Finite Math Examples

$4 x = 3 y + 2$ , $8 x = - 3 - 2 y$

Step 1

Divide each term in $4 x = 3 y + 2$ by $4$ and simplify.

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Step 1.1

Divide each term in $4 x = 3 y + 2$ by $4$ .

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

$8 x = - 3 - 2 y$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $4$ .

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Step 1.2.1.1

Cancel the common factor.

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

$8 x = - 3 - 2 y$

Step 1.2.1.2

Divide $x$ by $1$ .

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

$8 x = - 3 - 2 y$

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

$8 x = - 3 - 2 y$

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

$8 x = - 3 - 2 y$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $2$ and $4$ .

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Step 1.3.1.1

Factor $2$ out of $2$ .

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

$8 x = - 3 - 2 y$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $2$ out of $4$ .

$x = \frac{3 y}{4} + \frac{2 \cdot 1}{2 \cdot 2}$

$8 x = - 3 - 2 y$

Step 1.3.1.2.2

Cancel the common factor.

$x = \frac{3 y}{4} + \frac{2 \cdot 1}{2 \cdot 2}$

$8 x = - 3 - 2 y$

Step 1.3.1.2.3

Rewrite the expression.

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

$8 x = - 3 - 2 y$

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

$8 x = - 3 - 2 y$

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

$8 x = - 3 - 2 y$

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

$8 x = - 3 - 2 y$

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

$8 x = - 3 - 2 y$

Step 2

Replace all occurrences of $x$ with $\frac{3 y}{4} + \frac{1}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $8 x = - 3 - 2 y$ with $\frac{3 y}{4} + \frac{1}{2}$ .

$8 (\frac{3 y}{4} + \frac{1}{2}) = - 3 - 2 y$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $8 (\frac{3 y}{4} + \frac{1}{2})$ .

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Step 2.2.1.1

Apply the distributive property.

$8 (\frac{3 y}{4}) + 8 (\frac{1}{2}) = - 3 - 2 y$

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

Step 2.2.1.2

Cancel the common factor of $4$ .

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Step 2.2.1.2.1

Factor $4$ out of $8$ .

$4 (2) (\frac{3 y}{4}) + 8 (\frac{1}{2}) = - 3 - 2 y$

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

Step 2.2.1.2.2

Cancel the common factor.

$4 \cdot (2 (\frac{3 y}{4})) + 8 (\frac{1}{2}) = - 3 - 2 y$

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

Step 2.2.1.2.3

Rewrite the expression.

$2 (3 y) + 8 (\frac{1}{2}) = - 3 - 2 y$

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

$2 (3 y) + 8 (\frac{1}{2}) = - 3 - 2 y$

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

Step 2.2.1.3

Multiply $3$ by $2$ .

$6 y + 8 (\frac{1}{2}) = - 3 - 2 y$

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

Step 2.2.1.4

Cancel the common factor of $2$ .

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Step 2.2.1.4.1

Factor $2$ out of $8$ .

$6 y + 2 (4) (\frac{1}{2}) = - 3 - 2 y$

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

Step 2.2.1.4.2

Cancel the common factor.

$6 y + 2 \cdot (4 (\frac{1}{2})) = - 3 - 2 y$

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

Step 2.2.1.4.3

Rewrite the expression.

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

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

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

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

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

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

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

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

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

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

Step 3

Solve for $y$ in $6 y + 4 = - 3 - 2 y$ .

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Step 3.1

Move all terms containing $y$ to the left side of the equation.

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Step 3.1.1

Add $2 y$ to both sides of the equation.

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

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

Step 3.1.2

Add $6 y$ and $2 y$ .

$8 y + 4 = - 3$

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

$8 y + 4 = - 3$

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

Step 3.2

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

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Step 3.2.1

Subtract $4$ from both sides of the equation.

$8 y = - 3 - 4$

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

Step 3.2.2

Subtract $4$ from $- 3$ .

$8 y = - 7$

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

$8 y = - 7$

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

Step 3.3

Divide each term in $8 y = - 7$ by $8$ and simplify.

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Step 3.3.1

Divide each term in $8 y = - 7$ by $8$ .

