Precalculus Examples

$2 x + 2 y - z = 2$ , $x - 3 y + z = - 28$ , $- x + y = 4$

Step 1

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

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Step 1.1

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

$x + z = - 28 + 3 y$

$2 x + 2 y - z = 2$

$- x + y = 4$

Step 1.2

Subtract $z$ from both sides of the equation.

$x = - 28 + 3 y - z$

$2 x + 2 y - z = 2$

$- x + y = 4$

$x = - 28 + 3 y - z$

$2 x + 2 y - z = 2$

$- x + y = 4$

Step 2

Replace all occurrences of $x$ with $- 28 + 3 y - z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $2 x + 2 y - z = 2$ with $- 28 + 3 y - z$ .

$2 (- 28 + 3 y - z) + 2 y - z = 2$

$x = - 28 + 3 y - z$

$- x + y = 4$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (- 28 + 3 y - z) + 2 y - z$ .

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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.

$2 \cdot - 28 + 2 (3 y) + 2 (- z) + 2 y - z = 2$

$x = - 28 + 3 y - z$

$- x + y = 4$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $2$ by $- 28$ .

$- 56 + 2 (3 y) + 2 (- z) + 2 y - z = 2$

$x = - 28 + 3 y - z$

$- x + y = 4$

Step 2.2.1.1.2.2

Multiply $3$ by $2$ .

$- 56 + 6 y + 2 (- z) + 2 y - z = 2$

$x = - 28 + 3 y - z$

$- x + y = 4$

Step 2.2.1.1.2.3

Multiply $- 1$ by $2$ .

$- 56 + 6 y - 2 z + 2 y - z = 2$

$x = - 28 + 3 y - z$

$- x + y = 4$

$- 56 + 6 y - 2 z + 2 y - z = 2$

$x = - 28 + 3 y - z$

$- x + y = 4$

$- 56 + 6 y - 2 z + 2 y - z = 2$

$x = - 28 + 3 y - z$

$- x + y = 4$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $6 y$ and $2 y$ .

$- 56 + 8 y - 2 z - z = 2$

$x = - 28 + 3 y - z$

$- x + y = 4$

Step 2.2.1.2.2

Subtract $z$ from $- 2 z$ .

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

$- x + y = 4$

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

$- x + y = 4$

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

$- x + y = 4$

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

$- x + y = 4$

Step 2.3

Replace all occurrences of $x$ in $- x + y = 4$ with $- 28 + 3 y - z$ .

$- (- 28 + 3 y - z) + y = 4$

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $- (- 28 + 3 y - z) + y$ .

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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.

$28 - (3 y) + z + y = 4$

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 1$ by $- 28$ .

$28 - (3 y) + z + y = 4$

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

Step 2.4.1.1.2.2

Multiply $3$ by $- 1$ .

$28 - 3 y + z + y = 4$

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

Step 2.4.1.1.2.3

Multiply $- - z$ .

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Step 2.4.1.1.2.3.1

Multiply $- 1$ by $- 1$ .

$28 - 3 y + 1 z + y = 4$

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

Step 2.4.1.1.2.3.2

Multiply $z$ by $1$ .

$28 - 3 y + z + y = 4$

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

$28 - 3 y + z + y = 4$

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

$28 - 3 y + z + y = 4$

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

$28 - 3 y + z + y = 4$

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

Step 2.4.1.2

Add $- 3 y$ and $y$ .

$28 - 2 y + z = 4$

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

$28 - 2 y + z = 4$

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

$28 - 2 y + z = 4$

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

$28 - 2 y + z = 4$

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

Step 3

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

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Step 3.1

Subtract $28$ from both sides of the equation.

$- 2 y + z = 4 - 28$

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

Step 3.2

Add $2 y$ to both sides of the equation.

$z = 4 - 28 + 2 y$

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

Step 3.3

Subtract $28$ from $4$ .

$z = - 24 + 2 y$

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

$z = - 24 + 2 y$

$- 56 + 8 y - 3 z = 2$

$x = - 28 + 3 y - z$

Step 4

Replace all occurrences of $z$ with $- 24 + 2 y$ in each equation.

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Step 4.1

Replace all occurrences of $z$ in $- 56 + 8 y - 3 z = 2$ with $- 24 + 2 y$ .

$- 56 + 8 y - 3 (- 24 + 2 y) = 2$

$z = - 24 + 2 y$

$x = - 28 + 3 y - z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 56 + 8 y - 3 (- 24 + 2 y)$ .

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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.

$- 56 + 8 y - 3 \cdot - 24 - 3 (2 y) = 2$

$z = - 24 + 2 y$

$x = - 28 + 3 y - z$

Step 4.2.1.1.2

Multiply $- 3$ by $- 24$ .

$- 56 + 8 y + 72 - 3 (2 y) = 2$

$z = - 24 + 2 y$

$x = - 28 + 3 y - z$

Step 4.2.1.1.3

Multiply $2$ by $- 3$ .

$- 56 + 8 y + 72 - 6 y = 2$

$z = - 24 + 2 y$

$x = - 28 + 3 y - z$

$- 56 + 8 y + 72 - 6 y = 2$

$z = - 24 + 2 y$

$x = - 28 + 3 y - z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Add $- 56$ and $72$ .

