Finite Math Examples

$- 2 x - y = 1$ , $4 x + 2 y = - 2$ , $- 8 x - 4 y = 4$

Step 1

Solve for $x$ in $- 2 x - y = 1$ .

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Step 1.1

Add $y$ to both sides of the equation.

$- 2 x = 1 + y$

$4 x + 2 y = - 2$

$- 8 x - 4 y = 4$

Step 1.2

Divide each term in $- 2 x = 1 + y$ by $- 2$ and simplify.

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Step 1.2.1

Divide each term in $- 2 x = 1 + y$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{1}{- 2} + \frac{y}{- 2}$

$4 x + 2 y = - 2$

$- 8 x - 4 y = 4$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{1}{- 2} + \frac{y}{- 2}$

$4 x + 2 y = - 2$

$- 8 x - 4 y = 4$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{- 2} + \frac{y}{- 2}$

$4 x + 2 y = - 2$

$- 8 x - 4 y = 4$

$x = \frac{1}{- 2} + \frac{y}{- 2}$

$4 x + 2 y = - 2$

$- 8 x - 4 y = 4$

$x = \frac{1}{- 2} + \frac{y}{- 2}$

$4 x + 2 y = - 2$

$- 8 x - 4 y = 4$

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.

$x = - \frac{1}{2} + \frac{y}{- 2}$

$4 x + 2 y = - 2$

$- 8 x - 4 y = 4$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{1}{2} - \frac{y}{2}$

$4 x + 2 y = - 2$

$- 8 x - 4 y = 4$

$x = - \frac{1}{2} - \frac{y}{2}$

$4 x + 2 y = - 2$

$- 8 x - 4 y = 4$

$x = - \frac{1}{2} - \frac{y}{2}$

$4 x + 2 y = - 2$

$- 8 x - 4 y = 4$

$x = - \frac{1}{2} - \frac{y}{2}$

$4 x + 2 y = - 2$

$- 8 x - 4 y = 4$

$x = - \frac{1}{2} - \frac{y}{2}$

$4 x + 2 y = - 2$

$- 8 x - 4 y = 4$

Step 2

Replace all occurrences of $x$ with $- \frac{1}{2} - \frac{y}{2}$ in each equation.

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Step 2.1

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

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

$x = - \frac{1}{2} - \frac{y}{2}$

$- 8 x - 4 y = 4$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

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

$x = - \frac{1}{2} - \frac{y}{2}$

$- 8 x - 4 y = 4$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

$x = - \frac{1}{2} - \frac{y}{2}$

$- 8 x - 4 y = 4$

Step 2.2.1.1.2.2

Factor $2$ out of $4$ .

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

$x = - \frac{1}{2} - \frac{y}{2}$

$- 8 x - 4 y = 4$

Step 2.2.1.1.2.3

Cancel the common factor.

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

$x = - \frac{1}{2} - \frac{y}{2}$

$- 8 x - 4 y = 4$

Step 2.2.1.1.2.4

Rewrite the expression.

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

$x = - \frac{1}{2} - \frac{y}{2}$

$- 8 x - 4 y = 4$

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

$x = - \frac{1}{2} - \frac{y}{2}$

$- 8 x - 4 y = 4$

Step 2.2.1.1.3

Multiply $2$ by $- 1$ .

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

$x = - \frac{1}{2} - \frac{y}{2}$

$- 8 x - 4 y = 4$

Step 2.2.1.1.4

Cancel the common factor of $2$ .

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Step 2.2.1.1.4.1

Move the leading negative in $- \frac{y}{2}$ into the numerator.

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

$x = - \frac{1}{2} - \frac{y}{2}$

$- 8 x - 4 y = 4$

Step 2.2.1.1.4.2

Factor $2$ out of $4$ .

$- 2 + 2 (2) (\frac{- y}{2}) + 2 y = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

$- 8 x - 4 y = 4$

Step 2.2.1.1.4.3

Cancel the common factor.

