Algebra Examples

$2 \frac{1}{4} (4 x - 1) = 1 \frac{3}{5} (1 + 4 x)$

Step 1

Simplify the left side.

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Step 1.1

Simplify $2 \frac{1}{4} (4 x - 1)$ .

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Step 1.1.1

Convert $2 \frac{1}{4}$ to an improper fraction.

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Step 1.1.1.1

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

$(2 + \frac{1}{4}) (4 x - 1) = 1 \frac{3}{5} (1 + 4 x)$

Step 1.1.1.2

Add $2$ and $\frac{1}{4}$ .

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Step 1.1.1.2.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$(2 \cdot \frac{4}{4} + \frac{1}{4}) (4 x - 1) = 1 \frac{3}{5} (1 + 4 x)$

Step 1.1.1.2.2

Combine $2$ and $\frac{4}{4}$ .

$(\frac{2 \cdot 4}{4} + \frac{1}{4}) (4 x - 1) = 1 \frac{3}{5} (1 + 4 x)$

Step 1.1.1.2.3

Combine the numerators over the common denominator.

$\frac{2 \cdot 4 + 1}{4} (4 x - 1) = 1 \frac{3}{5} (1 + 4 x)$

Step 1.1.1.2.4

Simplify the numerator.

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Step 1.1.1.2.4.1

Multiply $2$ by $4$ .

$\frac{8 + 1}{4} (4 x - 1) = 1 \frac{3}{5} (1 + 4 x)$

Step 1.1.1.2.4.2

Add $8$ and $1$ .

$\frac{9}{4} (4 x - 1) = 1 \frac{3}{5} (1 + 4 x)$

Step 1.1.2

Apply the distributive property.

$\frac{9}{4} (4 x) + \frac{9}{4} \cdot - 1 = 1 \frac{3}{5} (1 + 4 x)$

Step 1.1.3

Cancel the common factor of $4$ .

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Step 1.1.3.1

Factor $4$ out of $4 x$ .

$\frac{9}{4} (4 (x)) + \frac{9}{4} \cdot - 1 = 1 \frac{3}{5} (1 + 4 x)$

Step 1.1.3.2

Cancel the common factor.

$\frac{9}{4} (4 x) + \frac{9}{4} \cdot - 1 = 1 \frac{3}{5} (1 + 4 x)$

Step 1.1.3.3

Rewrite the expression.

$9 x + \frac{9}{4} \cdot - 1 = 1 \frac{3}{5} (1 + 4 x)$

Step 1.1.4

Multiply $\frac{9}{4} \cdot - 1$ .

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Step 1.1.4.1

Combine $\frac{9}{4}$ and $- 1$ .

$9 x + \frac{9 \cdot - 1}{4} = 1 \frac{3}{5} (1 + 4 x)$

Step 1.1.4.2

Multiply $9$ by $- 1$ .

$9 x + \frac{- 9}{4} = 1 \frac{3}{5} (1 + 4 x)$

Step 1.1.5

Move the negative in front of the fraction.

$9 x - \frac{9}{4} = 1 \frac{3}{5} (1 + 4 x)$

Step 2

Simplify the right side.

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Step 2.1

Simplify $1 \frac{3}{5} (1 + 4 x)$ .

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Step 2.1.1

Convert $1 \frac{3}{5}$ to an improper fraction.

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Step 2.1.1.1

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

$9 x - \frac{9}{4} = (1 + \frac{3}{5}) (1 + 4 x)$

Step 2.1.1.2

Add $1$ and $\frac{3}{5}$ .

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Step 2.1.1.2.1

Write $1$ as a fraction with a common denominator.

$9 x - \frac{9}{4} = (\frac{5}{5} + \frac{3}{5}) (1 + 4 x)$

Step 2.1.1.2.2

Combine the numerators over the common denominator.

$9 x - \frac{9}{4} = \frac{5 + 3}{5} (1 + 4 x)$

Step 2.1.1.2.3

Add $5$ and $3$ .

$9 x - \frac{9}{4} = \frac{8}{5} (1 + 4 x)$

Step 2.1.2

Apply the distributive property.

$9 x - \frac{9}{4} = \frac{8}{5} \cdot 1 + \frac{8}{5} (4 x)$

Step 2.1.3

Multiply $\frac{8}{5}$ by $1$ .

$9 x - \frac{9}{4} = \frac{8}{5} + \frac{8}{5} (4 x)$

Step 2.1.4

Multiply $\frac{8}{5} (4 x)$ .

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Step 2.1.4.1

Combine $4$ and $\frac{8}{5}$ .

$9 x - \frac{9}{4} = \frac{8}{5} + \frac{4 \cdot 8}{5} x$

Step 2.1.4.2

Multiply $4$ by $8$ .

$9 x - \frac{9}{4} = \frac{8}{5} + \frac{32}{5} x$

Step 2.1.4.3

Combine $\frac{32}{5}$ and $x$ .

$9 x - \frac{9}{4} = \frac{8}{5} + \frac{32 x}{5}$

Step 3

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

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Step 3.1

Subtract $\frac{32 x}{5}$ from both sides of the equation.

$9 x - \frac{9}{4} - \frac{32 x}{5} = \frac{8}{5}$

Step 3.2

To write $9 x$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$9 x \cdot \frac{5}{5} - \frac{32 x}{5} - \frac{9}{4} = \frac{8}{5}$

Step 3.3

Combine $9 x$ and $\frac{5}{5}$ .

