Algebra Examples

$\frac{2 x + 9}{7} + \frac{5 - 2 x}{5} = 2$

Step 1

Simplify $\frac{2 x + 9}{7} + \frac{5 - 2 x}{5}$ .

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Step 1.1

To write $\frac{2 x + 9}{7}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{2 x + 9}{7} \cdot \frac{5}{5} + \frac{5 - 2 x}{5} = 2$

Step 1.2

To write $\frac{5 - 2 x}{5}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{2 x + 9}{7} \cdot \frac{5}{5} + \frac{5 - 2 x}{5} \cdot \frac{7}{7} = 2$

Step 1.3

Write each expression with a common denominator of $35$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{2 x + 9}{7}$ by $\frac{5}{5}$ .

$\frac{(2 x + 9) \cdot 5}{7 \cdot 5} + \frac{5 - 2 x}{5} \cdot \frac{7}{7} = 2$

Step 1.3.2

Multiply $7$ by $5$ .

$\frac{(2 x + 9) \cdot 5}{35} + \frac{5 - 2 x}{5} \cdot \frac{7}{7} = 2$

Step 1.3.3

Multiply $\frac{5 - 2 x}{5}$ by $\frac{7}{7}$ .

$\frac{(2 x + 9) \cdot 5}{35} + \frac{(5 - 2 x) \cdot 7}{5 \cdot 7} = 2$

Step 1.3.4

Multiply $5$ by $7$ .

$\frac{(2 x + 9) \cdot 5}{35} + \frac{(5 - 2 x) \cdot 7}{35} = 2$

Step 1.4

Combine the numerators over the common denominator.

$\frac{(2 x + 9) \cdot 5 + (5 - 2 x) \cdot 7}{35} = 2$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Apply the distributive property.

$\frac{2 x \cdot 5 + 9 \cdot 5 + (5 - 2 x) \cdot 7}{35} = 2$

Step 1.5.2

Multiply $5$ by $2$ .

$\frac{10 x + 9 \cdot 5 + (5 - 2 x) \cdot 7}{35} = 2$

Step 1.5.3

Multiply $9$ by $5$ .

$\frac{10 x + 45 + (5 - 2 x) \cdot 7}{35} = 2$

Step 1.5.4

Apply the distributive property.

$\frac{10 x + 45 + 5 \cdot 7 - 2 x \cdot 7}{35} = 2$

Step 1.5.5

Multiply $5$ by $7$ .

$\frac{10 x + 45 + 35 - 2 x \cdot 7}{35} = 2$

Step 1.5.6

Multiply $7$ by $- 2$ .

$\frac{10 x + 45 + 35 - 14 x}{35} = 2$

Step 1.5.7

Subtract $14 x$ from $10 x$ .

$\frac{- 4 x + 45 + 35}{35} = 2$

Step 1.5.8

Add $45$ and $35$ .

$\frac{- 4 x + 80}{35} = 2$

Step 1.5.9

Factor $4$ out of $- 4 x + 80$ .

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Step 1.5.9.1

Factor $4$ out of $- 4 x$ .

$\frac{4 (- x) + 80}{35} = 2$

Step 1.5.9.2

Factor $4$ out of $80$ .

$\frac{4 (- x) + 4 (20)}{35} = 2$

Step 1.5.9.3

Factor $4$ out of $4 (- x) + 4 (20)$ .

$\frac{4 (- x + 20)}{35} = 2$

Step 1.6

Simplify with factoring out.

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Step 1.6.1

Factor $- 1$ out of $- x$ .

$\frac{4 (- (x) + 20)}{35} = 2$

Step 1.6.2

Rewrite $20$ as $- 1 (- 20)$ .

$\frac{4 (- (x) - 1 (- 20))}{35} = 2$

Step 1.6.3

Factor $- 1$ out of $- (x) - 1 (- 20)$ .

$\frac{4 (- (x - 20))}{35} = 2$

Step 1.6.4

Simplify the expression.

