .

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Step 3.1

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

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Step 3.1.1

Subtract $7$ from both sides of the equation.

$- 2 y + 2 z = - 3 - 7$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

Step 3.1.2

Subtract $2 z$ from both sides of the equation.

$- 2 y = - 3 - 7 - 2 z$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

Step 3.1.3

Subtract $7$ from $- 3$ .

$- 2 y = - 10 - 2 z$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

$- 2 y = - 10 - 2 z$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

Step 3.2

Divide each term in $- 2 y = - 10 - 2 z$ by $- 2$ and simplify.

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Step 3.2.1

Divide each term in $- 2 y = - 10 - 2 z$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{- 10}{- 2} + \frac{- 2 z}{- 2}$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

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.

$\frac{- 2 y}{- 2} = \frac{- 10}{- 2} + \frac{- 2 z}{- 2}$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 10}{- 2} + \frac{- 2 z}{- 2}$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

$y = \frac{- 10}{- 2} + \frac{- 2 z}{- 2}$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

$y = \frac{- 10}{- 2} + \frac{- 2 z}{- 2}$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

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

Divide $- 10$ by $- 2$ .

$y = 5 + \frac{- 2 z}{- 2}$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

Step 3.2.3.1.2

Cancel the common factor of $- 2$ .

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Step 3.2.3.1.2.1

Cancel the common factor.

$y = 5 + \frac{- 2 z}{- 2}$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

Step 3.2.3.1.2.2

Divide $z$ by $1$ .

$y = 5 + z$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

$y = 5 + z$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

$y = 5 + z$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

$y = 5 + z$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

$y = 5 + z$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

$y = 5 + z$

$- 7 - 8 y + 2 z = - 5$

$x = 7 + 5 y + z$

Step 4

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

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Step 4.1

Replace all occurrences of $y$ in $- 7 - 8 y + 2 z = - 5$ with $5 + z$ .

$- 7 - 8 (5 + z) + 2 z = - 5$

$y = 5 + z$

$x = 7 + 5 y + z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 7 - 8 (5 + z) + 2 z$ .

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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.

$- 7 - 8 \cdot 5 - 8 z + 2 z = - 5$

$y = 5 + z$

$x = 7 + 5 y + z$

Step 4.2.1.1.2

Multiply $- 8$ by $5$ .

$- 7 - 40 - 8 z + 2 z = - 5$

$y = 5 + z$

$x = 7 + 5 y + z$

$- 7 - 40 - 8 z + 2 z = - 5$

$y = 5 + z$

$x = 7 + 5 y + z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Subtract $40$ from $- 7$ .

$- 47 - 8 z + 2 z = - 5$

$y = 5 + z$

$x = 7 + 5 y + z$

Step 4.2.1.2.2

Add $- 8 z$ and $2 z$ .

$- 47 - 6 z = - 5$

$y = 5 + z$

$x = 7 + 5 y + z$

$- 47 - 6 z = - 5$

$y = 5 + z$

$x = 7 + 5 y + z$

$- 47 - 6 z = - 5$

$y = 5 + z$

$x = 7 + 5 y + z$

$- 47 - 6 z = - 5$

$y = 5 + z$

$x = 7 + 5 y + z$

Step 4.3

Replace all occurrences of $y$ in $x = 7 + 5 y + z$ with $5 + z$ .

$x = 7 + 5 (5 + z) + z$

$- 47 - 6 z = - 5$

$y = 5 + z$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $7 + 5 (5 + z) + z$ .

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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.

$x = 7 + 5 \cdot 5 + 5 z + z$

$- 47 - 6 z = - 5$

$y = 5 + z$

Step 4.4.1.1.2

Multiply $5$ by $5$ .

$x = 7 + 25 + 5 z + z$

$- 47 - 6 z = - 5$

$y = 5 + z$

$x = 7 + 25 + 5 z + z$

$- 47 - 6 z = - 5$

$y = 5 + z$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Add $7$ and $25$ .

$x = 32 + 5 z + z$

$- 47 - 6 z = - 5$

$y = 5 + z$

Step 4.4.1.2.2

Add $5 z$ and $z$ .

$x = 32 + 6 z$

$- 47 - 6 z = - 5$

$y = 5 + z$

$x = 32 + 6 z$

$- 47 - 6 z = - 5$

$y = 5 + z$

$x = 32 + 6 z$

$- 47 - 6 z = - 5$

$y = 5 + z$

$x = 32 + 6 z$

$- 47 - 6 z = - 5$

$y = 5 + z$

$x = 32 + 6 z$

$- 47 - 6 z = - 5$

$y = 5 + z$

Step 5

Solve for $z$ in $- 47 - 6 z = - 5$ .

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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

Add $47$ to both sides of the equation.

$- 6 z = - 5 + 47$

$x = 32 + 6 z$

$y = 5 + z$

Step 5.1.2

Add $- 5$ and $47$ .

$- 6 z = 42$

$x = 32 + 6 z$

$y = 5 + z$

$- 6 z = 42$

$x = 32 + 6 z$

$y = 5 + z$

Step 5.2

Divide each term in $- 6 z = 42$ by $- 6$ and simplify.

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Step 5.2.1

Divide each term in $- 6 z = 42$ by $- 6$ .

$\frac{- 6 z}{- 6} = \frac{42}{- 6}$

$x = 32 + 6 z$

$y = 5 + z$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 6$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 6 z}{- 6} = \frac{42}{- 6}$

$x = 32 + 6 z$

$y = 5 + z$

Step 5.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{42}{- 6}$

$x = 32 + 6 z$

$y = 5 + z$

$z = \frac{42}{- 6}$

$x = 32 + 6 z$

$y = 5 + z$

$z = \frac{42}{- 6}$

$x = 32 + 6 z$

$y = 5 + z$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $42$ by $- 6$ .

$z = - 7$

$x = 32 + 6 z$

$y = 5 + z$

$z = - 7$

$x = 32 + 6 z$

$y = 5 + z$

$z = - 7$

$x = 32 + 6 z$

$y = 5 + z$

$z = - 7$

$x = 32 + 6 z$

$y = 5 + z$

Step 6

Replace all occurrences of $z$ with $- 7$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $x = 32 + 6 z$ with $- 7$ .

$x = 32 + 6 (- 7)$

$z = - 7$

$y = 5 + z$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $32 + 6 (- 7)$ .

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Step 6.2.1.1

Multiply $6$ by $- 7$ .

$x = 32 - 42$

$z = - 7$

$y = 5 + z$

Step 6.2.1.2

Subtract $42$ from $32$ .

$x = - 10$

$z = - 7$

$y = 5 + z$

$x = - 10$

$z = - 7$

$y = 5 + z$

$x = - 10$

$z = - 7$

$y = 5 + z$

Step 6.3

Replace all occurrences of $z$ in $y = 5 + z$ with $- 7$ .

$y = 5 - 7$

$x = - 10$

$z = - 7$

Step 6.4

Simplify $y = 5 - 7$ .

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Step 6.4.1

Simplify the left side.

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Step 6.4.1.1

Remove parentheses.

$y = 5 - 7$

$x = - 10$

$z = - 7$

$y = 5 - 7$

$x = - 10$

$z = - 7$

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Subtract $7$ from $5$ .

$y = - 2$

$x = - 10$

$z = - 7$

$y = - 2$

$x = - 10$

$z = - 7$

$y = - 2$

$x = - 10$

$z = - 7$

$y = - 2$

$x = - 10$

$z = - 7$

Step 7

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

$(- 10, - 2, - 7)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 10, - 2, - 7)$

Equation Form:

$x = - 10, y = - 2, z = - 7$