Finite Math Examples

y = 3 x

$y = 3 x$ ,

y = \frac{1}{2} x + 2 \frac{1}{2}

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

Step 1

Eliminate the equal sides of each equation and combine.

3 x = \frac{1}{2} x + 2 \frac{1}{2}

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

Step 2

Solve

3 x = \frac{1}{2} x + 2 \frac{1}{2}

$3 x = \frac{1}{2} x + 2 \frac{1}{2}$ for

x

$x$ .

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Step 2.1

Simplify the right side.

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Step 2.1.1

Simplify

\frac{1}{2} x + 2 \frac{1}{2}

$\frac{1}{2} x + 2 \frac{1}{2}$ .

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Step 2.1.1.1

Convert

2 \frac{1}{2}

$2 \frac{1}{2}$ to an improper fraction.

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Step 2.1.1.1.1

A mixed number is an addition of its whole and fractional parts.

3 x = \frac{1}{2} x + 2 + \frac{1}{2}

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

Step 2.1.1.1.2

Add

2

$2$ and

\frac{1}{2}

$\frac{1}{2}$ .

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Step 2.1.1.1.2.1

To write

2

$2$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

3 x = \frac{1}{2} x + 2 \cdot \frac{2}{2} + \frac{1}{2}

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

Step 2.1.1.1.2.2

Combine

2

$2$ and

\frac{2}{2}

$\frac{2}{2}$ .

3 x = \frac{1}{2} x + \frac{2 \cdot 2}{2} + \frac{1}{2}

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

Step 2.1.1.1.2.3

Combine the numerators over the common denominator.

3 x = \frac{1}{2} x + \frac{2 \cdot 2 + 1}{2}

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

Step 2.1.1.1.2.4

Simplify the numerator.

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Step 2.1.1.1.2.4.1

Multiply

2

$2$ by

2

$2$ .

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

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

Step 2.1.1.1.2.4.2

Add

4

$4$ and

1

$1$ .

3 x = \frac{1}{2} x + \frac{5}{2}

$3 x = \frac{1}{2} x + \frac{5}{2}$

3 x = \frac{1}{2} x + \frac{5}{2}

$3 x = \frac{1}{2} x + \frac{5}{2}$

3 x = \frac{1}{2} x + \frac{5}{2}

$3 x = \frac{1}{2} x + \frac{5}{2}$

3 x = \frac{1}{2} x + \frac{5}{2}

$3 x = \frac{1}{2} x + \frac{5}{2}$

Step 2.1.1.2

Combine

\frac{1}{2}

$\frac{1}{2}$ and

x

$x$ .

3 x = \frac{x}{2} + \frac{5}{2}

$3 x = \frac{x}{2} + \frac{5}{2}$

3 x = \frac{x}{2} + \frac{5}{2}

$3 x = \frac{x}{2} + \frac{5}{2}$

3 x = \frac{x}{2} + \frac{5}{2}

$3 x = \frac{x}{2} + \frac{5}{2}$

Step 2.2

Move all terms containing

x

$x$ to the left side of the equation.

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Step 2.2.1

Subtract

\frac{x}{2}

$\frac{x}{2}$ from both sides of the equation.

3 x - \frac{x}{2} = \frac{5}{2}

$3 x - \frac{x}{2} = \frac{5}{2}$

Step 2.2.2

To write

3 x

$3 x$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

3 x \cdot \frac{2}{2} - \frac{x}{2} = \frac{5}{2}

$3 x \cdot \frac{2}{2} - \frac{x}{2} = \frac{5}{2}$

Step 2.2.3

Combine

3 x

$3 x$ and

\frac{2}{2}

$\frac{2}{2}$ .

\frac{3 x \cdot 2}{2} - \frac{x}{2} = \frac{5}{2}

$\frac{3 x \cdot 2}{2} - \frac{x}{2} = \frac{5}{2}$

Step 2.2.4

Combine the numerators over the common denominator.

\frac{3 x \cdot 2 - x}{2} = \frac{5}{2}

$\frac{3 x \cdot 2 - x}{2} = \frac{5}{2}$

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Factor

x

$x$ out of

3 x \cdot 2 - x

$3 x \cdot 2 - x$ .

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Step 2.2.5.1.1

Factor

x

$x$ out of

3 x \cdot 2

$3 x \cdot 2$ .

