Finite Math Examples

$x + w = 6$ , $2 x + y + w = 16$ , $x - 2 z = 0$ , $z + w = 5$

Step 1

Subtract $w$ from both sides of the equation.

$x = 6 - w$

$2 x + y + w = 16$

$x - 2 z = 0$

$z + w = 5$

Step 2

Replace all occurrences of $x$ with $6 - w$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $2 x + y + w = 16$ with $6 - w$ .

$2 (6 - w) + y + w = 16$

$x = 6 - w$

$x - 2 z = 0$

$z + w = 5$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (6 - w) + y + w$ .

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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.

$2 \cdot 6 + 2 (- w) + y + w = 16$

$x = 6 - w$

$x - 2 z = 0$

$z + w = 5$

Step 2.2.1.1.2

Multiply $2$ by $6$ .

$12 + 2 (- w) + y + w = 16$

$x = 6 - w$

$x - 2 z = 0$

$z + w = 5$

Step 2.2.1.1.3

Multiply $- 1$ by $2$ .

$12 - 2 w + y + w = 16$

$x = 6 - w$

$x - 2 z = 0$

$z + w = 5$

$12 - 2 w + y + w = 16$

$x = 6 - w$

$x - 2 z = 0$

$z + w = 5$

Step 2.2.1.2

Add $- 2 w$ and $w$ .

$12 - w + y = 16$

$x = 6 - w$

$x - 2 z = 0$

$z + w = 5$

$12 - w + y = 16$

$x = 6 - w$

$x - 2 z = 0$

$z + w = 5$

$12 - w + y = 16$

$x = 6 - w$

$x - 2 z = 0$

$z + w = 5$

Step 2.3

Replace all occurrences of $x$ in $x - 2 z = 0$ with $6 - w$ .

$(6 - w) - 2 z = 0$

$12 - w + y = 16$

$x = 6 - w$

$z + w = 5$

Step 2.4

Simplify the left side.

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Step 2.4.1

Remove parentheses.

$6 - w - 2 z = 0$

$12 - w + y = 16$

$x = 6 - w$

$z + w = 5$

$6 - w - 2 z = 0$

$12 - w + y = 16$

$x = 6 - w$

$z + w = 5$

$6 - w - 2 z = 0$

$12 - w + y = 16$

$x = 6 - w$

$z + w = 5$

Step 3

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

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Step 3.1

Subtract $12$ from both sides of the equation.

$- w + y = 16 - 12$

$6 - w - 2 z = 0$

$x = 6 - w$

$z + w = 5$

Step 3.2

Add $w$ to both sides of the equation.

$y = 16 - 12 + w$

$6 - w - 2 z = 0$

$x = 6 - w$

$z + w = 5$

Step 3.3

Subtract $12$ from $16$ .

$y = 4 + w$

$6 - w - 2 z = 0$

$x = 6 - w$

$z + w = 5$

$y = 4 + w$

$6 - w - 2 z = 0$

$x = 6 - w$

$z + w = 5$

Step 4

Reorder $4$ and $w$ .

$y = w + 4$

$6 - w - 2 z = 0$

$x = 6 - w$

$z + w = 5$

Step 5

Reorder $6$ and $- w$ .

$x = - w + 6$

$y = w + 4$

$6 - w - 2 z = 0$

$z + w = 5$

Step 6

Subtract $w$ from both sides of the equation.

$z = 5 - w$

$x = - w + 6$

$y = w + 4$

$6 - w - 2 z = 0$

Step 7

Replace all occurrences of $z$ with $5 - w$ in each equation.

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Step 7.1

Replace all occurrences of $z$ in $6 - w - 2 z = 0$ with $5 - w$ .

$6 - w - 2 (5 - w) = 0$

$z = 5 - w$

$x = - w + 6$

$y = w + 4$

Step 7.2

Simplify the left side.

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Step 7.2.1

Simplify $6 - w - 2 (5 - w)$ .

