Finite Math Examples

$3 x - y = 3$ , $6 x + k y = 3$

Step 1

Solve the equation for $x$ .

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Step 1.1

Add $y$ to both sides of the equation.

$3 x = 3 + y$

$6 x + k y = 3$

Step 1.2

Divide each term in $3 x = 3 + y$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = 3 + y$ by $3$ .

$\frac{3 x}{3} = \frac{3}{3} + \frac{y}{3}$

$6 x + k y = 3$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{3}{3} + \frac{y}{3}$

$6 x + k y = 3$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{3} + \frac{y}{3}$

$6 x + k y = 3$

$x = \frac{3}{3} + \frac{y}{3}$

$6 x + k y = 3$

$x = \frac{3}{3} + \frac{y}{3}$

$6 x + k y = 3$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $3$ by $3$ .

$x = 1 + \frac{y}{3}$

$6 x + k y = 3$

$x = 1 + \frac{y}{3}$

$6 x + k y = 3$

$x = 1 + \frac{y}{3}$

$6 x + k y = 3$

$x = 1 + \frac{y}{3}$

$6 x + k y = 3$

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.

$6 \cdot 1 + 6 (\frac{y}{3}) + k y = 3$

$x = 1 + \frac{y}{3}$

Step 2.1.2

Multiply $6$ by $1$ .

$6 + 6 (\frac{y}{3}) + k y = 3$

$x = 1 + \frac{y}{3}$

Step 2.1.3

Cancel the common factor of $3$ .

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Step 2.1.3.1

Factor $3$ out of $6$ .

$6 + 3 (2) (\frac{y}{3}) + k y = 3$

$x = 1 + \frac{y}{3}$

Step 2.1.3.2

Cancel the common factor.

$6 + 3 \cdot (2 (\frac{y}{3})) + k y = 3$

$x = 1 + \frac{y}{3}$

Step 2.1.3.3

Rewrite the expression.

$6 + 2 y + k y = 3$

$x = 1 + \frac{y}{3}$

$6 + 2 y + k y = 3$

$x = 1 + \frac{y}{3}$

$6 + 2 y + k y = 3$

$x = 1 + \frac{y}{3}$

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 $6$ from both sides of the equation.

$2 y + k y = 3 - 6$

$x = 1 + \frac{y}{3}$

Step 2.2.2

Subtract $6$ from $3$ .

$2 y + k y = - 3$

$x = 1 + \frac{y}{3}$

$2 y + k y = - 3$

$x = 1 + \frac{y}{3}$

Step 2.3

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

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Step 2.3.1

Factor $y$ out of $2 y$ .

$y \cdot 2 + k y = - 3$

$x = 1 + \frac{y}{3}$

Step 2.3.2

Factor $y$ out of $k y$ .

$y \cdot 2 + y k = - 3$

$x = 1 + \frac{y}{3}$

Step 2.3.3

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

$y \cdot (2 + k) = - 3$

$x = 1 + \frac{y}{3}$

Step 2.3.4

Multiply $y$ by $2 + k$ .

$y (2 + k) = - 3$

$x = 1 + \frac{y}{3}$

$y (2 + k) = - 3$

$x = 1 + \frac{y}{3}$

Step 2.4

Divide each term in $y (2 + k) = - 3$ by $2 + k$ and simplify.

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Step 2.4.1

Divide each term in $y (2 + k) = - 3$ by $2 + k$ .

$\frac{y (2 + k)}{2 + k} = \frac{- 3}{2 + k}$

$x = 1 + \frac{y}{3}$

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.

$\frac{y (2 + k)}{2 + k} = \frac{- 3}{2 + k}$

$x = 1 + \frac{y}{3}$

Step 2.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 3}{2 + k}$

$x = 1 + \frac{y}{3}$

$y = \frac{- 3}{2 + k}$

$x = 1 + \frac{y}{3}$

$y = \frac{- 3}{2 + k}$

$x = 1 + \frac{y}{3}$

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.

$y = - \frac{3}{2 + k}$

$x = 1 + \frac{y}{3}$

$y = - \frac{3}{2 + k}$

$x = 1 + \frac{y}{3}$

$y = - \frac{3}{2 + k}$

$x = 1 + \frac{y}{3}$

$y = - \frac{3}{2 + k}$

$x = 1 + \frac{y}{3}$

Step 3

Simplify the right side.

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Step 3.1

Simplify $1 + \frac{- \frac{3}{2 + k}}{3}$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = 1 - \frac{3}{2 + k} \cdot \frac{1}{3}$

$y = - \frac{3}{2 + k}$

Step 3.1.1.2

Cancel the common factor of $3$ .

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Step 3.1.1.2.1

Move the leading negative in $- \frac{3}{2 + k}$ into the numerator.

$x = 1 + \frac{- 3}{2 + k} \cdot \frac{1}{3}$

$y = - \frac{3}{2 + k}$

Step 3.1.1.2.2

Factor $3$ out of $- 3$ .

$x = 1 + \frac{3 (- 1)}{2 + k} \cdot \frac{1}{3}$

$y = - \frac{3}{2 + k}$

Step 3.1.1.2.3

Cancel the common factor.

$x = 1 + \frac{3 \cdot - 1}{2 + k} \cdot \frac{1}{3}$

$y = - \frac{3}{2 + k}$

Step 3.1.1.2.4

Rewrite the expression.

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

$y = - \frac{3}{2 + k}$

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

$y = - \frac{3}{2 + k}$

Step 3.1.1.3

Move the negative in front of the fraction.

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

$y = - \frac{3}{2 + k}$

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

$y = - \frac{3}{2 + k}$

Step 3.1.2

Combine into one fraction.

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Step 3.1.2.1

Write $1$ as a fraction with a common denominator.

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

$y = - \frac{3}{2 + k}$

Step 3.1.2.2

Combine the numerators over the common denominator.

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

$y = - \frac{3}{2 + k}$

Step 3.1.2.3

Subtract $1$ from $2$ .

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

$y = - \frac{3}{2 + k}$

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

$y = - \frac{3}{2 + k}$

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

$y = - \frac{3}{2 + k}$

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

$y = - \frac{3}{2 + k}$