Precalculus Examples

$4 x - 3 y - 4 z = - 1$ , $12 x - 9 y - 12 z = - 1$ , $2 x - 4 y - 3 z = 4$

Step 1

Solve for $x$ in $4 x - 3 y - 4 z = - 1$ .

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Step 1.1

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

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Step 1.1.1

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

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

$12 x - 9 y - 12 z = - 1$

$2 x - 4 y - 3 z = 4$

Step 1.1.2

Add $4 z$ to both sides of the equation.

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

$12 x - 9 y - 12 z = - 1$

$2 x - 4 y - 3 z = 4$

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

$12 x - 9 y - 12 z = - 1$

$2 x - 4 y - 3 z = 4$

Step 1.2

Divide each term in $4 x = - 1 + 3 y + 4 z$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x = - 1 + 3 y + 4 z$ by $4$ .

$\frac{4 x}{4} = \frac{- 1}{4} + \frac{3 y}{4} + \frac{4 z}{4}$

$12 x - 9 y - 12 z = - 1$

$2 x - 4 y - 3 z = 4$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 1}{4} + \frac{3 y}{4} + \frac{4 z}{4}$

$12 x - 9 y - 12 z = - 1$

$2 x - 4 y - 3 z = 4$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{4} + \frac{3 y}{4} + \frac{4 z}{4}$

$12 x - 9 y - 12 z = - 1$

$2 x - 4 y - 3 z = 4$

$x = \frac{- 1}{4} + \frac{3 y}{4} + \frac{4 z}{4}$

$12 x - 9 y - 12 z = - 1$

$2 x - 4 y - 3 z = 4$

$x = \frac{- 1}{4} + \frac{3 y}{4} + \frac{4 z}{4}$

$12 x - 9 y - 12 z = - 1$

$2 x - 4 y - 3 z = 4$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Move the negative in front of the fraction.

$x = - \frac{1}{4} + \frac{3 y}{4} + \frac{4 z}{4}$

$12 x - 9 y - 12 z = - 1$

$2 x - 4 y - 3 z = 4$

Step 1.2.3.1.2

Cancel the common factor of $4$ .

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Step 1.2.3.1.2.1

Cancel the common factor.

$x = - \frac{1}{4} + \frac{3 y}{4} + \frac{4 z}{4}$

$12 x - 9 y - 12 z = - 1$

$2 x - 4 y - 3 z = 4$

Step 1.2.3.1.2.2

Divide $z$ by $1$ .

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$12 x - 9 y - 12 z = - 1$

$2 x - 4 y - 3 z = 4$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$12 x - 9 y - 12 z = - 1$

$2 x - 4 y - 3 z = 4$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$12 x - 9 y - 12 z = - 1$

$2 x - 4 y - 3 z = 4$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$12 x - 9 y - 12 z = - 1$

$2 x - 4 y - 3 z = 4$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$12 x - 9 y - 12 z = - 1$

$2 x - 4 y - 3 z = 4$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$12 x - 9 y - 12 z = - 1$

$2 x - 4 y - 3 z = 4$

Step 2

Replace all occurrences of $x$ with $- \frac{1}{4} + \frac{3 y}{4} + z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $12 x - 9 y - 12 z = - 1$ with $- \frac{1}{4} + \frac{3 y}{4} + z$ .

$12 (- \frac{1}{4} + \frac{3 y}{4} + z) - 9 y - 12 z = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $12 (- \frac{1}{4} + \frac{3 y}{4} + z) - 9 y - 12 z$ .

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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.

$12 (- \frac{1}{4}) + 12 (\frac{3 y}{4}) + 12 z - 9 y - 12 z = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1.2.1.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$12 (\frac{- 1}{4}) + 12 (\frac{3 y}{4}) + 12 z - 9 y - 12 z = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

Step 2.2.1.1.2.1.2

Factor $4$ out of $12$ .