$\frac{8 y}{8} = \frac{- 7}{8}$

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $8$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{8 y}{8} = \frac{- 7}{8}$

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

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 7}{8}$

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

$y = \frac{- 7}{8}$

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

$y = \frac{- 7}{8}$

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

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Move the negative in front of the fraction.

$y = - \frac{7}{8}$

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

$y = - \frac{7}{8}$

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

$y = - \frac{7}{8}$

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

$y = - \frac{7}{8}$

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

Step 4

Replace all occurrences of $y$ with $- \frac{7}{8}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{3 y}{4} + \frac{1}{2}$ with $- \frac{7}{8}$ .

$x = \frac{3 (- \frac{7}{8})}{4} + \frac{1}{2}$

$y = - \frac{7}{8}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{3 (- \frac{7}{8})}{4} + \frac{1}{2}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Simplify the numerator.

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Step 4.2.1.1.1.1

Multiply $- 1$ by $3$ .

$x = \frac{- 3 (\frac{7}{8})}{4} + \frac{1}{2}$

$y = - \frac{7}{8}$

Step 4.2.1.1.1.2

Combine $- 3$ and $\frac{7}{8}$ .

$x = \frac{\frac{- 3 \cdot 7}{8}}{4} + \frac{1}{2}$

$y = - \frac{7}{8}$

$x = \frac{\frac{- 3 \cdot 7}{8}}{4} + \frac{1}{2}$

$y = - \frac{7}{8}$

Step 4.2.1.1.2

Multiply $- 3$ by $7$ .

$x = \frac{\frac{- 21}{8}}{4} + \frac{1}{2}$

$y = - \frac{7}{8}$

Step 4.2.1.1.3

Move the negative in front of the fraction.

$x = \frac{- \frac{21}{8}}{4} + \frac{1}{2}$

$y = - \frac{7}{8}$

Step 4.2.1.1.4

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{21}{8} \cdot \frac{1}{4} + \frac{1}{2}$

$y = - \frac{7}{8}$

Step 4.2.1.1.5

Multiply $- \frac{21}{8} \cdot \frac{1}{4}$ .

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Step 4.2.1.1.5.1

Multiply $\frac{1}{4}$ by $\frac{21}{8}$ .

$x = - \frac{21}{4 \cdot 8} + \frac{1}{2}$

$y = - \frac{7}{8}$

Step 4.2.1.1.5.2

Multiply $4$ by $8$ .

$x = - \frac{21}{32} + \frac{1}{2}$

$y = - \frac{7}{8}$

$x = - \frac{21}{32} + \frac{1}{2}$

$y = - \frac{7}{8}$

$x = - \frac{21}{32} + \frac{1}{2}$

$y = - \frac{7}{8}$

Step 4.2.1.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$x = - \frac{21}{32} + \frac{1}{2} \cdot \frac{16}{16}$

$y = - \frac{7}{8}$

Step 4.2.1.3

Write each expression with a common denominator of $32$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1.3.1

Multiply $\frac{1}{2}$ by $\frac{16}{16}$ .

$x = - \frac{21}{32} + \frac{16}{2 \cdot 16}$

$y = - \frac{7}{8}$

Step 4.2.1.3.2

Multiply $2$ by $16$ .

$x = - \frac{21}{32} + \frac{16}{32}$

$y = - \frac{7}{8}$

$x = - \frac{21}{32} + \frac{16}{32}$

$y = - \frac{7}{8}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x = \frac{- 21 + 16}{32}$

$y = - \frac{7}{8}$

Step 4.2.1.5

Add $- 21$ and $16$ .

$x = \frac{- 5}{32}$

$y = - \frac{7}{8}$

Step 4.2.1.6

Move the negative in front of the fraction.

$x = - \frac{5}{32}$

$y = - \frac{7}{8}$

$x = - \frac{5}{32}$

$y = - \frac{7}{8}$

$x = - \frac{5}{32}$

$y = - \frac{7}{8}$

$x = - \frac{5}{32}$

$y = - \frac{7}{8}$

Step 5

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

$(- \frac{5}{32}, - \frac{7}{8})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- \frac{5}{32}, - \frac{7}{8})$

Equation Form:

$x = - \frac{5}{32}, y = - \frac{7}{8}$

Step 7