Finite Math Examples

$- \frac{1}{10} x + \frac{1}{2} y = 2$ , $2 x - 10 y = - 40$

Step 1

Solve for $x$ in $2 x - 10 y = - 40$ .

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Step 1.1

Add $10 y$ to both sides of the equation.

$2 x = - 40 + 10 y$

$- \frac{1}{10} x + \frac{1}{2} y = 2$

Step 1.2

Divide each term in $2 x = - 40 + 10 y$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 x = - 40 + 10 y$ by $2$ .

$\frac{2 x}{2} = \frac{- 40}{2} + \frac{10 y}{2}$

$- \frac{1}{10} x + \frac{1}{2} y = 2$

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.

$\frac{2 x}{2} = \frac{- 40}{2} + \frac{10 y}{2}$

$- \frac{1}{10} x + \frac{1}{2} y = 2$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 40}{2} + \frac{10 y}{2}$

$- \frac{1}{10} x + \frac{1}{2} y = 2$

$x = \frac{- 40}{2} + \frac{10 y}{2}$

$- \frac{1}{10} x + \frac{1}{2} y = 2$

$x = \frac{- 40}{2} + \frac{10 y}{2}$

$- \frac{1}{10} x + \frac{1}{2} y = 2$

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

Divide $- 40$ by $2$ .

$x = - 20 + \frac{10 y}{2}$

$- \frac{1}{10} x + \frac{1}{2} y = 2$

Step 1.2.3.1.2

Cancel the common factor of $10$ and $2$ .

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Step 1.2.3.1.2.1

Factor $2$ out of $10 y$ .

$x = - 20 + \frac{2 (5 y)}{2}$

$- \frac{1}{10} x + \frac{1}{2} y = 2$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $2$ out of $2$ .

$x = - 20 + \frac{2 (5 y)}{2 (1)}$

$- \frac{1}{10} x + \frac{1}{2} y = 2$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$x = - 20 + \frac{2 (5 y)}{2 \cdot 1}$

$- \frac{1}{10} x + \frac{1}{2} y = 2$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$x = - 20 + \frac{5 y}{1}$

$- \frac{1}{10} x + \frac{1}{2} y = 2$

Step 1.2.3.1.2.2.4

Divide $5 y$ by $1$ .

$x = - 20 + 5 y$

$- \frac{1}{10} x + \frac{1}{2} y = 2$

$x = - 20 + 5 y$

$- \frac{1}{10} x + \frac{1}{2} y = 2$

$x = - 20 + 5 y$

$- \frac{1}{10} x + \frac{1}{2} y = 2$

$x = - 20 + 5 y$

$- \frac{1}{10} x + \frac{1}{2} y = 2$

$x = - 20 + 5 y$

$- \frac{1}{10} x + \frac{1}{2} y = 2$

$x = - 20 + 5 y$

$- \frac{1}{10} x + \frac{1}{2} y = 2$

$x = - 20 + 5 y$

$- \frac{1}{10} x + \frac{1}{2} y = 2$

Step 2

Replace all occurrences of $x$ with $- 20 + 5 y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- \frac{1}{10} x + \frac{1}{2} y = 2$ with $- 20 + 5 y$ .

$- \frac{1}{10} \cdot (- 20 + 5 y) + \frac{1}{2} y = 2$

$x = - 20 + 5 y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- \frac{1}{10} \cdot (- 20 + 5 y) + \frac{1}{2} y$ .

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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.

$- \frac{1}{10} \cdot - 20 - \frac{1}{10} \cdot (5 y) + \frac{1}{2} y = 2$

$x = - 20 + 5 y$

Step 2.2.1.1.2

Cancel the common factor of $10$ .

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Step 2.2.1.1.2.1

Move the leading negative in $- \frac{1}{10}$ into the numerator.

$\frac{- 1}{10} \cdot - 20 - \frac{1}{10} \cdot (5 y) + \frac{1}{2} y = 2$

$x = - 20 + 5 y$

Step 2.2.1.1.2.2

Factor $10$ out of $- 20$ .

$\frac{- 1}{10} \cdot (10 (- 2)) - \frac{1}{10} \cdot (5 y) + \frac{1}{2} y = 2$

$x = - 20 + 5 y$

Step 2.2.1.1.2.3

Cancel the common factor.

$\frac{- 1}{10} \cdot (10 \cdot - 2) - \frac{1}{10} \cdot (5 y) + \frac{1}{2} y = 2$

$x = - 20 + 5 y$

Step 2.2.1.1.2.4

Rewrite the expression.

$- 1 \cdot - 2 - \frac{1}{10} \cdot (5 y) + \frac{1}{2} y = 2$

$x = - 20 + 5 y$

$- 1 \cdot - 2 - \frac{1}{10} \cdot (5 y) + \frac{1}{2} y = 2$

$x = - 20 + 5 y$

Step 2.2.1.1.3

Multiply $- 1$ by $- 2$ .

$2 - \frac{1}{10} \cdot (5 y) + \frac{1}{2} y = 2$

$x = - 20 + 5 y$

Step 2.2.1.1.4

Cancel the common factor of $5$ .

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Step 2.2.1.1.4.1

Move the leading negative in $- \frac{1}{10}$ into the numerator.

$2 + \frac{- 1}{10} \cdot (5 y) + \frac{1}{2} y = 2$

$x = - 20 + 5 y$

Step 2.2.1.1.4.2

Factor $5$ out of $10$ .

$2 + \frac{- 1}{5 (2)} \cdot (5 y) + \frac{1}{2} y = 2$

$x = - 20 + 5 y$

Step 2.2.1.1.4.3

Factor $5$ out of $5 y$ .

$2 + \frac{- 1}{5 (2)} \cdot (5 (y)) + \frac{1}{2} y = 2$

$x = - 20 + 5 y$

Step 2.2.1.1.4.4

Cancel the common factor.

$2 + \frac{- 1}{5 \cdot 2} \cdot (5 y) + \frac{1}{2} y = 2$

$x = - 20 + 5 y$

Step 2.2.1.1.4.5

Rewrite the expression.

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

$x = - 20 + 5 y$

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

$x = - 20 + 5 y$

Step 2.2.1.1.5

Combine $\frac{- 1}{2}$ and $y$ .

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

$x = - 20 + 5 y$

Step 2.2.1.1.6

Move the negative in front of the fraction.

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

$x = - 20 + 5 y$

Step 2.2.1.1.7

Combine $\frac{1}{2}$ and $y$ .

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

$x = - 20 + 5 y$

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

$x = - 20 + 5 y$

Step 2.2.1.2

Combine the opposite terms in $2 - \frac{y}{2} + \frac{y}{2}$ .

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Step 2.2.1.2.1

Add $- \frac{y}{2}$ and $\frac{y}{2}$ .

$2 + 0 = 2$

$x = - 20 + 5 y$

Step 2.2.1.2.2

Add $2$ and $0$ .

$2 = 2$

$x = - 20 + 5 y$

$2 = 2$

$x = - 20 + 5 y$

$2 = 2$

$x = - 20 + 5 y$

$2 = 2$

$x = - 20 + 5 y$

$2 = 2$

$x = - 20 + 5 y$

Step 3

Remove any equations from the system that are always true.

$x = - 20 + 5 y$

Step 4