Precalculus Examples

$x + y + 5 z = 7$ , $3 x + 2 y + 14 z = 5$ , $x + 4 z = - 9$

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.

$x + 5 z = 7 - y$

$3 x + 2 y + 14 z = 5$

$x + 4 z = - 9$

Step 1.2

Subtract $5 z$ from both sides of the equation.

$x = 7 - y - 5 z$

$3 x + 2 y + 14 z = 5$

$x + 4 z = - 9$

$x = 7 - y - 5 z$

$3 x + 2 y + 14 z = 5$

$x + 4 z = - 9$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $3 x + 2 y + 14 z = 5$ with $7 - y - 5 z$ .

$3 (7 - y - 5 z) + 2 y + 14 z = 5$

$x = 7 - y - 5 z$

$x + 4 z = - 9$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (7 - y - 5 z) + 2 y + 14 z$ .

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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.

$3 \cdot 7 + 3 (- y) + 3 (- 5 z) + 2 y + 14 z = 5$

$x = 7 - y - 5 z$

$x + 4 z = - 9$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $3$ by $7$ .

$21 + 3 (- y) + 3 (- 5 z) + 2 y + 14 z = 5$

$x = 7 - y - 5 z$

$x + 4 z = - 9$

Step 2.2.1.1.2.2

Multiply $- 1$ by $3$ .

$21 - 3 y + 3 (- 5 z) + 2 y + 14 z = 5$

$x = 7 - y - 5 z$

$x + 4 z = - 9$

Step 2.2.1.1.2.3

Multiply $- 5$ by $3$ .

$21 - 3 y - 15 z + 2 y + 14 z = 5$

$x = 7 - y - 5 z$

$x + 4 z = - 9$

$21 - 3 y - 15 z + 2 y + 14 z = 5$

$x = 7 - y - 5 z$

$x + 4 z = - 9$

$21 - 3 y - 15 z + 2 y + 14 z = 5$

$x = 7 - y - 5 z$

$x + 4 z = - 9$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $- 3 y$ and $2 y$ .

$21 - y - 15 z + 14 z = 5$

$x = 7 - y - 5 z$

$x + 4 z = - 9$

Step 2.2.1.2.2

Add $- 15 z$ and $14 z$ .

$21 - y - z = 5$

$x = 7 - y - 5 z$

$x + 4 z = - 9$

$21 - y - z = 5$

$x = 7 - y - 5 z$

$x + 4 z = - 9$

$21 - y - z = 5$

$x = 7 - y - 5 z$

$x + 4 z = - 9$

$21 - y - z = 5$

$x = 7 - y - 5 z$

$x + 4 z = - 9$

Step 2.3

Replace all occurrences of $x$ in $x + 4 z = - 9$ with $7 - y - 5 z$ .

$(7 - y - 5 z) + 4 z = - 9$

$21 - y - z = 5$

$x = 7 - y - 5 z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Add $- 5 z$ and $4 z$ .

$7 - y - z = - 9$

$21 - y - z = 5$

$x = 7 - y - 5 z$

$7 - y - z = - 9$

$21 - y - z = 5$

$x = 7 - y - 5 z$

$7 - y - z = - 9$

$21 - y - z = 5$

$x = 7 - y - 5 z$

Step 3

Solve for $y$ in $7 - y - z = - 9$ .

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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.

$- y - z = - 9 - 7$

$21 - y - z = 5$

$x = 7 - y - 5 z$

Step 3.1.2

Add $z$ to both sides of the equation.

$- y = - 9 - 7 + z$

$21 - y - z = 5$

$x = 7 - y - 5 z$

Step 3.1.3

Subtract $7$ from $- 9$ .

$- y = - 16 + z$

$21 - y - z = 5$

$x = 7 - y - 5 z$

$- y = - 16 + z$

$21 - y - z = 5$

$x = 7 - y - 5 z$

Step 3.2

Divide each term in $- y = - 16 + z$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- y = - 16 + z$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 16}{- 1} + \frac{z}{- 1}$

$21 - y - z = 5$

$x = 7 - y - 5 z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 16}{- 1} + \frac{z}{- 1}$

$21 - y - z = 5$

$x = 7 - y - 5 z$

Step 3.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 16}{- 1} + \frac{z}{- 1}$

$21 - y - z = 5$

$x = 7 - y - 5 z$

$y = \frac{- 16}{- 1} + \frac{z}{- 1}$

$21 - y - z = 5$

$x = 7 - y - 5 z$

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 $- 16$ by $- 1$ .

$y = 16 + \frac{z}{- 1}$

$21 - y - z = 5$

$x = 7 - y - 5 z$

Step 3.2.3.1.2

Move the negative one from the denominator of $\frac{z}{- 1}$ .

