Finite Math Examples

$x + y + 5 z + w = 1$ , $y + 2 z + w = 0$ , $x + 3 y + 7 z + 2 w = 1$ , $x + y + 5 z + 2 w = 1$

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.

$x + 5 z + w = 1 - y$

$y + 2 z + w = 0$

$x + 3 y + 7 z + 2 w = 1$

$x + y + 5 z + 2 w = 1$

Step 1.2

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

$x + w = 1 - y - 5 z$

$y + 2 z + w = 0$

$x + 3 y + 7 z + 2 w = 1$

$x + y + 5 z + 2 w = 1$

Step 1.3

Subtract $w$ from both sides of the equation.

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

$x + 3 y + 7 z + 2 w = 1$

$x + y + 5 z + 2 w = 1$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

$x + 3 y + 7 z + 2 w = 1$

$x + y + 5 z + 2 w = 1$

Step 2

Replace all occurrences of $x$ with $1 - y - 5 z - w$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x + 3 y + 7 z + 2 w = 1$ with $1 - y - 5 z - w$ .

$(1 - y - 5 z - w) + 3 y + 7 z + 2 w = 1$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

$x + y + 5 z + 2 w = 1$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $(1 - y - 5 z - w) + 3 y + 7 z + 2 w$ .

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Step 2.2.1.1

Add $- y$ and $3 y$ .

$1 + 2 y - 5 z - w + 7 z + 2 w = 1$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

$x + y + 5 z + 2 w = 1$

Step 2.2.1.2

Add $- 5 z$ and $7 z$ .

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

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

$x + y + 5 z + 2 w = 1$

Step 2.2.1.3

Add $- w$ and $2 w$ .

$1 + 2 y + w + 2 z = 1$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

$x + y + 5 z + 2 w = 1$

$1 + 2 y + w + 2 z = 1$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

$x + y + 5 z + 2 w = 1$

$1 + 2 y + w + 2 z = 1$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

$x + y + 5 z + 2 w = 1$

Step 2.3

Replace all occurrences of $x$ in $x + y + 5 z + 2 w = 1$ with $1 - y - 5 z - w$ .

$(1 - y - 5 z - w) + y + 5 z + 2 w = 1$

$1 + 2 y + w + 2 z = 1$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $(1 - y - 5 z - w) + y + 5 z + 2 w$ .

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Step 2.4.1.1

Combine the opposite terms in $(1 - y - 5 z - w) + y + 5 z + 2 w$ .

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Step 2.4.1.1.1

Add $- y$ and $y$ .

$1 - 5 z - w + 0 + 5 z + 2 w = 1$

$1 + 2 y + w + 2 z = 1$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

Step 2.4.1.1.2

Add $1 - 5 z - w$ and $0$ .

$1 - 5 z - w + 5 z + 2 w = 1$

$1 + 2 y + w + 2 z = 1$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

Step 2.4.1.1.3

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

$1 - w + 0 + 2 w = 1$

$1 + 2 y + w + 2 z = 1$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

Step 2.4.1.1.4

Add $1 - w$ and $0$ .

$1 - w + 2 w = 1$

$1 + 2 y + w + 2 z = 1$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

$1 - w + 2 w = 1$

$1 + 2 y + w + 2 z = 1$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

Step 2.4.1.2

Add $- w$ and $2 w$ .

$1 + w = 1$

$1 + 2 y + w + 2 z = 1$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

$1 + w = 1$

$1 + 2 y + w + 2 z = 1$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

$1 + w = 1$

$1 + 2 y + w + 2 z = 1$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

$1 + w = 1$

$1 + 2 y + w + 2 z = 1$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

Step 3

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

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Step 3.1

Subtract $1$ from both sides of the equation.

$w = 1 - 1$

$1 + 2 y + w + 2 z = 1$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

Step 3.2

Subtract $1$ from $1$ .