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Step 1.6.4.1

Rewrite $- (x - 20)$ as $- 1 (x - 20)$ .

$\frac{4 (- 1 (x - 20))}{35} = 2$

Step 1.6.4.2

Move the negative in front of the fraction.

$- \frac{4 (x - 20)}{35} = 2$

Step 2

Multiply both sides of the equation by $- \frac{35}{4}$ .

$- \frac{35}{4} (- \frac{4 (x - 20)}{35}) = - \frac{35}{4} \cdot 2$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $- \frac{35}{4} (- \frac{4 (x - 20)}{35})$ .

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Step 3.1.1.1

Cancel the common factor of $35$ .

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Step 3.1.1.1.1

Move the leading negative in $- \frac{35}{4}$ into the numerator.

$\frac{- 35}{4} (- \frac{4 (x - 20)}{35}) = - \frac{35}{4} \cdot 2$

Step 3.1.1.1.2

Move the leading negative in $- \frac{4 (x - 20)}{35}$ into the numerator.

$\frac{- 35}{4} \cdot \frac{- 4 (x - 20)}{35} = - \frac{35}{4} \cdot 2$

Step 3.1.1.1.3

Factor $35$ out of $- 35$ .

$\frac{35 (- 1)}{4} \cdot \frac{- 4 (x - 20)}{35} = - \frac{35}{4} \cdot 2$

Step 3.1.1.1.4

Cancel the common factor.

$\frac{35 \cdot - 1}{4} \cdot \frac{- 4 (x - 20)}{35} = - \frac{35}{4} \cdot 2$

Step 3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{4} (- 4 (x - 20)) = - \frac{35}{4} \cdot 2$

Step 3.1.1.2

Cancel the common factor of $4$ .

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Step 3.1.1.2.1

Factor $4$ out of $- 4 (x - 20)$ .

$\frac{- 1}{4} (4 (- (x - 20))) = - \frac{35}{4} \cdot 2$

Step 3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{4} (4 (- (x - 20))) = - \frac{35}{4} \cdot 2$

Step 3.1.1.2.3

Rewrite the expression.

$- - (x - 20) = - \frac{35}{4} \cdot 2$

Step 3.1.1.3

Multiply.

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Step 3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 (x - 20) = - \frac{35}{4} \cdot 2$

Step 3.1.1.3.2

Multiply $x - 20$ by $1$ .

$x - 20 = - \frac{35}{4} \cdot 2$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $- \frac{35}{4} \cdot 2$ .

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Step 3.2.1.1

Cancel the common factor of $2$ .

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Step 3.2.1.1.1

Move the leading negative in $- \frac{35}{4}$ into the numerator.

$x - 20 = \frac{- 35}{4} \cdot 2$

Step 3.2.1.1.2

Factor $2$ out of $4$ .

$x - 20 = \frac{- 35}{2 (2)} \cdot 2$

Step 3.2.1.1.3

Cancel the common factor.

$x - 20 = \frac{- 35}{2 \cdot 2} \cdot 2$

Step 3.2.1.1.4

Rewrite the expression.

$x - 20 = \frac{- 35}{2}$

Step 3.2.1.2

Move the negative in front of the fraction.

$x - 20 = - \frac{35}{2}$

Step 4

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

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Step 4.1

Add $20$ to both sides of the equation.

$x = - \frac{35}{2} + 20$

Step 4.2

To write $20$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = - \frac{35}{2} + 20 \cdot \frac{2}{2}$

Step 4.3

Combine $20$ and $\frac{2}{2}$ .

$x = - \frac{35}{2} + \frac{20 \cdot 2}{2}$

Step 4.4

Combine the numerators over the common denominator.

$x = \frac{- 35 + 20 \cdot 2}{2}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $20$ by $2$ .

$x = \frac{- 35 + 40}{2}$

Step 4.5.2

Add $- 35$ and $40$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 2.5$

Mixed Number Form:

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