$y = 16 - 1 \cdot z$

$21 - y - z = 5$

$x = 7 - y - 5 z$

Step 3.2.3.1.3

Rewrite $- 1 \cdot z$ as $- z$ .

$y = 16 - z$

$21 - y - z = 5$

$x = 7 - y - 5 z$

$y = 16 - z$

$21 - y - z = 5$

$x = 7 - y - 5 z$

$y = 16 - z$

$21 - y - z = 5$

$x = 7 - y - 5 z$

$y = 16 - z$

$21 - y - z = 5$

$x = 7 - y - 5 z$

$y = 16 - z$

$21 - y - z = 5$

$x = 7 - y - 5 z$

Step 4

Replace all occurrences of $y$ with $16 - z$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $21 - y - z = 5$ with $16 - z$ .

$21 - (16 - z) - z = 5$

$y = 16 - z$

$x = 7 - y - 5 z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $21 - (16 - z) - z$ .

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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.

$21 - 1 \cdot 16 + z - z = 5$

$y = 16 - z$

$x = 7 - y - 5 z$

Step 4.2.1.1.2

Multiply $- 1$ by $16$ .

$21 - 16 + z - z = 5$

$y = 16 - z$

$x = 7 - y - 5 z$

Step 4.2.1.1.3

Multiply $- - z$ .

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Step 4.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$21 - 16 + 1 z - z = 5$

$y = 16 - z$

$x = 7 - y - 5 z$

Step 4.2.1.1.3.2

Multiply $z$ by $1$ .

$21 - 16 + z - z = 5$

$y = 16 - z$

$x = 7 - y - 5 z$

$21 - 16 + z - z = 5$

$y = 16 - z$

$x = 7 - y - 5 z$

$21 - 16 + z - z = 5$

$y = 16 - z$

$x = 7 - y - 5 z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Combine the opposite terms in $21 - 16 + z - z$ .

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Step 4.2.1.2.1.1

Subtract $z$ from $z$ .

$21 - 16 + 0 = 5$

$y = 16 - z$

$x = 7 - y - 5 z$

Step 4.2.1.2.1.2

Add $21 - 16$ and $0$ .

$21 - 16 = 5$

$y = 16 - z$

$x = 7 - y - 5 z$

$21 - 16 = 5$

$y = 16 - z$

$x = 7 - y - 5 z$

Step 4.2.1.2.2

Subtract $16$ from $21$ .

$5 = 5$

$y = 16 - z$

$x = 7 - y - 5 z$

$5 = 5$

$y = 16 - z$

$x = 7 - y - 5 z$

$5 = 5$

$y = 16 - z$

$x = 7 - y - 5 z$

$5 = 5$

$y = 16 - z$

$x = 7 - y - 5 z$

Step 4.3

Replace all occurrences of $y$ in $x = 7 - y - 5 z$ with $16 - z$ .

$x = 7 - (16 - z) - 5 z$

$5 = 5$

$y = 16 - z$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $7 - (16 - z) - 5 z$ .

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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.

$x = 7 - 1 \cdot 16 + z - 5 z$

$5 = 5$

$y = 16 - z$

Step 4.4.1.1.2

Multiply $- 1$ by $16$ .

$x = 7 - 16 + z - 5 z$

$5 = 5$

$y = 16 - z$

Step 4.4.1.1.3

Multiply $- - z$ .

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Step 4.4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$x = 7 - 16 + 1 z - 5 z$

$5 = 5$

$y = 16 - z$

Step 4.4.1.1.3.2

Multiply $z$ by $1$ .

$x = 7 - 16 + z - 5 z$

$5 = 5$

$y = 16 - z$

$x = 7 - 16 + z - 5 z$

$5 = 5$

$y = 16 - z$

$x = 7 - 16 + z - 5 z$

$5 = 5$

$y = 16 - z$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Subtract $16$ from $7$ .

$x = - 9 + z - 5 z$

$5 = 5$

$y = 16 - z$

Step 4.4.1.2.2

Subtract $5 z$ from $z$ .

$x = - 9 - 4 z$

$5 = 5$

$y = 16 - z$

$x = - 9 - 4 z$

$5 = 5$

$y = 16 - z$

$x = - 9 - 4 z$

$5 = 5$

$y = 16 - z$

$x = - 9 - 4 z$

$5 = 5$

$y = 16 - z$

$x = - 9 - 4 z$

$5 = 5$

$y = 16 - z$

Step 5

Remove any equations from the system that are always true.

$x = - 9 - 4 z$

$y = 16 - z$