\frac{x (3 \cdot 2) - x}{2} = \frac{5}{2}

$\frac{x (3 \cdot 2) - x}{2} = \frac{5}{2}$

Step 2.2.5.1.2

Factor

x

$x$ out of

- x

$- x$ .

\frac{x (3 \cdot 2) + x \cdot - 1}{2} = \frac{5}{2}

$\frac{x (3 \cdot 2) + x \cdot - 1}{2} = \frac{5}{2}$

Step 2.2.5.1.3

Factor

x

$x$ out of

x (3 \cdot 2) + x \cdot - 1

$x (3 \cdot 2) + x \cdot - 1$ .

\frac{x (3 \cdot 2 - 1)}{2} = \frac{5}{2}

$\frac{x (3 \cdot 2 - 1)}{2} = \frac{5}{2}$

\frac{x (3 \cdot 2 - 1)}{2} = \frac{5}{2}

$\frac{x (3 \cdot 2 - 1)}{2} = \frac{5}{2}$

Step 2.2.5.2

Multiply

3

$3$ by

2

$2$ .

\frac{x (6 - 1)}{2} = \frac{5}{2}

$\frac{x (6 - 1)}{2} = \frac{5}{2}$

Step 2.2.5.3

Subtract

1

$1$ from

6

$6$ .

\frac{x \cdot 5}{2} = \frac{5}{2}

$\frac{x \cdot 5}{2} = \frac{5}{2}$

\frac{x \cdot 5}{2} = \frac{5}{2}

$\frac{x \cdot 5}{2} = \frac{5}{2}$

Step 2.2.6

Move

5

$5$ to the left of

x

$x$ .

\frac{5 x}{2} = \frac{5}{2}

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

\frac{5 x}{2} = \frac{5}{2}

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

Step 2.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

5 x = 5

$5 x = 5$

Step 2.4

Divide each term in

5 x = 5

$5 x = 5$ by

5

$5$ and simplify.

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Step 2.4.1

Divide each term in

5 x = 5

$5 x = 5$ by

5

$5$ .

\frac{5 x}{5} = \frac{5}{5}

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of

5

$5$ .

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Step 2.4.2.1.1

Cancel the common factor.

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

Step 2.4.2.1.2

Divide

$x$ by

$1$ .

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

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Divide

$5$ by

$5$ .

$x = 1$

Step 3

Evaluate

$y$ when

$x = 1$ .

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Step 3.1

Substitute

$1$ for

$x$ .

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

Step 3.2

Substitute

$1$ for

$x$ in

$y = \frac{1}{2} \cdot (1) + 2 \frac{1}{2}$ and solve for

$y$ .

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Step 3.2.1

Multiply

$\frac{1}{2}$ by

$1$ .

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

Step 3.2.2

Simplify

$\frac{1}{2} \cdot 1 + 2 \frac{1}{2}$ .

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Step 3.2.2.1

Convert

$2 \frac{1}{2}$ to an improper fraction.

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Step 3.2.2.1.1

A mixed number is an addition of its whole and fractional parts.

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

Step 3.2.2.1.2

Add

$2$ and

$\frac{1}{2}$ .

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Step 3.2.2.1.2.1

To write

$2$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

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

Step 3.2.2.1.2.2

Combine

$2$ and

$\frac{2}{2}$ .

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

Step 3.2.2.1.2.3

Combine the numerators over the common denominator.

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

Step 3.2.2.1.2.4

Simplify the numerator.

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Step 3.2.2.1.2.4.1

Multiply

$2$ by

$2$ .

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

Step 3.2.2.1.2.4.2

Add

$4$ and

$1$ .

$y = \frac{1}{2} \cdot 1 + \frac{5}{2}$

Step 3.2.2.2

Multiply

$\frac{1}{2}$ by

$1$ .

$y = \frac{1}{2} + \frac{5}{2}$

Step 3.2.2.3

Combine the numerators over the common denominator.

$y = \frac{1 + 5}{2}$

Step 3.2.2.4

Simplify the expression.

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Step 3.2.2.4.1

Add

$1$ and

$5$ .

$y = \frac{6}{2}$

Step 3.2.2.4.2

Divide

$6$ by

$2$ .

$y = 3$

Step 4

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

$(1, 3)$

Step 5

The result can be shown in multiple forms.

Point Form:

$(1, 3)$

Equation Form:

$x = 1, y = 3$

Step 6