$4 (3) (\frac{- 1}{4}) + 12 (\frac{3 y}{4}) + 12 z - 9 y - 12 z = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

Step 2.2.1.1.2.1.3

Cancel the common factor.

$4 \cdot (3 (\frac{- 1}{4})) + 12 (\frac{3 y}{4}) + 12 z - 9 y - 12 z = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

Step 2.2.1.1.2.1.4

Rewrite the expression.

$3 \cdot - 1 + 12 (\frac{3 y}{4}) + 12 z - 9 y - 12 z = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

$3 \cdot - 1 + 12 (\frac{3 y}{4}) + 12 z - 9 y - 12 z = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

Step 2.2.1.1.2.2

Multiply $3$ by $- 1$ .

$- 3 + 12 (\frac{3 y}{4}) + 12 z - 9 y - 12 z = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

Step 2.2.1.1.2.3

Cancel the common factor of $4$ .

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Step 2.2.1.1.2.3.1

Factor $4$ out of $12$ .

$- 3 + 4 (3) (\frac{3 y}{4}) + 12 z - 9 y - 12 z = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

Step 2.2.1.1.2.3.2

Cancel the common factor.

$- 3 + 4 \cdot (3 (\frac{3 y}{4})) + 12 z - 9 y - 12 z = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

Step 2.2.1.1.2.3.3

Rewrite the expression.

$- 3 + 3 (3 y) + 12 z - 9 y - 12 z = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

$- 3 + 3 (3 y) + 12 z - 9 y - 12 z = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

Step 2.2.1.1.2.4

Multiply $3$ by $3$ .

$- 3 + 9 y + 12 z - 9 y - 12 z = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

$- 3 + 9 y + 12 z - 9 y - 12 z = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

$- 3 + 9 y + 12 z - 9 y - 12 z = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

Step 2.2.1.2

Combine the opposite terms in $- 3 + 9 y + 12 z - 9 y - 12 z$ .

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Step 2.2.1.2.1

Subtract $9 y$ from $9 y$ .

$- 3 + 12 z + 0 - 12 z = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

Step 2.2.1.2.2

Add $- 3 + 12 z$ and $0$ .

$- 3 + 12 z - 12 z = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

Step 2.2.1.2.3

Subtract $12 z$ from $12 z$ .

$- 3 + 0 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

Step 2.2.1.2.4

Add $- 3$ and $0$ .

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$2 x - 4 y - 3 z = 4$

Step 2.3

Replace all occurrences of $x$ in $2 x - 4 y - 3 z = 4$ with $- \frac{1}{4} + \frac{3 y}{4} + z$ .

$2 (- \frac{1}{4} + \frac{3 y}{4} + z) - 4 y - 3 z = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $2 (- \frac{1}{4} + \frac{3 y}{4} + z) - 4 y - 3 z$ .

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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.

$2 (- \frac{1}{4}) + 2 (\frac{3 y}{4}) + 2 z - 4 y - 3 z = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Cancel the common factor of $2$ .

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Step 2.4.1.1.2.1.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$2 (\frac{- 1}{4}) + 2 (\frac{3 y}{4}) + 2 z - 4 y - 3 z = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.1.2.1.2

Factor $2$ out of $4$ .

$2 (\frac{- 1}{2 (2)}) + 2 (\frac{3 y}{4}) + 2 z - 4 y - 3 z = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.1.2.1.3

Cancel the common factor.

$2 (\frac{- 1}{2 \cdot 2}) + 2 (\frac{3 y}{4}) + 2 z - 4 y - 3 z = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.1.2.1.4

Rewrite the expression.

$\frac{- 1}{2} + 2 (\frac{3 y}{4}) + 2 z - 4 y - 3 z = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$\frac{- 1}{2} + 2 (\frac{3 y}{4}) + 2 z - 4 y - 3 z = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.1.2.2

Cancel the common factor of $2$ .

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Step 2.4.1.1.2.2.1

Factor $2$ out of $4$ .