$8 y + 16 - 6 y = 2$

$z = - 24 + 2 y$

$x = - 28 + 3 y - z$

Step 4.2.1.2.2

Subtract $6 y$ from $8 y$ .

$2 y + 16 = 2$

$z = - 24 + 2 y$

$x = - 28 + 3 y - z$

$2 y + 16 = 2$

$z = - 24 + 2 y$

$x = - 28 + 3 y - z$

$2 y + 16 = 2$

$z = - 24 + 2 y$

$x = - 28 + 3 y - z$

$2 y + 16 = 2$

$z = - 24 + 2 y$

$x = - 28 + 3 y - z$

Step 4.3

Replace all occurrences of $z$ in $x = - 28 + 3 y - z$ with $- 24 + 2 y$ .

$x = - 28 + 3 y - (- 24 + 2 y)$

$2 y + 16 = 2$

$z = - 24 + 2 y$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $- 28 + 3 y - (- 24 + 2 y)$ .

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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.

$x = - 28 + 3 y + 24 - (2 y)$

$2 y + 16 = 2$

$z = - 24 + 2 y$

Step 4.4.1.1.2

Multiply $- 1$ by $- 24$ .

$x = - 28 + 3 y + 24 - (2 y)$

$2 y + 16 = 2$

$z = - 24 + 2 y$

Step 4.4.1.1.3

Multiply $2$ by $- 1$ .

$x = - 28 + 3 y + 24 - 2 y$

$2 y + 16 = 2$

$z = - 24 + 2 y$

$x = - 28 + 3 y + 24 - 2 y$

$2 y + 16 = 2$

$z = - 24 + 2 y$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Add $- 28$ and $24$ .

$x = 3 y - 4 - 2 y$

$2 y + 16 = 2$

$z = - 24 + 2 y$

Step 4.4.1.2.2

Subtract $2 y$ from $3 y$ .

$x = y - 4$

$2 y + 16 = 2$

$z = - 24 + 2 y$

$x = y - 4$

$2 y + 16 = 2$

$z = - 24 + 2 y$

$x = y - 4$

$2 y + 16 = 2$

$z = - 24 + 2 y$

$x = y - 4$

$2 y + 16 = 2$

$z = - 24 + 2 y$

$x = y - 4$

$2 y + 16 = 2$

$z = - 24 + 2 y$

Step 5

Solve for $y$ in $2 y + 16 = 2$ .

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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 $16$ from both sides of the equation.

$2 y = 2 - 16$

$x = y - 4$

$z = - 24 + 2 y$

Step 5.1.2

Subtract $16$ from $2$ .

$2 y = - 14$

$x = y - 4$

$z = - 24 + 2 y$

$2 y = - 14$

$x = y - 4$

$z = - 24 + 2 y$

Step 5.2

Divide each term in $2 y = - 14$ by $2$ and simplify.

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Step 5.2.1

Divide each term in $2 y = - 14$ by $2$ .

$\frac{2 y}{2} = \frac{- 14}{2}$

$x = y - 4$

$z = - 24 + 2 y$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{- 14}{2}$

$x = y - 4$

$z = - 24 + 2 y$

Step 5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 14}{2}$

$x = y - 4$

$z = - 24 + 2 y$

$y = \frac{- 14}{2}$

$x = y - 4$

$z = - 24 + 2 y$

$y = \frac{- 14}{2}$

$x = y - 4$

$z = - 24 + 2 y$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $- 14$ by $2$ .

$y = - 7$

$x = y - 4$

$z = - 24 + 2 y$

$y = - 7$

$x = y - 4$

$z = - 24 + 2 y$

$y = - 7$

$x = y - 4$

$z = - 24 + 2 y$

$y = - 7$

$x = y - 4$

$z = - 24 + 2 y$

Step 6

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

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Step 6.1

Replace all occurrences of $y$ in $x = y - 4$ with $- 7$ .

$x = (- 7) - 4$

$y = - 7$

$z = - 24 + 2 y$

Step 6.2

Simplify the right side.

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Step 6.2.1

Subtract $4$ from $- 7$ .

$x = - 11$

$y = - 7$

$z = - 24 + 2 y$

$x = - 11$

$y = - 7$

$z = - 24 + 2 y$

Step 6.3

Replace all occurrences of $y$ in $z = - 24 + 2 y$ with $- 7$ .

$z = - 24 + 2 (- 7)$

$x = - 11$

$y = - 7$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- 24 + 2 (- 7)$ .

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Step 6.4.1.1

Multiply $2$ by $- 7$ .

$z = - 24 - 14$

$x = - 11$

$y = - 7$

Step 6.4.1.2

Subtract $14$ from $- 24$ .

$z = - 38$

$x = - 11$

$y = - 7$

$z = - 38$

$x = - 11$

$y = - 7$

$z = - 38$

$x = - 11$

$y = - 7$

$z = - 38$

$x = - 11$

$y = - 7$

Step 7

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

$(- 11, - 7, - 38)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 11, - 7, - 38)$

Equation Form:

$x = - 11, y = - 7, z = - 38$