Finite Math Examples

$3 y + 3 x = 4$ , $3 y = 6 - 2 x$

Step 1

Divide each term in $3 y = 6 - 2 x$ by $3$ and simplify.

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Step 1.1

Divide each term in $3 y = 6 - 2 x$ by $3$ .

$\frac{3 y}{3} = \frac{6}{3} + \frac{- 2 x}{3}$

$3 y + 3 x = 4$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $3$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{6}{3} + \frac{- 2 x}{3}$

$3 y + 3 x = 4$

Step 1.2.1.2

Divide $y$ by $1$ .

$y = \frac{6}{3} + \frac{- 2 x}{3}$

$3 y + 3 x = 4$

$y = \frac{6}{3} + \frac{- 2 x}{3}$

$3 y + 3 x = 4$

$y = \frac{6}{3} + \frac{- 2 x}{3}$

$3 y + 3 x = 4$

Step 1.3

Simplify the right side.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Divide $6$ by $3$ .

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

$3 y + 3 x = 4$

Step 1.3.1.2

Move the negative in front of the fraction.

$y = 2 - \frac{2 x}{3}$

$3 y + 3 x = 4$

$y = 2 - \frac{2 x}{3}$

$3 y + 3 x = 4$

$y = 2 - \frac{2 x}{3}$

$3 y + 3 x = 4$

$y = 2 - \frac{2 x}{3}$

$3 y + 3 x = 4$

Step 2

Replace all occurrences of $y$ with $2 - \frac{2 x}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $3 y + 3 x = 4$ with $2 - \frac{2 x}{3}$ .

$3 (2 - \frac{2 x}{3}) + 3 x = 4$

$y = 2 - \frac{2 x}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (2 - \frac{2 x}{3}) + 3 x$ .

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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.

$3 \cdot 2 + 3 (- \frac{2 x}{3}) + 3 x = 4$

$y = 2 - \frac{2 x}{3}$

Step 2.2.1.1.2

Multiply $3$ by $2$ .

$6 + 3 (- \frac{2 x}{3}) + 3 x = 4$

$y = 2 - \frac{2 x}{3}$

Step 2.2.1.1.3

Cancel the common factor of $3$ .

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Step 2.2.1.1.3.1

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

$6 + 3 (\frac{- 2 x}{3}) + 3 x = 4$

$y = 2 - \frac{2 x}{3}$

Step 2.2.1.1.3.2

Cancel the common factor.

$6 + 3 (\frac{- 2 x}{3}) + 3 x = 4$

$y = 2 - \frac{2 x}{3}$

Step 2.2.1.1.3.3

Rewrite the expression.

$6 - 2 x + 3 x = 4$

$y = 2 - \frac{2 x}{3}$

$6 - 2 x + 3 x = 4$

$y = 2 - \frac{2 x}{3}$

$6 - 2 x + 3 x = 4$

$y = 2 - \frac{2 x}{3}$

Step 2.2.1.2

Add $- 2 x$ and $3 x$ .

$6 + x = 4$

$y = 2 - \frac{2 x}{3}$

$6 + x = 4$

$y = 2 - \frac{2 x}{3}$

$6 + x = 4$

$y = 2 - \frac{2 x}{3}$

$6 + x = 4$

$y = 2 - \frac{2 x}{3}$

Step 3

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

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Step 3.1

Subtract $6$ from both sides of the equation.

$x = 4 - 6$

$y = 2 - \frac{2 x}{3}$

Step 3.2

Subtract $6$ from $4$ .

$x = - 2$

$y = 2 - \frac{2 x}{3}$

$x = - 2$

$y = 2 - \frac{2 x}{3}$

Step 4

Replace all occurrences of $x$ with $- 2$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = 2 - \frac{2 x}{3}$ with $- 2$ .

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

$x = - 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $2 - \frac{2 (- 2)}{3}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Multiply $2$ by $- 2$ .

$y = 2 - \frac{- 4}{3}$

$x = - 2$

Step 4.2.1.1.2

Move the negative in front of the fraction.

$y = 2 + \frac{4}{3}$

$x = - 2$

Step 4.2.1.1.3

Multiply $- - \frac{4}{3}$ .

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Step 4.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$y = 2 + 1 (\frac{4}{3})$

$x = - 2$

Step 4.2.1.1.3.2

Multiply $\frac{4}{3}$ by $1$ .

$y = 2 + \frac{4}{3}$

$x = - 2$

$y = 2 + \frac{4}{3}$

$x = - 2$

$y = 2 + \frac{4}{3}$

$x = - 2$

Step 4.2.1.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = 2 \cdot \frac{3}{3} + \frac{4}{3}$

$x = - 2$

Step 4.2.1.3

Combine $2$ and $\frac{3}{3}$ .

$y = \frac{2 \cdot 3}{3} + \frac{4}{3}$

$x = - 2$

Step 4.2.1.4

Combine the numerators over the common denominator.

$y = \frac{2 \cdot 3 + 4}{3}$

$x = - 2$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $2$ by $3$ .

$y = \frac{6 + 4}{3}$

$x = - 2$

Step 4.2.1.5.2

Add $6$ and $4$ .

$y = \frac{10}{3}$

$x = - 2$

$y = \frac{10}{3}$

$x = - 2$

$y = \frac{10}{3}$

$x = - 2$

$y = \frac{10}{3}$

$x = - 2$

$y = \frac{10}{3}$

$x = - 2$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(- 2, \frac{10}{3})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 2, \frac{10}{3})$

Equation Form:

$x = - 2, y = \frac{10}{3}$

Step 7