Finite Math Examples

$8 x = 2 y$ , $2 x + 3 y = 42$

Step 1

Divide each term in $8 x = 2 y$ by $8$ and simplify.

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Step 1.1

Divide each term in $8 x = 2 y$ by $8$ .

$\frac{8 x}{8} = \frac{2 y}{8}$

$2 x + 3 y = 42$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $8$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{2 y}{8}$

$2 x + 3 y = 42$

Step 1.2.1.2

Divide $x$ by $1$ .

$x = \frac{2 y}{8}$

$2 x + 3 y = 42$

$x = \frac{2 y}{8}$

$2 x + 3 y = 42$

$x = \frac{2 y}{8}$

$2 x + 3 y = 42$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $2$ and $8$ .

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Step 1.3.1.1

Factor $2$ out of $2 y$ .

$x = \frac{2 (y)}{8}$

$2 x + 3 y = 42$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $2$ out of $8$ .

$x = \frac{2 y}{2 \cdot 4}$

$2 x + 3 y = 42$

Step 1.3.1.2.2

Cancel the common factor.

$x = \frac{2 y}{2 \cdot 4}$

$2 x + 3 y = 42$

Step 1.3.1.2.3

Rewrite the expression.

$x = \frac{y}{4}$

$2 x + 3 y = 42$

$x = \frac{y}{4}$

$2 x + 3 y = 42$

$x = \frac{y}{4}$

$2 x + 3 y = 42$

$x = \frac{y}{4}$

$2 x + 3 y = 42$

$x = \frac{y}{4}$

$2 x + 3 y = 42$

Step 2

Replace all occurrences of $x$ with $\frac{y}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $2 x + 3 y = 42$ with $\frac{y}{4}$ .

$2 (\frac{y}{4}) + 3 y = 42$

$x = \frac{y}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (\frac{y}{4}) + 3 y$ .

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Step 2.2.1.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.1

Factor $2$ out of $4$ .

$2 (\frac{y}{2 (2)}) + 3 y = 42$

$x = \frac{y}{4}$

Step 2.2.1.1.2

Cancel the common factor.

$2 (\frac{y}{2 \cdot 2}) + 3 y = 42$

$x = \frac{y}{4}$

Step 2.2.1.1.3

Rewrite the expression.

$\frac{y}{2} + 3 y = 42$

$x = \frac{y}{4}$

$\frac{y}{2} + 3 y = 42$

$x = \frac{y}{4}$

Step 2.2.1.2

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

$\frac{y}{2} + 3 y \cdot \frac{2}{2} = 42$

$x = \frac{y}{4}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

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

$\frac{y}{2} + \frac{3 y \cdot 2}{2} = 42$

$x = \frac{y}{4}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$\frac{y + 3 y \cdot 2}{2} = 42$

$x = \frac{y}{4}$

$\frac{y + 3 y \cdot 2}{2} = 42$

$x = \frac{y}{4}$

Step 2.2.1.4

Simplify the numerator.

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Step 2.2.1.4.1

Factor $y$ out of $y + 3 y \cdot 2$ .

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Step 2.2.1.4.1.1

Raise $y$ to the power of $1$ .

$\frac{y + 3 y \cdot 2}{2} = 42$

$x = \frac{y}{4}$

Step 2.2.1.4.1.2

Factor $y$ out of $y^{1}$ .

$\frac{y \cdot 1 + 3 y \cdot 2}{2} = 42$

$x = \frac{y}{4}$

Step 2.2.1.4.1.3

Factor $y$ out of $3 y \cdot 2$ .

$\frac{y \cdot 1 + y (3 \cdot 2)}{2} = 42$

$x = \frac{y}{4}$

Step 2.2.1.4.1.4

Factor $y$ out of $y \cdot 1 + y (3 \cdot 2)$ .

$\frac{y (1 + 3 \cdot 2)}{2} = 42$

$x = \frac{y}{4}$

$\frac{y (1 + 3 \cdot 2)}{2} = 42$

$x = \frac{y}{4}$

Step 2.2.1.4.2

Multiply $3$ by $2$ .

