Finite Math Examples

$x + 2 y + z = - 4$ , $- 2 x - 3 y - z = 1$ , $4 x + 8 y + 4 z = - 16$

Step 1

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

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Step 1.1

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

$x + z = - 4 - 2 y$

$- 2 x - 3 y - z = 1$

$4 x + 8 y + 4 z = - 16$

Step 1.2

Subtract $z$ from both sides of the equation.

$x = - 4 - 2 y - z$

$- 2 x - 3 y - z = 1$

$4 x + 8 y + 4 z = - 16$

$x = - 4 - 2 y - z$

$- 2 x - 3 y - z = 1$

$4 x + 8 y + 4 z = - 16$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $- 2 x - 3 y - z = 1$ with $- 4 - 2 y - z$ .

$- 2 (- 4 - 2 y - z) - 3 y - z = 1$

$x = - 4 - 2 y - z$

$4 x + 8 y + 4 z = - 16$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 2 (- 4 - 2 y - z) - 3 y - z$ .

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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.

$- 2 \cdot - 4 - 2 (- 2 y) - 2 (- z) - 3 y - z = 1$

$x = - 4 - 2 y - z$

$4 x + 8 y + 4 z = - 16$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 2$ by $- 4$ .

$8 - 2 (- 2 y) - 2 (- z) - 3 y - z = 1$

$x = - 4 - 2 y - z$

$4 x + 8 y + 4 z = - 16$

Step 2.2.1.1.2.2

Multiply $- 2$ by $- 2$ .

$8 + 4 y - 2 (- z) - 3 y - z = 1$

$x = - 4 - 2 y - z$

$4 x + 8 y + 4 z = - 16$

Step 2.2.1.1.2.3

Multiply $- 1$ by $- 2$ .

$8 + 4 y + 2 z - 3 y - z = 1$

$x = - 4 - 2 y - z$

$4 x + 8 y + 4 z = - 16$

$8 + 4 y + 2 z - 3 y - z = 1$

$x = - 4 - 2 y - z$

$4 x + 8 y + 4 z = - 16$

$8 + 4 y + 2 z - 3 y - z = 1$

$x = - 4 - 2 y - z$

$4 x + 8 y + 4 z = - 16$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $3 y$ from $4 y$ .

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

$x = - 4 - 2 y - z$

$4 x + 8 y + 4 z = - 16$

Step 2.2.1.2.2

Subtract $z$ from $2 z$ .

$8 + y + z = 1$

$x = - 4 - 2 y - z$

$4 x + 8 y + 4 z = - 16$

$8 + y + z = 1$

$x = - 4 - 2 y - z$

$4 x + 8 y + 4 z = - 16$

$8 + y + z = 1$

$x = - 4 - 2 y - z$

$4 x + 8 y + 4 z = - 16$

$8 + y + z = 1$

$x = - 4 - 2 y - z$

$4 x + 8 y + 4 z = - 16$

Step 2.3

Replace all occurrences of $x$ in $4 x + 8 y + 4 z = - 16$ with $- 4 - 2 y - z$ .

$4 (- 4 - 2 y - z) + 8 y + 4 z = - 16$

$8 + y + z = 1$

$x = - 4 - 2 y - z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $4 (- 4 - 2 y - z) + 8 y + 4 z$ .

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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.

$4 \cdot - 4 + 4 (- 2 y) + 4 (- z) + 8 y + 4 z = - 16$

$8 + y + z = 1$

$x = - 4 - 2 y - z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $4$ by $- 4$ .

$- 16 + 4 (- 2 y) + 4 (- z) + 8 y + 4 z = - 16$

$8 + y + z = 1$

$x = - 4 - 2 y - z$

Step 2.4.1.1.2.2

Multiply $- 2$ by $4$ .

$- 16 - 8 y + 4 (- z) + 8 y + 4 z = - 16$

$8 + y + z = 1$

$x = - 4 - 2 y - z$

Step 2.4.1.1.2.3

Multiply $- 1$ by $4$ .

$- 16 - 8 y - 4 z + 8 y + 4 z = - 16$

$8 + y + z = 1$

$x = - 4 - 2 y - z$

$- 16 - 8 y - 4 z + 8 y + 4 z = - 16$

$8 + y + z = 1$

$x = - 4 - 2 y - z$

$- 16 - 8 y - 4 z + 8 y + 4 z = - 16$

$8 + y + z = 1$

$x = - 4 - 2 y - z$

Step 2.4.1.2

Combine the opposite terms in $- 16 - 8 y - 4 z + 8 y + 4 z$ .

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Step 2.4.1.2.1

Add $- 8 y$ and $8 y$ .

$- 16 - 4 z + 0 + 4 z = - 16$

$8 + y + z = 1$

$x = - 4 - 2 y - z$

Step 2.4.1.2.2

Add $- 16 - 4 z$ and $0$ .

$- 16 - 4 z + 4 z = - 16$

$8 + y + z = 1$

$x = - 4 - 2 y - z$

Step 2.4.1.2.3

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

$- 16 + 0 = - 16$

$8 + y + z = 1$

$x = - 4 - 2 y - z$

Step 2.4.1.2.4

Add $- 16$ and $0$ .

$- 16 = - 16$

$8 + y + z = 1$

$x = - 4 - 2 y - z$

$- 16 = - 16$

$8 + y + z = 1$

$x = - 4 - 2 y - z$

$- 16 = - 16$

$8 + y + z = 1$

$x = - 4 - 2 y - z$

$- 16 = - 16$

$8 + y + z = 1$

$x = - 4 - 2 y - z$

$- 16 = - 16$

$8 + y + z = 1$

$x = - 4 - 2 y - z$

Step 3

Remove any equations from the system that are always true.

$8 + y + z = 1$

$x = - 4 - 2 y - z$