Algebra Examples

$2 x - 1 + \frac{x + 5}{2} = \frac{3 x - 7}{10}$

Step 1

Multiply both sides by $10$ .

$(2 x - 1 + \frac{x + 5}{2}) \cdot 10 = \frac{3 x - 7}{10} \cdot 10$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(2 x - 1 + \frac{x + 5}{2}) \cdot 10$ .

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Step 2.1.1.1

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

$(2 x \cdot \frac{2}{2} + \frac{x + 5}{2} - 1) \cdot 10 = \frac{3 x - 7}{10} \cdot 10$

Step 2.1.1.2

Simplify terms.

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Step 2.1.1.2.1

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

$(\frac{2 x \cdot 2}{2} + \frac{x + 5}{2} - 1) \cdot 10 = \frac{3 x - 7}{10} \cdot 10$

Step 2.1.1.2.2

Combine the numerators over the common denominator.

$(\frac{2 x \cdot 2 + x + 5}{2} - 1) \cdot 10 = \frac{3 x - 7}{10} \cdot 10$

Step 2.1.1.3

Simplify the numerator.

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Step 2.1.1.3.1

Multiply $2$ by $2$ .

$(\frac{4 x + x + 5}{2} - 1) \cdot 10 = \frac{3 x - 7}{10} \cdot 10$

Step 2.1.1.3.2

Add $4 x$ and $x$ .

$(\frac{5 x + 5}{2} - 1) \cdot 10 = \frac{3 x - 7}{10} \cdot 10$

Step 2.1.1.3.3

Factor $5$ out of $5 x + 5$ .

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Step 2.1.1.3.3.1

Factor $5$ out of $5 x$ .

$(\frac{5 (x) + 5}{2} - 1) \cdot 10 = \frac{3 x - 7}{10} \cdot 10$

Step 2.1.1.3.3.2

Factor $5$ out of $5$ .

$(\frac{5 (x) + 5 (1)}{2} - 1) \cdot 10 = \frac{3 x - 7}{10} \cdot 10$

Step 2.1.1.3.3.3

Factor $5$ out of $5 (x) + 5 (1)$ .

$(\frac{5 (x + 1)}{2} - 1) \cdot 10 = \frac{3 x - 7}{10} \cdot 10$

Step 2.1.1.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$(\frac{5 (x + 1)}{2} - 1 \cdot \frac{2}{2}) \cdot 10 = \frac{3 x - 7}{10} \cdot 10$

Step 2.1.1.5

Simplify terms.

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Step 2.1.1.5.1

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

$(\frac{5 (x + 1)}{2} + \frac{- 1 \cdot 2}{2}) \cdot 10 = \frac{3 x - 7}{10} \cdot 10$

Step 2.1.1.5.2

Combine the numerators over the common denominator.

$\frac{5 (x + 1) - 1 \cdot 2}{2} \cdot 10 = \frac{3 x - 7}{10} \cdot 10$

Step 2.1.1.6

Simplify the numerator.

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Step 2.1.1.6.1

Apply the distributive property.

$\frac{5 x + 5 \cdot 1 - 1 \cdot 2}{2} \cdot 10 = \frac{3 x - 7}{10} \cdot 10$

Step 2.1.1.6.2

Multiply $5$ by $1$ .

$\frac{5 x + 5 - 1 \cdot 2}{2} \cdot 10 = \frac{3 x - 7}{10} \cdot 10$

Step 2.1.1.6.3

Multiply $- 1$ by $2$ .

$\frac{5 x + 5 - 2}{2} \cdot 10 = \frac{3 x - 7}{10} \cdot 10$

Step 2.1.1.6.4

Subtract $2$ from $5$ .

$\frac{5 x + 3}{2} \cdot 10 = \frac{3 x - 7}{10} \cdot 10$

Step 2.1.1.7

Simplify terms.

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Step 2.1.1.7.1

Cancel the common factor of $2$ .

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Step 2.1.1.7.1.1

Factor $2$ out of $10$ .

$\frac{5 x + 3}{2} \cdot (2 (5)) = \frac{3 x - 7}{10} \cdot 10$

Step 2.1.1.7.1.2

Cancel the common factor.

$\frac{5 x + 3}{2} \cdot (2 \cdot 5) = \frac{3 x - 7}{10} \cdot 10$

Step 2.1.1.7.1.3

Rewrite the expression.

$(5 x + 3) \cdot 5 = \frac{3 x - 7}{10} \cdot 10$

Step 2.1.1.7.2

Apply the distributive property.

$5 x \cdot 5 + 3 \cdot 5 = \frac{3 x - 7}{10} \cdot 10$

Step 2.1.1.7.3

Multiply.

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Step 2.1.1.7.3.1

Multiply $5$ by $5$ .

$25 x + 3 \cdot 5 = \frac{3 x - 7}{10} \cdot 10$

Step 2.1.1.7.3.2

Multiply $3$ by $5$ .

$25 x + 15 = \frac{3 x - 7}{10} \cdot 10$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $10$ .

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Step 2.2.1.1

Cancel the common factor.

$25 x + 15 = \frac{3 x - 7}{10} \cdot 10$

Step 2.2.1.2

Rewrite the expression.

$25 x + 15 = 3 x - 7$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $3 x$ from both sides of the equation.

$25 x + 15 - 3 x = - 7$

Step 3.1.2

Subtract $3 x$ from $25 x$ .

$22 x + 15 = - 7$

Step 3.2

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

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Step 3.2.1

Subtract $15$ from both sides of the equation.

$22 x = - 7 - 15$

Step 3.2.2

Subtract $15$ from $- 7$ .

$22 x = - 22$

Step 3.3

Divide each term in $22 x = - 22$ by $22$ and simplify.

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Step 3.3.1

Divide each term in $22 x = - 22$ by $22$ .

$\frac{22 x}{22} = \frac{- 22}{22}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $22$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{22 x}{22} = \frac{- 22}{22}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 22}{22}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $- 22$ by $22$ .

$x = - 1$