Precalculus Examples

$x + y + 2 z = 11$ , $x + y + 3 z = 14$ , $x + 2 y - z = 5$

Step 1

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

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Step 1.1

Subtract $y$ from both sides of the equation.

$x + 2 z = 11 - y$

$x + y + 3 z = 14$

$x + 2 y - z = 5$

Step 1.2

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

$x = 11 - y - 2 z$

$x + y + 3 z = 14$

$x + 2 y - z = 5$

$x = 11 - y - 2 z$

$x + y + 3 z = 14$

$x + 2 y - z = 5$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $x + y + 3 z = 14$ with $11 - y - 2 z$ .

$(11 - y - 2 z) + y + 3 z = 14$

$x = 11 - y - 2 z$

$x + 2 y - z = 5$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Combine the opposite terms in $(11 - y - 2 z) + y + 3 z$ .

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Step 2.2.1.1.1

Add $- y$ and $y$ .

$11 - 2 z + 0 + 3 z = 14$

$x = 11 - y - 2 z$

$x + 2 y - z = 5$

Step 2.2.1.1.2

Add $11 - 2 z$ and $0$ .

$11 - 2 z + 3 z = 14$

$x = 11 - y - 2 z$

$x + 2 y - z = 5$

$11 - 2 z + 3 z = 14$

$x = 11 - y - 2 z$

$x + 2 y - z = 5$

Step 2.2.1.2

Add $- 2 z$ and $3 z$ .

$11 + z = 14$

$x = 11 - y - 2 z$

$x + 2 y - z = 5$

$11 + z = 14$

$x = 11 - y - 2 z$

$x + 2 y - z = 5$

$11 + z = 14$

$x = 11 - y - 2 z$

$x + 2 y - z = 5$

Step 2.3

Replace all occurrences of $x$ in $x + 2 y - z = 5$ with $11 - y - 2 z$ .

$(11 - y - 2 z) + 2 y - z = 5$

$11 + z = 14$

$x = 11 - y - 2 z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $(11 - y - 2 z) + 2 y - z$ .

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Step 2.4.1.1

Add $- y$ and $2 y$ .

$11 + y - 2 z - z = 5$

$11 + z = 14$

$x = 11 - y - 2 z$

Step 2.4.1.2

Subtract $z$ from $- 2 z$ .

$11 + y - 3 z = 5$

$11 + z = 14$

$x = 11 - y - 2 z$

$11 + y - 3 z = 5$

$11 + z = 14$

$x = 11 - y - 2 z$

$11 + y - 3 z = 5$

$11 + z = 14$

$x = 11 - y - 2 z$

$11 + y - 3 z = 5$

$11 + z = 14$

$x = 11 - y - 2 z$

Step 3

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

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Step 3.1

Subtract $11$ from both sides of the equation.

$z = 14 - 11$

$11 + y - 3 z = 5$

$x = 11 - y - 2 z$

Step 3.2

Subtract $11$ from $14$ .

$z = 3$

$11 + y - 3 z = 5$

$x = 11 - y - 2 z$

$z = 3$

$11 + y - 3 z = 5$

$x = 11 - y - 2 z$

Step 4

Replace all occurrences of $z$ with $3$ in each equation.

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Step 4.1

Replace all occurrences of $z$ in $11 + y - 3 z = 5$ with $3$ .

$11 + y - 3 \cdot 3 = 5$

$z = 3$

$x = 11 - y - 2 z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $11 + y - 3 \cdot 3$ .

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Step 4.2.1.1

Multiply $- 3$ by $3$ .

$11 + y - 9 = 5$

$z = 3$

$x = 11 - y - 2 z$

Step 4.2.1.2

Subtract $9$ from $11$ .

$y + 2 = 5$

$z = 3$

$x = 11 - y - 2 z$

$y + 2 = 5$

$z = 3$

$x = 11 - y - 2 z$

$y + 2 = 5$

$z = 3$

$x = 11 - y - 2 z$

Step 4.3

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

$x = 11 - y - 2 \cdot 3$

$y + 2 = 5$

$z = 3$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $11 - y - 2 \cdot 3$ .

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Step 4.4.1.1

Multiply $- 2$ by $3$ .

$x = 11 - y - 6$

$y + 2 = 5$

$z = 3$

Step 4.4.1.2

Subtract $6$ from $11$ .

$x = - y + 5$

$y + 2 = 5$

$z = 3$

$x = - y + 5$

$y + 2 = 5$

$z = 3$

$x = - y + 5$

$y + 2 = 5$

$z = 3$

$x = - y + 5$

$y + 2 = 5$

$z = 3$

Step 5

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

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Step 5.1

Subtract $2$ from both sides of the equation.

$y = 5 - 2$

$x = - y + 5$

$z = 3$

Step 5.2

Subtract $2$ from $5$ .

$y = 3$

$x = - y + 5$

$z = 3$

$y = 3$

$x = - y + 5$

$z = 3$

Step 6

Replace all occurrences of $y$ with $3$ in each equation.

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Step 6.1

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

$x = - (3) + 5$

$y = 3$

$z = 3$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- (3) + 5$ .

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Step 6.2.1.1

Multiply $- 1$ by $3$ .

$x = - 3 + 5$

$y = 3$

$z = 3$

Step 6.2.1.2

Add $- 3$ and $5$ .

$x = 2$

$y = 3$

$z = 3$

$x = 2$

$y = 3$

$z = 3$

$x = 2$

$y = 3$

$z = 3$

$x = 2$

$y = 3$

$z = 3$

Step 7

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

$(2, 3, 3)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(2, 3, 3)$

Equation Form:

$x = 2, y = 3, z = 3$