Precalculus Examples

2 x + 2 y - z = 2

$2 x + 2 y - z = 2$ ,

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

$- x + y = 4$

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:

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