Finite Math Examples

$\frac{7}{4} x - \frac{5}{2} = 2$ , $\frac{1}{4} x + \frac{7}{2} = 8$

Step 1

Solve for $x$ in $\frac{7}{4} x - \frac{5}{2} = 2$ .

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Step 1.1

Combine $\frac{7}{4}$ and $x$ .

$\frac{7 x}{4} - \frac{5}{2} = 2$

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

Step 1.2

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

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Step 1.2.1

Add $\frac{5}{2}$ to both sides of the equation.

$\frac{7 x}{4} = 2 + \frac{5}{2}$

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

Step 1.2.2

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

$\frac{7 x}{4} = 2 \cdot \frac{2}{2} + \frac{5}{2}$

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

Step 1.2.3

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

$\frac{7 x}{4} = \frac{2 \cdot 2}{2} + \frac{5}{2}$

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

Step 1.2.4

Combine the numerators over the common denominator.

$\frac{7 x}{4} = \frac{2 \cdot 2 + 5}{2}$

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

Step 1.2.5

Simplify the numerator.

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Step 1.2.5.1

Multiply $2$ by $2$ .

$\frac{7 x}{4} = \frac{4 + 5}{2}$

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

Step 1.2.5.2

Add $4$ and $5$ .

$\frac{7 x}{4} = \frac{9}{2}$

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

$\frac{7 x}{4} = \frac{9}{2}$

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

$\frac{7 x}{4} = \frac{9}{2}$

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

Step 1.3

Multiply both sides of the equation by $\frac{4}{7}$ .

$\frac{4}{7} \cdot \frac{7 x}{4} = \frac{4}{7} \cdot \frac{9}{2}$

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

Step 1.4

Simplify both sides of the equation.

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Step 1.4.1

Simplify the left side.

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Step 1.4.1.1

Simplify $\frac{4}{7} \cdot \frac{7 x}{4}$ .

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Step 1.4.1.1.1

Cancel the common factor of $4$ .

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Step 1.4.1.1.1.1

Cancel the common factor.

$\frac{4}{7} \cdot \frac{7 x}{4} = \frac{4}{7} \cdot \frac{9}{2}$

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

Step 1.4.1.1.1.2

Rewrite the expression.

$\frac{1}{7} \cdot (7 x) = \frac{4}{7} \cdot \frac{9}{2}$

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

$\frac{1}{7} \cdot (7 x) = \frac{4}{7} \cdot \frac{9}{2}$

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

Step 1.4.1.1.2

Cancel the common factor of $7$ .

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Step 1.4.1.1.2.1

Factor $7$ out of $7 x$ .

$\frac{1}{7} \cdot (7 (x)) = \frac{4}{7} \cdot \frac{9}{2}$

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

Step 1.4.1.1.2.2

Cancel the common factor.

$\frac{1}{7} \cdot (7 x) = \frac{4}{7} \cdot \frac{9}{2}$

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

Step 1.4.1.1.2.3

Rewrite the expression.

$x = \frac{4}{7} \cdot \frac{9}{2}$

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

$x = \frac{4}{7} \cdot \frac{9}{2}$

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

$x = \frac{4}{7} \cdot \frac{9}{2}$

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

$x = \frac{4}{7} \cdot \frac{9}{2}$

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

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Simplify $\frac{4}{7} \cdot \frac{9}{2}$ .

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Step 1.4.2.1.1

Cancel the common factor of $2$ .

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Step 1.4.2.1.1.1

Factor $2$ out of $4$ .

$x = \frac{2 (2)}{7} \cdot \frac{9}{2}$

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

Step 1.4.2.1.1.2

Cancel the common factor.

$x = \frac{2 \cdot 2}{7} \cdot \frac{9}{2}$

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

Step 1.4.2.1.1.3

Rewrite the expression.

$x = \frac{2}{7} \cdot 9$

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

$x = \frac{2}{7} \cdot 9$

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

Step 1.4.2.1.2

Combine $\frac{2}{7}$ and $9$ .

$x = \frac{2 \cdot 9}{7}$

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

Step 1.4.2.1.3

Multiply $2$ by $9$ .

$x = \frac{18}{7}$

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

$x = \frac{18}{7}$

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

$x = \frac{18}{7}$

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

$x = \frac{18}{7}$

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

$x = \frac{18}{7}$

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

Step 2

Replace all occurrences of $x$ with $\frac{18}{7}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $\frac{1}{4} x + \frac{7}{2} = 8$ with $\frac{18}{7}$ .