$- 2 + 2 \cdot (2 (\frac{- y}{2})) + 2 y = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

$- 8 x - 4 y = 4$

Step 2.2.1.1.4.4

Rewrite the expression.

$- 2 + 2 (- y) + 2 y = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

$- 8 x - 4 y = 4$

$- 2 + 2 (- y) + 2 y = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

$- 8 x - 4 y = 4$

Step 2.2.1.1.5

Multiply $- 1$ by $2$ .

$- 2 - 2 y + 2 y = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

$- 8 x - 4 y = 4$

$- 2 - 2 y + 2 y = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

$- 8 x - 4 y = 4$

Step 2.2.1.2

Combine the opposite terms in $- 2 - 2 y + 2 y$ .

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Step 2.2.1.2.1

Add $- 2 y$ and $2 y$ .

$- 2 + 0 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

$- 8 x - 4 y = 4$

Step 2.2.1.2.2

Add $- 2$ and $0$ .

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

$- 8 x - 4 y = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

$- 8 x - 4 y = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

$- 8 x - 4 y = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

$- 8 x - 4 y = 4$

Step 2.3

Replace all occurrences of $x$ in $- 8 x - 4 y = 4$ with $- \frac{1}{2} - \frac{y}{2}$ .

$- 8 (- \frac{1}{2} - \frac{y}{2}) - 4 y = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $- 8 (- \frac{1}{2} - \frac{y}{2}) - 4 y$ .

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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.

$- 8 (- \frac{1}{2}) - 8 (- \frac{y}{2}) - 4 y = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

Step 2.4.1.1.2

Cancel the common factor of $2$ .

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Step 2.4.1.1.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$- 8 (\frac{- 1}{2}) - 8 (- \frac{y}{2}) - 4 y = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

Step 2.4.1.1.2.2

Factor $2$ out of $- 8$ .

$2 (- 4) (\frac{- 1}{2}) - 8 (- \frac{y}{2}) - 4 y = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

Step 2.4.1.1.2.3

Cancel the common factor.

$2 \cdot (- 4 (\frac{- 1}{2})) - 8 (- \frac{y}{2}) - 4 y = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

Step 2.4.1.1.2.4

Rewrite the expression.

$- 4 \cdot - 1 - 8 (- \frac{y}{2}) - 4 y = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

$- 4 \cdot - 1 - 8 (- \frac{y}{2}) - 4 y = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

Step 2.4.1.1.3

Multiply $- 4$ by $- 1$ .

$4 - 8 (- \frac{y}{2}) - 4 y = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

Step 2.4.1.1.4

Cancel the common factor of $2$ .

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Step 2.4.1.1.4.1

Move the leading negative in $- \frac{y}{2}$ into the numerator.

$4 - 8 \frac{- y}{2} - 4 y = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

Step 2.4.1.1.4.2

Factor $2$ out of $- 8$ .

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

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

Step 2.4.1.1.4.3

Cancel the common factor.

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

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

Step 2.4.1.1.4.4

Rewrite the expression.

$4 - 4 (- y) - 4 y = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

$4 - 4 (- y) - 4 y = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

Step 2.4.1.1.5

Multiply $- 1$ by $- 4$ .

$4 + 4 y - 4 y = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

$4 + 4 y - 4 y = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

Step 2.4.1.2

Combine the opposite terms in $4 + 4 y - 4 y$ .

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Step 2.4.1.2.1

Subtract $4 y$ from $4 y$ .

$4 + 0 = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

Step 2.4.1.2.2

Add $4$ and $0$ .

$4 = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

$4 = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

$4 = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

$4 = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

$4 = 4$

$- 2 = - 2$

$x = - \frac{1}{2} - \frac{y}{2}$

Step 3

Remove any equations from the system that are always true.

$x = - \frac{1}{2} - \frac{y}{2}$

Step 4