Question

REASONING Solve the equation below. Describe each step.

$t=2-2\left[2t-3\left(1-t\right)\right]$

Short answer

The solution of the given equation is $t=\frac{8}{11}$.

Step-by-step solution

Step 1. Write the given equation.

The given equation is “$t=2-2\left[2t-3\left(1-t\right)\right]$”.

Step 2. Solve the given equation.

Solving,

$t=2-2\left[2t-3\left(1-t\right)\right]$ (Given equation)

$t=2-2\left[2t-3+3t\right]$ (Distributive property)

$t=2-2\left[5t-3\right]$ (Simplify)

$t=2-10t+6$ (Distributive property)

$t=-10t+8$ (Simplify)

$t+10t=-10t+8+10t$ (Add $10t$to both sides)

$11t=8$ (Simplify)

$\frac{11t}{11}=\frac{8}{11}$ (Divide both sides by 11)

$t=\frac{8}{11}$ (Simplify)

So, $t=\frac{8}{11}$is the solution of the given equation.

Step 3. Check the obtained solution.

Put $t=\frac{8}{11}$in the given equation $t=2-2\left[2t-3\left(1-t\right)\right]$to get,

$\begin{array}{l}\frac{8}{11}=2-2\left[2\left(\frac{8}{11}\right)-3\left(1-\frac{8}{11}\right)\right]\\ \frac{8}{11}=2-2\left[\frac{16}{11}-3\left(\frac{3}{11}\right)\right]\\ \frac{8}{11}=2-2\left[\frac{16}{11}-\frac{9}{11}\right]\\ \frac{8}{11}=2-2\left[\frac{7}{11}\right]\end{array}$

On further simplification,

$\begin{array}{l}\frac{8}{11}=2-\frac{14}{11}\\ \frac{8}{11}=\frac{22-14}{11}\\ \frac{8}{11}=\frac{8}{11}\end{array}$

Since the value obtained on the left-hand side and right-hand side are equal, the obtained value is indeed a solution of the given equation.