$\frac{y (1 + 6)}{2} = 42$

$x = \frac{y}{4}$

Step 2.2.1.4.3

Add $1$ and $6$ .

$\frac{y \cdot 7}{2} = 42$

$x = \frac{y}{4}$

$\frac{y \cdot 7}{2} = 42$

$x = \frac{y}{4}$

Step 2.2.1.5

Move $7$ to the left of $y$ .

$\frac{7 y}{2} = 42$

$x = \frac{y}{4}$

$\frac{7 y}{2} = 42$

$x = \frac{y}{4}$

$\frac{7 y}{2} = 42$

$x = \frac{y}{4}$

$\frac{7 y}{2} = 42$

$x = \frac{y}{4}$

Step 3

Solve for $y$ in $\frac{7 y}{2} = 42$ .

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Step 3.1

Multiply both sides of the equation by $\frac{2}{7}$ .

$\frac{2}{7} \cdot \frac{7 y}{2} = \frac{2}{7} \cdot 42$

$x = \frac{y}{4}$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $\frac{2}{7} \cdot \frac{7 y}{2}$ .

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Step 3.2.1.1.1

Cancel the common factor of $2$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{2}{7} \cdot \frac{7 y}{2} = \frac{2}{7} \cdot 42$

$x = \frac{y}{4}$

Step 3.2.1.1.1.2

Rewrite the expression.

$\frac{1}{7} \cdot (7 y) = \frac{2}{7} \cdot 42$

$x = \frac{y}{4}$

$\frac{1}{7} \cdot (7 y) = \frac{2}{7} \cdot 42$

$x = \frac{y}{4}$

Step 3.2.1.1.2

Cancel the common factor of $7$ .

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Step 3.2.1.1.2.1

Factor $7$ out of $7 y$ .

$\frac{1}{7} \cdot (7 (y)) = \frac{2}{7} \cdot 42$

$x = \frac{y}{4}$

Step 3.2.1.1.2.2

Cancel the common factor.

$\frac{1}{7} \cdot (7 y) = \frac{2}{7} \cdot 42$

$x = \frac{y}{4}$

Step 3.2.1.1.2.3

Rewrite the expression.

$y = \frac{2}{7} \cdot 42$

$x = \frac{y}{4}$

$y = \frac{2}{7} \cdot 42$

$x = \frac{y}{4}$

$y = \frac{2}{7} \cdot 42$

$x = \frac{y}{4}$

$y = \frac{2}{7} \cdot 42$

$x = \frac{y}{4}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\frac{2}{7} \cdot 42$ .

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Step 3.2.2.1.1

Cancel the common factor of $7$ .

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Step 3.2.2.1.1.1

Factor $7$ out of $42$ .

$y = \frac{2}{7} \cdot (7 (6))$

$x = \frac{y}{4}$

Step 3.2.2.1.1.2

Cancel the common factor.

$y = \frac{2}{7} \cdot (7 \cdot 6)$

$x = \frac{y}{4}$

Step 3.2.2.1.1.3

Rewrite the expression.

$y = 2 \cdot 6$

$x = \frac{y}{4}$

$y = 2 \cdot 6$

$x = \frac{y}{4}$

Step 3.2.2.1.2

Multiply $2$ by $6$ .

$y = 12$

$x = \frac{y}{4}$

$y = 12$

$x = \frac{y}{4}$

$y = 12$

$x = \frac{y}{4}$

$y = 12$

$x = \frac{y}{4}$

$y = 12$

$x = \frac{y}{4}$

Step 4

Replace all occurrences of $y$ with $12$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{y}{4}$ with $12$ .

$x = \frac{12}{4}$

$y = 12$

Step 4.2

Simplify the right side.

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Step 4.2.1

Divide $12$ by $4$ .

$x = 3$

$y = 12$

$x = 3$

$y = 12$

$x = 3$

$y = 12$

Step 5

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

$(3, 12)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(3, 12)$

Equation Form:

$x = 3, y = 12$

Step 7