$w = 0$

$1 + 2 y + w + 2 z = 1$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

$w = 0$

$1 + 2 y + w + 2 z = 1$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

Step 4

Replace all occurrences of $w$ with $0$ in each equation.

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Step 4.1

Replace all occurrences of $w$ in $1 + 2 y + w + 2 z = 1$ with $0$ .

$1 + 2 y + 0 + 2 z = 1$

$w = 0$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $1 + 2 y + 0 + 2 z$ .

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Step 4.2.1.1

Remove parentheses.

$1 + 2 y + 0 + 2 z = 1$

$w = 0$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

Step 4.2.1.2

Add $1 + 2 y$ and $0$ .

$1 + 2 y + 2 z = 1$

$w = 0$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

$1 + 2 y + 2 z = 1$

$w = 0$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

$1 + 2 y + 2 z = 1$

$w = 0$

$x = 1 - y - 5 z - w$

$y + 2 z + w = 0$

Step 4.3

Replace all occurrences of $w$ in $x = 1 - y - 5 z - w$ with $0$ .

$x = 1 - y - 5 z - (0)$

$1 + 2 y + 2 z = 1$

$w = 0$

$y + 2 z + w = 0$

Step 4.4

Simplify the right side.

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Step 4.4.1

Subtract $0$ from $1 - y - 5 z$ .

$x = 1 - y - 5 z$

$1 + 2 y + 2 z = 1$

$w = 0$

$y + 2 z + w = 0$

$x = 1 - y - 5 z$

$1 + 2 y + 2 z = 1$

$w = 0$

$y + 2 z + w = 0$

Step 4.5

Replace all occurrences of $w$ in $y + 2 z + w = 0$ with $0$ .

$y + 2 z + 0 = 0$

$x = 1 - y - 5 z$

$1 + 2 y + 2 z = 1$

$w = 0$

Step 4.6

Simplify the left side.

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Step 4.6.1

Simplify $y + 2 z + 0$ .

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Step 4.6.1.1

Remove parentheses.

$y + 2 z + 0 = 0$

$x = 1 - y - 5 z$

$1 + 2 y + 2 z = 1$

$w = 0$

Step 4.6.1.2

Add $y + 2 z$ and $0$ .

$y + 2 z = 0$

$x = 1 - y - 5 z$

$1 + 2 y + 2 z = 1$

$w = 0$

$y + 2 z = 0$

$x = 1 - y - 5 z$

$1 + 2 y + 2 z = 1$

$w = 0$

$y + 2 z = 0$

$x = 1 - y - 5 z$

$1 + 2 y + 2 z = 1$

$w = 0$

$y + 2 z = 0$

$x = 1 - y - 5 z$

$1 + 2 y + 2 z = 1$

$w = 0$

Step 5

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

$y = - 2 z$

$x = 1 - y - 5 z$

$1 + 2 y + 2 z = 1$

$w = 0$

Step 6

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

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Step 6.1

Replace all occurrences of $y$ in $x = 1 - y - 5 z$ with $- 2 z$ .

$x = 1 - (- 2 z) - 5 z$

$y = - 2 z$

$1 + 2 y + 2 z = 1$

$w = 0$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $1 - (- 2 z) - 5 z$ .

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Step 6.2.1.1

Multiply $- 2$ by $- 1$ .

$x = 1 + 2 z - 5 z$

$y = - 2 z$

$1 + 2 y + 2 z = 1$

$w = 0$

Step 6.2.1.2

Subtract $5 z$ from $2 z$ .

$x = 1 - 3 z$

$y = - 2 z$

$1 + 2 y + 2 z = 1$

$w = 0$

$x = 1 - 3 z$

$y = - 2 z$

$1 + 2 y + 2 z = 1$

$w = 0$

$x = 1 - 3 z$

$y = - 2 z$

$1 + 2 y + 2 z = 1$

$w = 0$

Step 6.3

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

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

$x = 1 - 3 z$

$y = - 2 z$

$w = 0$

Step 6.4

Simplify the left side.