$\frac{9 x \cdot 5}{5} - \frac{32 x}{5} - \frac{9}{4} = \frac{8}{5}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{9 x \cdot 5 - 32 x}{5} - \frac{9}{4} = \frac{8}{5}$

Step 3.5

Simplify each term.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Factor $x$ out of $9 x \cdot 5 - 32 x$ .

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Step 3.5.1.1.1

Factor $x$ out of $9 x \cdot 5$ .

$\frac{x (9 \cdot 5) - 32 x}{5} - \frac{9}{4} = \frac{8}{5}$

Step 3.5.1.1.2

Factor $x$ out of $- 32 x$ .

$\frac{x (9 \cdot 5) + x \cdot - 32}{5} - \frac{9}{4} = \frac{8}{5}$

Step 3.5.1.1.3

Factor $x$ out of $x (9 \cdot 5) + x \cdot - 32$ .

$\frac{x (9 \cdot 5 - 32)}{5} - \frac{9}{4} = \frac{8}{5}$

Step 3.5.1.2

Multiply $9$ by $5$ .

$\frac{x (45 - 32)}{5} - \frac{9}{4} = \frac{8}{5}$

Step 3.5.1.3

Subtract $32$ from $45$ .

$\frac{x \cdot 13}{5} - \frac{9}{4} = \frac{8}{5}$

Step 3.5.2

Move $13$ to the left of $x$ .

$\frac{13 x}{5} - \frac{9}{4} = \frac{8}{5}$

Step 4

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

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Step 4.1

Add $\frac{9}{4}$ to both sides of the equation.

$\frac{13 x}{5} = \frac{8}{5} + \frac{9}{4}$

Step 4.2

To write $\frac{8}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{13 x}{5} = \frac{8}{5} \cdot \frac{4}{4} + \frac{9}{4}$

Step 4.3

To write $\frac{9}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{13 x}{5} = \frac{8}{5} \cdot \frac{4}{4} + \frac{9}{4} \cdot \frac{5}{5}$

Step 4.4

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.4.1

Multiply $\frac{8}{5}$ by $\frac{4}{4}$ .

$\frac{13 x}{5} = \frac{8 \cdot 4}{5 \cdot 4} + \frac{9}{4} \cdot \frac{5}{5}$

Step 4.4.2

Multiply $5$ by $4$ .

$\frac{13 x}{5} = \frac{8 \cdot 4}{20} + \frac{9}{4} \cdot \frac{5}{5}$

Step 4.4.3

Multiply $\frac{9}{4}$ by $\frac{5}{5}$ .

$\frac{13 x}{5} = \frac{8 \cdot 4}{20} + \frac{9 \cdot 5}{4 \cdot 5}$

Step 4.4.4

Multiply $4$ by $5$ .

$\frac{13 x}{5} = \frac{8 \cdot 4}{20} + \frac{9 \cdot 5}{20}$

Step 4.5

Combine the numerators over the common denominator.

$\frac{13 x}{5} = \frac{8 \cdot 4 + 9 \cdot 5}{20}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $8$ by $4$ .

$\frac{13 x}{5} = \frac{32 + 9 \cdot 5}{20}$

Step 4.6.2

Multiply $9$ by $5$ .

$\frac{13 x}{5} = \frac{32 + 45}{20}$

Step 4.6.3

Add $32$ and $45$ .

$\frac{13 x}{5} = \frac{77}{20}$

Step 5

Multiply both sides of the equation by $\frac{5}{13}$ .

$\frac{5}{13} \cdot \frac{13 x}{5} = \frac{5}{13} \cdot \frac{77}{20}$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $\frac{5}{13} \cdot \frac{13 x}{5}$ .

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Step 6.1.1.1

Cancel the common factor of $5$ .

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Step 6.1.1.1.1

Cancel the common factor.

$\frac{5}{13} \cdot \frac{13 x}{5} = \frac{5}{13} \cdot \frac{77}{20}$

Step 6.1.1.1.2

Rewrite the expression.

$\frac{1}{13} (13 x) = \frac{5}{13} \cdot \frac{77}{20}$

Step 6.1.1.2

Cancel the common factor of $13$ .

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Step 6.1.1.2.1

Factor $13$ out of $13 x$ .

$\frac{1}{13} (13 (x)) = \frac{5}{13} \cdot \frac{77}{20}$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{1}{13} (13 x) = \frac{5}{13} \cdot \frac{77}{20}$

Step 6.1.1.2.3

Rewrite the expression.

$x = \frac{5}{13} \cdot \frac{77}{20}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{5}{13} \cdot \frac{77}{20}$ .

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Step 6.2.1.1

Cancel the common factor of $5$ .

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Step 6.2.1.1.1

Factor $5$ out of $20$ .

$x = \frac{5}{13} \cdot \frac{77}{5 (4)}$

Step 6.2.1.1.2

Cancel the common factor.

$x = \frac{5}{13} \cdot \frac{77}{5 \cdot 4}$

Step 6.2.1.1.3

Rewrite the expression.

$x = \frac{1}{13} \cdot \frac{77}{4}$

Step 6.2.1.2

Multiply $\frac{1}{13}$ by $\frac{77}{4}$ .

$x = \frac{77}{13 \cdot 4}$

Step 6.2.1.3

Multiply $13$ by $4$ .

$x = \frac{77}{52}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{77}{52}$

Decimal Form:

$x = 1.48076923 \dots$

Mixed Number Form:

$x = 1 \frac{25}{52}$