Finite Math Examples

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

Step 1

Eliminate the equal sides of each equation and combine.

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

Step 2

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

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Step 2.1

Simplify the right side.

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Step 2.1.1

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

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Step 2.1.1.1

Convert $2 \frac{1}{2}$ 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.

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

Step 2.1.1.1.2

Add $2$ and $\frac{1}{2}$ .

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Step 2.1.1.1.2.1

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

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

Step 2.1.1.1.2.2

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

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

Step 2.1.1.1.2.3

Combine the numerators over the common denominator.

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

Step 2.1.1.1.2.4

Simplify the numerator.

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Step 2.1.1.1.2.4.1

Multiply $2$ by $2$ .

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

Step 2.1.1.1.2.4.2

Add $4$ and $1$ .

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

Step 2.1.1.2

Combine $\frac{1}{2}$ and $x$ .

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Combine the numerators over the common denominator.

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

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Factor $x$ out of $3 x \cdot 2 - x$ .

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Step 2.2.5.1.1

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

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

Step 2.2.5.1.2

Factor $x$ out of $- x$ .

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

Step 2.2.5.1.3

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

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

Step 2.2.5.2

Multiply $3$ by $2$ .

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

Step 2.2.5.3

Subtract $1$ from $6$ .

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

Step 2.2.6

Move $5$ to the left of $x$ .

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

Step 2.3

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

$5 x = 5$

Step 2.4

Divide each term in $5 x = 5$ by $5$ and simplify.

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Step 2.4.1

Divide each term in $5 x = 5$ by $5$ .

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $5$ .

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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