Finite Math Examples

$2 x - y = 5$ , $4 x + k y = 9$

Step 1

Solve the equation for $x$ .

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Step 1.1

Add $y$ to both sides of the equation.

$2 x = 5 + y$

$4 x + k y = 9$

Step 1.2

Divide each term in $2 x = 5 + y$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 x = 5 + y$ by $2$ .

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

$4 x + k y = 9$

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.

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

$4 x + k y = 9$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$4 x + k y = 9$

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

$4 x + k y = 9$

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

$4 x + k y = 9$

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

$4 x + k y = 9$

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

$4 x + k y = 9$

Step 2

Solve the equation for $y$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$4 (\frac{5}{2}) + 4 (\frac{y}{2}) + k y = 9$

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

Step 2.1.2

Cancel the common factor of $2$ .

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Step 2.1.2.1

Factor $2$ out of $4$ .

$2 (2) (\frac{5}{2}) + 4 (\frac{y}{2}) + k y = 9$

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

Step 2.1.2.2

Cancel the common factor.

$2 \cdot (2 (\frac{5}{2})) + 4 (\frac{y}{2}) + k y = 9$

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

Step 2.1.2.3

Rewrite the expression.

$2 \cdot 5 + 4 (\frac{y}{2}) + k y = 9$

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

$2 \cdot 5 + 4 (\frac{y}{2}) + k y = 9$

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

Step 2.1.3

Multiply $2$ by $5$ .

$10 + 4 (\frac{y}{2}) + k y = 9$

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

Step 2.1.4

Cancel the common factor of $2$ .

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Step 2.1.4.1

Factor $2$ out of $4$ .

$10 + 2 (2) (\frac{y}{2}) + k y = 9$

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

Step 2.1.4.2

Cancel the common factor.

$10 + 2 \cdot (2 (\frac{y}{2})) + k y = 9$

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

Step 2.1.4.3

Rewrite the expression.

$10 + 2 y + k y = 9$

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

$10 + 2 y + k y = 9$

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

$10 + 2 y + k y = 9$

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

Step 2.2

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

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Step 2.2.1

Subtract $10$ from both sides of the equation.

$2 y + k y = 9 - 10$

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

Step 2.2.2

Subtract $10$ from $9$ .

$2 y + k y = - 1$

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

$2 y + k y = - 1$

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

Step 2.3

Factor $y$ out of $2 y + k y$ .

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Step 2.3.1

Factor $y$ out of $2 y$ .

$y \cdot 2 + k y = - 1$

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

Step 2.3.2

Factor $y$ out of $k y$ .

$y \cdot 2 + y k = - 1$

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

Step 2.3.3

Factor $y$ out of $y \cdot 2 + y k$ .

$y \cdot (2 + k) = - 1$

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

Step 2.3.4

Multiply $y$ by $2 + k$ .

$y (2 + k) = - 1$

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

$y (2 + k) = - 1$

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

Step 2.4

Divide each term in $y (2 + k) = - 1$ by $2 + k$ and simplify.

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Step 2.4.1

Divide each term in $y (2 + k) = - 1$ by $2 + k$ .

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

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $2 + k$ .

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Step 2.4.2.1.1

Cancel the common factor.

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

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

Step 2.4.2.1.2

Divide $y$ by $1$ .

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

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

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

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

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

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

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Move the negative in front of the fraction.

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

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

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

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

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

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

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

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

Step 3

Simplify the right side.

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Step 3.1

Simplify $\frac{5}{2} + \frac{- \frac{1}{2 + k}}{2}$ .

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Step 3.1.1

Combine the numerators over the common denominator.

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

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

Step 3.1.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{2 + k}{2 + k}$ .

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

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

Step 3.1.3

Simplify terms.

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Step 3.1.3.1

Combine $5$ and $\frac{2 + k}{2 + k}$ .

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

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

Step 3.1.3.2

Combine the numerators over the common denominator.

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

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

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

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

Step 3.1.4

Simplify the numerator.

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Step 3.1.4.1

Apply the distributive property.

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

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

Step 3.1.4.2

Multiply $5$ by $2$ .

$x = \frac{\frac{10 + 5 k - 1}{2 + k}}{2}$

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

Step 3.1.4.3

Subtract $1$ from $10$ .

$x = \frac{\frac{5 k + 9}{2 + k}}{2}$

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

$x = \frac{\frac{5 k + 9}{2 + k}}{2}$

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

Step 3.1.5

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5 k + 9}{2 + k} \cdot \frac{1}{2}$

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

Step 3.1.6

Multiply $\frac{5 k + 9}{2 + k}$ by $\frac{1}{2}$ .

$x = \frac{5 k + 9}{(2 + k) \cdot 2}$

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

Step 3.1.7

Move $2$ to the left of $2 + k$ .

$x = \frac{5 k + 9}{2 (2 + k)}$

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

$x = \frac{5 k + 9}{2 (2 + k)}$

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

$x = \frac{5 k + 9}{2 (2 + k)}$

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