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Step 6.4.1

Simplify $1 + 2 (- 2 z) + 2 z$ .

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Step 6.4.1.1

Multiply $- 2$ by $2$ .

$1 - 4 z + 2 z = 1$

$x = 1 - 3 z$

$y = - 2 z$

$w = 0$

Step 6.4.1.2

Add $- 4 z$ and $2 z$ .

$1 - 2 z = 1$

$x = 1 - 3 z$

$y = - 2 z$

$w = 0$

$1 - 2 z = 1$

$x = 1 - 3 z$

$y = - 2 z$

$w = 0$

$1 - 2 z = 1$

$x = 1 - 3 z$

$y = - 2 z$

$w = 0$

$1 - 2 z = 1$

$x = 1 - 3 z$

$y = - 2 z$

$w = 0$

Step 7

Solve for $z$ in $1 - 2 z = 1$ .

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Step 7.1

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

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Step 7.1.1

Subtract $1$ from both sides of the equation.

$- 2 z = 1 - 1$

$x = 1 - 3 z$

$y = - 2 z$

$w = 0$

Step 7.1.2

Subtract $1$ from $1$ .

$- 2 z = 0$

$x = 1 - 3 z$

$y = - 2 z$

$w = 0$

$- 2 z = 0$

$x = 1 - 3 z$

$y = - 2 z$

$w = 0$

Step 7.2

Divide each term in $- 2 z = 0$ by $- 2$ and simplify.

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Step 7.2.1

Divide each term in $- 2 z = 0$ by $- 2$ .

$\frac{- 2 z}{- 2} = \frac{0}{- 2}$

$x = 1 - 3 z$

$y = - 2 z$

$w = 0$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $- 2$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{- 2 z}{- 2} = \frac{0}{- 2}$

$x = 1 - 3 z$

$y = - 2 z$

$w = 0$

Step 7.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{0}{- 2}$

$x = 1 - 3 z$

$y = - 2 z$

$w = 0$

$z = \frac{0}{- 2}$

$x = 1 - 3 z$

$y = - 2 z$

$w = 0$

$z = \frac{0}{- 2}$

$x = 1 - 3 z$

$y = - 2 z$

$w = 0$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $0$ by $- 2$ .

$z = 0$

$x = 1 - 3 z$

$y = - 2 z$

$w = 0$

$z = 0$

$x = 1 - 3 z$

$y = - 2 z$

$w = 0$

$z = 0$

$x = 1 - 3 z$

$y = - 2 z$

$w = 0$

$z = 0$

$x = 1 - 3 z$

$y = - 2 z$

$w = 0$

Step 8

Replace all occurrences of $z$ with $0$ in each equation.

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Step 8.1

Replace all occurrences of $z$ in $x = 1 - 3 z$ with $0$ .

$x = 1 - 3 \cdot 0$

$z = 0$

$y = - 2 z$

$w = 0$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $1 - 3 \cdot 0$ .

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Step 8.2.1.1

Multiply $- 3$ by $0$ .

$x = 1 + 0$

$z = 0$

$y = - 2 z$

$w = 0$

Step 8.2.1.2

Add $1$ and $0$ .

$x = 1$

$z = 0$

$y = - 2 z$

$w = 0$

$x = 1$

$z = 0$

$y = - 2 z$

$w = 0$

$x = 1$

$z = 0$

$y = - 2 z$

$w = 0$

Step 8.3

Replace all occurrences of $z$ in $y = - 2 z$ with $0$ .

$y = - 2 \cdot 0$

$x = 1$

$z = 0$

$w = 0$

Step 8.4

Simplify the right side.

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Step 8.4.1

Multiply $- 2$ by $0$ .

$y = 0$

$x = 1$

$z = 0$

$w = 0$

$y = 0$

$x = 1$

$z = 0$

$w = 0$

$y = 0$

$x = 1$

$z = 0$

$w = 0$

Step 9

List all of the solutions.

$y = 0, x = 1, z = 0, w = 0$