Finite Math Examples

$10 x - 4 y = - 2$ , $- 25 x + 10 y = 5$

Step 1

Solve for $x$ in $10 x - 4 y = - 2$ .

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Step 1.1

Add $4 y$ to both sides of the equation.

$10 x = - 2 + 4 y$

$- 25 x + 10 y = 5$

Step 1.2

Divide each term in $10 x = - 2 + 4 y$ by $10$ and simplify.

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Step 1.2.1

Divide each term in $10 x = - 2 + 4 y$ by $10$ .

$\frac{10 x}{10} = \frac{- 2}{10} + \frac{4 y}{10}$

$- 25 x + 10 y = 5$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $10$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{- 2}{10} + \frac{4 y}{10}$

$- 25 x + 10 y = 5$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{10} + \frac{4 y}{10}$

$- 25 x + 10 y = 5$

$x = \frac{- 2}{10} + \frac{4 y}{10}$

$- 25 x + 10 y = 5$

$x = \frac{- 2}{10} + \frac{4 y}{10}$

$- 25 x + 10 y = 5$

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

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

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Step 1.2.3.1.1.1

Factor $2$ out of $- 2$ .

$x = \frac{2 (- 1)}{10} + \frac{4 y}{10}$

$- 25 x + 10 y = 5$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $2$ out of $10$ .

$x = \frac{2 \cdot - 1}{2 \cdot 5} + \frac{4 y}{10}$

$- 25 x + 10 y = 5$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot - 1}{2 \cdot 5} + \frac{4 y}{10}$

$- 25 x + 10 y = 5$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{- 1}{5} + \frac{4 y}{10}$

$- 25 x + 10 y = 5$

$x = \frac{- 1}{5} + \frac{4 y}{10}$

$- 25 x + 10 y = 5$

$x = \frac{- 1}{5} + \frac{4 y}{10}$

$- 25 x + 10 y = 5$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{1}{5} + \frac{4 y}{10}$

$- 25 x + 10 y = 5$

Step 1.2.3.1.3

Cancel the common factor of $4$ and $10$ .

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Step 1.2.3.1.3.1

Factor $2$ out of $4 y$ .

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

$- 25 x + 10 y = 5$

Step 1.2.3.1.3.2

Cancel the common factors.

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Step 1.2.3.1.3.2.1

Factor $2$ out of $10$ .

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

$- 25 x + 10 y = 5$

Step 1.2.3.1.3.2.2

Cancel the common factor.

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

$- 25 x + 10 y = 5$

Step 1.2.3.1.3.2.3

Rewrite the expression.

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

$- 25 x + 10 y = 5$

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

$- 25 x + 10 y = 5$

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

$- 25 x + 10 y = 5$

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

$- 25 x + 10 y = 5$

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

$- 25 x + 10 y = 5$

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

$- 25 x + 10 y = 5$

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

$- 25 x + 10 y = 5$

Step 2

Replace all occurrences of $x$ with $- \frac{1}{5} + \frac{2 y}{5}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- 25 x + 10 y = 5$ with $- \frac{1}{5} + \frac{2 y}{5}$ .

$- 25 (- \frac{1}{5} + \frac{2 y}{5}) + 10 y = 5$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 25 (- \frac{1}{5} + \frac{2 y}{5}) + 10 y$ .

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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.

$- 25 (- \frac{1}{5}) - 25 \frac{2 y}{5} + 10 y = 5$

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

Step 2.2.1.1.2

Cancel the common factor of $5$ .

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Step 2.2.1.1.2.1

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$- 25 (\frac{- 1}{5}) - 25 \frac{2 y}{5} + 10 y = 5$

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

Step 2.2.1.1.2.2

Factor $5$ out of $- 25$ .

$5 (- 5) (\frac{- 1}{5}) - 25 \frac{2 y}{5} + 10 y = 5$

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

Step 2.2.1.1.2.3

Cancel the common factor.

$5 \cdot (- 5 (\frac{- 1}{5})) - 25 \frac{2 y}{5} + 10 y = 5$

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

Step 2.2.1.1.2.4

Rewrite the expression.

$- 5 \cdot - 1 - 25 \frac{2 y}{5} + 10 y = 5$

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

$- 5 \cdot - 1 - 25 \frac{2 y}{5} + 10 y = 5$

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

Step 2.2.1.1.3

Multiply $- 5$ by $- 1$ .

$5 - 25 \frac{2 y}{5} + 10 y = 5$

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

Step 2.2.1.1.4

Cancel the common factor of $5$ .

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Step 2.2.1.1.4.1

Factor $5$ out of $- 25$ .

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

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

Step 2.2.1.1.4.2

Cancel the common factor.

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

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

Step 2.2.1.1.4.3

Rewrite the expression.

$5 - 5 (2 y) + 10 y = 5$

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

$5 - 5 (2 y) + 10 y = 5$

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

Step 2.2.1.1.5

Multiply $2$ by $- 5$ .

$5 - 10 y + 10 y = 5$

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

$5 - 10 y + 10 y = 5$

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

Step 2.2.1.2

Combine the opposite terms in $5 - 10 y + 10 y$ .

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Step 2.2.1.2.1

Add $- 10 y$ and $10 y$ .

$5 + 0 = 5$

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

Step 2.2.1.2.2

Add $5$ and $0$ .

$5 = 5$

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

$5 = 5$

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

$5 = 5$

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

$5 = 5$

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

$5 = 5$

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

Step 3

Remove any equations from the system that are always true.

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

Step 4