$\frac{1}{4} \cdot (\frac{18}{7}) + \frac{7}{2} = 8$

$x = \frac{18}{7}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $\frac{1}{4} \cdot (\frac{18}{7}) + \frac{7}{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.1.1

Factor $2$ out of $4$ .

$\frac{1}{2 (2)} \cdot \frac{18}{7} + \frac{7}{2} = 8$

$x = \frac{18}{7}$

Step 2.2.1.1.1.2

Factor $2$ out of $18$ .

$\frac{1}{2 \cdot 2} \cdot \frac{2 \cdot 9}{7} + \frac{7}{2} = 8$

$x = \frac{18}{7}$

Step 2.2.1.1.1.3

Cancel the common factor.

$\frac{1}{2 \cdot 2} \cdot \frac{2 \cdot 9}{7} + \frac{7}{2} = 8$

$x = \frac{18}{7}$

Step 2.2.1.1.1.4

Rewrite the expression.

$\frac{1}{2} \cdot \frac{9}{7} + \frac{7}{2} = 8$

$x = \frac{18}{7}$

$\frac{1}{2} \cdot \frac{9}{7} + \frac{7}{2} = 8$

$x = \frac{18}{7}$

Step 2.2.1.1.2

Multiply $\frac{1}{2}$ by $\frac{9}{7}$ .

$\frac{9}{2 \cdot 7} + \frac{7}{2} = 8$

$x = \frac{18}{7}$

Step 2.2.1.1.3

Multiply $2$ by $7$ .

$\frac{9}{14} + \frac{7}{2} = 8$

$x = \frac{18}{7}$

$\frac{9}{14} + \frac{7}{2} = 8$

$x = \frac{18}{7}$

Step 2.2.1.2

To write $\frac{7}{2}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{9}{14} + \frac{7}{2} \cdot \frac{7}{7} = 8$

$x = \frac{18}{7}$

Step 2.2.1.3

Write each expression with a common denominator of $14$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.3.1

Multiply $\frac{7}{2}$ by $\frac{7}{7}$ .

$\frac{9}{14} + \frac{7 \cdot 7}{2 \cdot 7} = 8$

$x = \frac{18}{7}$

Step 2.2.1.3.2

Multiply $2$ by $7$ .

$\frac{9}{14} + \frac{7 \cdot 7}{14} = 8$

$x = \frac{18}{7}$

$\frac{9}{14} + \frac{7 \cdot 7}{14} = 8$

$x = \frac{18}{7}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{9 + 7 \cdot 7}{14} = 8$

$x = \frac{18}{7}$

Step 2.2.1.5

Simplify the numerator.

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Step 2.2.1.5.1

Multiply $7$ by $7$ .

$\frac{9 + 49}{14} = 8$

$x = \frac{18}{7}$

Step 2.2.1.5.2

Add $9$ and $49$ .

$\frac{58}{14} = 8$

$x = \frac{18}{7}$

$\frac{58}{14} = 8$

$x = \frac{18}{7}$

Step 2.2.1.6

Cancel the common factor of $58$ and $14$ .

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Step 2.2.1.6.1

Factor $2$ out of $58$ .

$\frac{2 (29)}{14} = 8$

$x = \frac{18}{7}$

Step 2.2.1.6.2

Cancel the common factors.

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Step 2.2.1.6.2.1

Factor $2$ out of $14$ .

$\frac{2 \cdot 29}{2 \cdot 7} = 8$

$x = \frac{18}{7}$

Step 2.2.1.6.2.2

Cancel the common factor.

$\frac{2 \cdot 29}{2 \cdot 7} = 8$

$x = \frac{18}{7}$

Step 2.2.1.6.2.3

Rewrite the expression.

$\frac{29}{7} = 8$

$x = \frac{18}{7}$

$\frac{29}{7} = 8$

$x = \frac{18}{7}$

$\frac{29}{7} = 8$

$x = \frac{18}{7}$

$\frac{29}{7} = 8$

$x = \frac{18}{7}$

$\frac{29}{7} = 8$

$x = \frac{18}{7}$

$\frac{29}{7} = 8$

$x = \frac{18}{7}$

Step 3

Since $\frac{29}{7} = 8$ is not true, there is no solution.

No solution

Step 4