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Step 7.2.1.1

Simplify each term.

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Step 7.2.1.1.1

Apply the distributive property.

$6 - w - 2 \cdot 5 - 2 (- w) = 0$

$z = 5 - w$

$x = - w + 6$

$y = w + 4$

Step 7.2.1.1.2

Multiply $- 2$ by $5$ .

$6 - w - 10 - 2 (- w) = 0$

$z = 5 - w$

$x = - w + 6$

$y = w + 4$

Step 7.2.1.1.3

Multiply $- 1$ by $- 2$ .

$6 - w - 10 + 2 w = 0$

$z = 5 - w$

$x = - w + 6$

$y = w + 4$

$6 - w - 10 + 2 w = 0$

$z = 5 - w$

$x = - w + 6$

$y = w + 4$

Step 7.2.1.2

Simplify by adding terms.

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Step 7.2.1.2.1

Subtract $10$ from $6$ .

$- w - 4 + 2 w = 0$

$z = 5 - w$

$x = - w + 6$

$y = w + 4$

Step 7.2.1.2.2

Add $- w$ and $2 w$ .

$w - 4 = 0$

$z = 5 - w$

$x = - w + 6$

$y = w + 4$

$w - 4 = 0$

$z = 5 - w$

$x = - w + 6$

$y = w + 4$

$w - 4 = 0$

$z = 5 - w$

$x = - w + 6$

$y = w + 4$

$w - 4 = 0$

$z = 5 - w$

$x = - w + 6$

$y = w + 4$

$w - 4 = 0$

$z = 5 - w$

$x = - w + 6$

$y = w + 4$

Step 8

Add $4$ to both sides of the equation.

$w = 4$

$z = 5 - w$

$x = - w + 6$

$y = w + 4$

Step 9

Replace all occurrences of $w$ with $4$ in each equation.

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Step 9.1

Replace all occurrences of $w$ in $z = 5 - w$ with $4$ .

$z = 5 - (4)$

$w = 4$

$x = - w + 6$

$y = w + 4$

Step 9.2

Simplify the right side.

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Step 9.2.1

Simplify $5 - (4)$ .

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Step 9.2.1.1

Multiply $- 1$ by $4$ .

$z = 5 - 4$

$w = 4$

$x = - w + 6$

$y = w + 4$

Step 9.2.1.2

Subtract $4$ from $5$ .

$z = 1$

$w = 4$

$x = - w + 6$

$y = w + 4$

$z = 1$

$w = 4$

$x = - w + 6$

$y = w + 4$

$z = 1$

$w = 4$

$x = - w + 6$

$y = w + 4$

Step 9.3

Replace all occurrences of $w$ in $x = - w + 6$ with $4$ .

$x = - (4) + 6$

$z = 1$

$w = 4$

$y = w + 4$

Step 9.4

Simplify the right side.

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Step 9.4.1

Simplify $- (4) + 6$ .

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Step 9.4.1.1

Multiply $- 1$ by $4$ .

$x = - 4 + 6$

$z = 1$

$w = 4$

$y = w + 4$

Step 9.4.1.2

Add $- 4$ and $6$ .

$x = 2$

$z = 1$

$w = 4$

$y = w + 4$

$x = 2$

$z = 1$

$w = 4$

$y = w + 4$

$x = 2$

$z = 1$

$w = 4$

$y = w + 4$

Step 9.5

Replace all occurrences of $w$ in $y = w + 4$ with $4$ .

$y = (4) + 4$

$x = 2$

$z = 1$

$w = 4$

Step 9.6

Simplify the right side.

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Step 9.6.1

Add $4$ and $4$ .

$y = 8$

$x = 2$

$z = 1$

$w = 4$

$y = 8$

$x = 2$

$z = 1$

$w = 4$

$y = 8$

$x = 2$

$z = 1$

$w = 4$

Step 10

List all of the solutions.

$y = 8, x = 2, z = 1, w = 4$