$\frac{- 1}{2} + 2 (\frac{3 y}{2 (2)}) + 2 z - 4 y - 3 z = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.1.2.2.2

Cancel the common factor.

$\frac{- 1}{2} + 2 (\frac{3 y}{2 \cdot 2}) + 2 z - 4 y - 3 z = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.1.2.2.3

Rewrite the expression.

$\frac{- 1}{2} + \frac{3 y}{2} + 2 z - 4 y - 3 z = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$\frac{- 1}{2} + \frac{3 y}{2} + 2 z - 4 y - 3 z = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$\frac{- 1}{2} + \frac{3 y}{2} + 2 z - 4 y - 3 z = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.1.3

Move the negative in front of the fraction.

$- \frac{1}{2} + \frac{3 y}{2} + 2 z - 4 y - 3 z = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$- \frac{1}{2} + \frac{3 y}{2} + 2 z - 4 y - 3 z = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.2

To write $- 4 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{1}{2} + 2 z + \frac{3 y}{2} - 4 y \cdot \frac{2}{2} - 3 z = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.3

Combine $- 4 y$ and $\frac{2}{2}$ .

$- \frac{1}{2} + 2 z + \frac{3 y}{2} + \frac{- 4 y \cdot 2}{2} - 3 z = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.4

Combine the numerators over the common denominator.

$- \frac{1}{2} + 2 z + \frac{3 y - 4 y \cdot 2}{2} - 3 z = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.5

Combine the numerators over the common denominator.

$2 z - 3 z + \frac{- 1 + 3 y - 4 y \cdot 2}{2} = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.6

Multiply $2$ by $- 4$ .

$2 z - 3 z + \frac{- 1 + 3 y - 8 y}{2} = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.7

Subtract $8 y$ from $3 y$ .

$2 z - 3 z + \frac{- 1 - 5 y}{2} = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.8

Subtract $3 z$ from $2 z$ .

$- z + \frac{- 1 - 5 y}{2} = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.9

To write $- z$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- z \cdot \frac{2}{2} + \frac{- 1 - 5 y}{2} = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.10

Combine $- z$ and $\frac{2}{2}$ .

$\frac{- z \cdot 2}{2} + \frac{- 1 - 5 y}{2} = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.11

Combine the numerators over the common denominator.

$\frac{- z \cdot 2 - 1 - 5 y}{2} = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.12

Multiply $2$ by $- 1$ .

$\frac{- 2 z - 1 - 5 y}{2} = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.13

Factor $- 1$ out of $- 2 z$ .

$\frac{- (2 z) - 1 - 5 y}{2} = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.14

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{- (2 z) - 1 \cdot 1 - 5 y}{2} = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.15

Factor $- 1$ out of $- (2 z) - 1 (1)$ .

$\frac{- (2 z + 1) - 5 y}{2} = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.16

Factor $- 1$ out of $- 5 y$ .

$\frac{- (2 z + 1) - (5 y)}{2} = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.17

Factor $- 1$ out of $- (2 z + 1) - (5 y)$ .

$\frac{- (2 z + 1 + 5 y)}{2} = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.18

Simplify the expression.

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Step 2.4.1.18.1

Rewrite $- (2 z + 1 + 5 y)$ as $- 1 (2 z + 1 + 5 y)$ .

$\frac{- 1 (2 z + 1 + 5 y)}{2} = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 2.4.1.18.2

Move the negative in front of the fraction.

$- \frac{2 z + 1 + 5 y}{2} = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$- \frac{2 z + 1 + 5 y}{2} = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$- \frac{2 z + 1 + 5 y}{2} = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$- \frac{2 z + 1 + 5 y}{2} = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

$- \frac{2 z + 1 + 5 y}{2} = 4$

$- 3 = - 1$

$x = - \frac{1}{4} + \frac{3 y}{4} + z$

Step 3

Since $- 3 = - 1$ is not true, there is no solution.

No solution