Precalculus Examples

$5 x - 3 y + 2 z = - 5$ , $2 x + y - z = 4$ , $- 4 x - 2 y + 2 z = - 1$

Step 1

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

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Step 1.1

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

$y - z = 4 - 2 x$

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

$- 4 x - 2 y + 2 z = - 1$

Step 1.2

Add $z$ to both sides of the equation.

$y = 4 - 2 x + z$

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

$- 4 x - 2 y + 2 z = - 1$

$y = 4 - 2 x + z$

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

$- 4 x - 2 y + 2 z = - 1$

Step 2

Replace all occurrences of $y$ with $4 - 2 x + z$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $5 x - 3 y + 2 z = - 5$ with $4 - 2 x + z$ .

$5 x - 3 (4 - 2 x + z) + 2 z = - 5$

$y = 4 - 2 x + z$

$- 4 x - 2 y + 2 z = - 1$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $5 x - 3 (4 - 2 x + z) + 2 z$ .

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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.

$5 x - 3 \cdot 4 - 3 (- 2 x) - 3 z + 2 z = - 5$

$y = 4 - 2 x + z$

$- 4 x - 2 y + 2 z = - 1$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 3$ by $4$ .

$5 x - 12 - 3 (- 2 x) - 3 z + 2 z = - 5$

$y = 4 - 2 x + z$

$- 4 x - 2 y + 2 z = - 1$

Step 2.2.1.1.2.2

Multiply $- 2$ by $- 3$ .

$5 x - 12 + 6 x - 3 z + 2 z = - 5$

$y = 4 - 2 x + z$

$- 4 x - 2 y + 2 z = - 1$

$5 x - 12 + 6 x - 3 z + 2 z = - 5$

$y = 4 - 2 x + z$

$- 4 x - 2 y + 2 z = - 1$

$5 x - 12 + 6 x - 3 z + 2 z = - 5$

$y = 4 - 2 x + z$

$- 4 x - 2 y + 2 z = - 1$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $5 x$ and $6 x$ .

$11 x - 12 - 3 z + 2 z = - 5$

$y = 4 - 2 x + z$

$- 4 x - 2 y + 2 z = - 1$

Step 2.2.1.2.2

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

$11 x - 12 - z = - 5$

$y = 4 - 2 x + z$

$- 4 x - 2 y + 2 z = - 1$

$11 x - 12 - z = - 5$

$y = 4 - 2 x + z$

$- 4 x - 2 y + 2 z = - 1$

$11 x - 12 - z = - 5$

$y = 4 - 2 x + z$

$- 4 x - 2 y + 2 z = - 1$

$11 x - 12 - z = - 5$

$y = 4 - 2 x + z$

$- 4 x - 2 y + 2 z = - 1$

Step 2.3

Replace all occurrences of $y$ in $- 4 x - 2 y + 2 z = - 1$ with $4 - 2 x + z$ .

$- 4 x - 2 (4 - 2 x + z) + 2 z = - 1$

$11 x - 12 - z = - 5$

$y = 4 - 2 x + z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $- 4 x - 2 (4 - 2 x + z) + 2 z$ .

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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.

$- 4 x - 2 \cdot 4 - 2 (- 2 x) - 2 z + 2 z = - 1$

$11 x - 12 - z = - 5$

$y = 4 - 2 x + z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 2$ by $4$ .

$- 4 x - 8 - 2 (- 2 x) - 2 z + 2 z = - 1$

$11 x - 12 - z = - 5$

$y = 4 - 2 x + z$

Step 2.4.1.1.2.2

Multiply $- 2$ by $- 2$ .

$- 4 x - 8 + 4 x - 2 z + 2 z = - 1$

$11 x - 12 - z = - 5$

$y = 4 - 2 x + z$

$- 4 x - 8 + 4 x - 2 z + 2 z = - 1$

$11 x - 12 - z = - 5$

$y = 4 - 2 x + z$

$- 4 x - 8 + 4 x - 2 z + 2 z = - 1$

$11 x - 12 - z = - 5$

$y = 4 - 2 x + z$

Step 2.4.1.2

Combine the opposite terms in $- 4 x - 8 + 4 x - 2 z + 2 z$ .

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Step 2.4.1.2.1

Add $- 4 x$ and $4 x$ .

$0 - 8 - 2 z + 2 z = - 1$

$11 x - 12 - z = - 5$

$y = 4 - 2 x + z$

Step 2.4.1.2.2

Subtract $8$ from $0$ .

$- 8 - 2 z + 2 z = - 1$

$11 x - 12 - z = - 5$

$y = 4 - 2 x + z$

Step 2.4.1.2.3

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

$- 8 + 0 = - 1$

$11 x - 12 - z = - 5$

$y = 4 - 2 x + z$

Step 2.4.1.2.4

Add $- 8$ and $0$ .

$- 8 = - 1$

$11 x - 12 - z = - 5$

$y = 4 - 2 x + z$

$- 8 = - 1$

$11 x - 12 - z = - 5$

$y = 4 - 2 x + z$

$- 8 = - 1$

$11 x - 12 - z = - 5$

$y = 4 - 2 x + z$

$- 8 = - 1$

$11 x - 12 - z = - 5$

$y = 4 - 2 x + z$

$- 8 = - 1$

$11 x - 12 - z = - 5$

$y = 4 - 2 x + z$

Step 3

Since $- 8 = - 1$ is not true, there is no solution.

No solution