Basic Math Examples

$\frac{t - 5}{5} + 1 = \frac{2 t - 3}{2} - \frac{6 t + 5}{10}$

Step 1

Since $t$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{2 t - 3}{2} - \frac{6 t + 5}{10} = \frac{t - 5}{5} + 1$

Step 2

Simplify $\frac{2 t - 3}{2} - \frac{6 t + 5}{10}$ .

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Step 2.1

To write $\frac{2 t - 3}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{2 t - 3}{2} \cdot \frac{5}{5} - \frac{6 t + 5}{10} = \frac{t - 5}{5} + 1$

Step 2.2

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1

Multiply $\frac{2 t - 3}{2}$ by $\frac{5}{5}$ .

$\frac{(2 t - 3) \cdot 5}{2 \cdot 5} - \frac{6 t + 5}{10} = \frac{t - 5}{5} + 1$

Step 2.2.2

Multiply $2$ by $5$ .

$\frac{(2 t - 3) \cdot 5}{10} - \frac{6 t + 5}{10} = \frac{t - 5}{5} + 1$

Step 2.3

Combine the numerators over the common denominator.

$\frac{(2 t - 3) \cdot 5 - (6 t + 5)}{10} = \frac{t - 5}{5} + 1$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Apply the distributive property.

$\frac{2 t \cdot 5 - 3 \cdot 5 - (6 t + 5)}{10} = \frac{t - 5}{5} + 1$

Step 2.4.2

Multiply $5$ by $2$ .

$\frac{10 t - 3 \cdot 5 - (6 t + 5)}{10} = \frac{t - 5}{5} + 1$

Step 2.4.3

Multiply $- 3$ by $5$ .

$\frac{10 t - 15 - (6 t + 5)}{10} = \frac{t - 5}{5} + 1$

Step 2.4.4

Apply the distributive property.

$\frac{10 t - 15 - (6 t) - 1 \cdot 5}{10} = \frac{t - 5}{5} + 1$

Step 2.4.5

Multiply $6$ by $- 1$ .

$\frac{10 t - 15 - 6 t - 1 \cdot 5}{10} = \frac{t - 5}{5} + 1$

Step 2.4.6

Multiply $- 1$ by $5$ .

$\frac{10 t - 15 - 6 t - 5}{10} = \frac{t - 5}{5} + 1$

Step 2.4.7

Subtract $6 t$ from $10 t$ .

$\frac{4 t - 15 - 5}{10} = \frac{t - 5}{5} + 1$

Step 2.4.8

Subtract $5$ from $- 15$ .

$\frac{4 t - 20}{10} = \frac{t - 5}{5} + 1$

Step 2.4.9

Factor $4$ out of $4 t - 20$ .

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Step 2.4.9.1

Factor $4$ out of $4 t$ .

$\frac{4 (t) - 20}{10} = \frac{t - 5}{5} + 1$

Step 2.4.9.2

Factor $4$ out of $- 20$ .

$\frac{4 t + 4 \cdot - 5}{10} = \frac{t - 5}{5} + 1$

Step 2.4.9.3

Factor $4$ out of $4 t + 4 \cdot - 5$ .

$\frac{4 (t - 5)}{10} = \frac{t - 5}{5} + 1$

Step 2.5

Cancel the common factor of $4$ and $10$ .

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Step 2.5.1

Factor $2$ out of $4 (t - 5)$ .

$\frac{2 (2 (t - 5))}{10} = \frac{t - 5}{5} + 1$

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

Factor $2$ out of $10$ .

$\frac{2 (2 (t - 5))}{2 (5)} = \frac{t - 5}{5} + 1$

Step 2.5.2.2

Cancel the common factor.

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

Step 2.5.2.3

Rewrite the expression.

$\frac{2 (t - 5)}{5} = \frac{t - 5}{5} + 1$

Step 3

Simplify $\frac{t - 5}{5} + 1$ .

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Step 3.1

Combine into one fraction.

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Step 3.1.1

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

$\frac{2 (t - 5)}{5} = \frac{t - 5}{5} + \frac{5}{5}$

Step 3.1.2

Combine the numerators over the common denominator.

$\frac{2 (t - 5)}{5} = \frac{t - 5 + 5}{5}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Add $- 5$ and $5$ .

$\frac{2 (t - 5)}{5} = \frac{t + 0}{5}$

Step 3.2.2

Add $t$ and $0$ .

$\frac{2 (t - 5)}{5} = \frac{t}{5}$

Step 4

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

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Step 4.1

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

$\frac{2 (t - 5)}{5} - \frac{t}{5} = 0$

Step 4.2

Combine the numerators over the common denominator.

$\frac{2 (t - 5) - t}{5} = 0$

Step 4.3

Simplify each term.

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Step 4.3.1

Apply the distributive property.

$\frac{2 t + 2 \cdot - 5 - t}{5} = 0$

Step 4.3.2

Multiply $2$ by $- 5$ .

$\frac{2 t - 10 - t}{5} = 0$

Step 4.4

Subtract $t$ from $2 t$ .

$\frac{t - 10}{5} = 0$

Step 5

Set the numerator equal to zero.

$t - 10 = 0$

Step 6

Add $10$ to both sides of the equation.

$t = 10$