Solve for t 1/2+2/5t-1=1/5t+t

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

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.

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

Step 2

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

Combine $\frac{1}{5}$ and $t$ .

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

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

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

Simplify terms.

Combine $t$ and $\frac{5}{5}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Move $5$ to the left of $t$ .

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

Add $t$ and $5 t$ .

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

Step 3

Simplify $\frac{1}{2} + \frac{2}{5} t - 1$ .

Combine $\frac{2}{5}$ and $t$ .

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Move the negative in front of the fraction.

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

Step 4

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

Subtract $\frac{2 t}{5}$ from both sides of the equation.

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

Combine the numerators over the common denominator.

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

Subtract $2 t$ from $6 t$ .

$\frac{4 t}{5} = - \frac{1}{2}$

Step 5

Multiply both sides of the equation by $\frac{5}{4}$ .

$\frac{5}{4} \cdot \frac{4 t}{5} = \frac{5}{4} (- \frac{1}{2})$

Step 6

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{5}{4} \cdot \frac{4 t}{5}$ .

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5}{4} \cdot \frac{4 t}{5} = \frac{5}{4} (- \frac{1}{2})$

Rewrite the expression.

$\frac{1}{4} (4 t) = \frac{5}{4} (- \frac{1}{2})$

Cancel the common factor of $4$ .

Factor $4$ out of $4 t$ .

$\frac{1}{4} (4 (t)) = \frac{5}{4} (- \frac{1}{2})$

Cancel the common factor.

$\frac{1}{4} (4 t) = \frac{5}{4} (- \frac{1}{2})$

Rewrite the expression.

$t = \frac{5}{4} (- \frac{1}{2})$

Simplify the right side.

Multiply $\frac{5}{4} (- \frac{1}{2})$ .

Multiply $\frac{5}{4}$ by $\frac{1}{2}$ .

$t = - \frac{5}{4 \cdot 2}$

Multiply $4$ by $2$ .

$t = - \frac{5}{8}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$t = - \frac{5}{8}$

Decimal Form:

$t = - 0.625$

Solve for t 1/2+2/5t-1=1/5t+t

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