Question

Justify each step.

$\frac{2}{3}b=8-2b$

$2b=3\left(8-2b\right)$

$2b=24-6b$

$8b=24$

$b=3$

Short answer

$\frac{2}{3}b=8-2b$Given

$2b=3\left(8-2b\right)$Multiplication property of equality

$2b=24-6b$Distributive property

$8b=24$Addition property of equality.

$b=3$ Division property of equality

Step-by-step solution

Step 1. Apply the multiplication property of equality.

If $p=q$then, $rp=rq$.

Step 2. Description of step.

Multiply each side of the $\frac{2}{3}b=8-2b$by 3 and simplify.

$\begin{array}{c}3\times \frac{2}{3}b=3\left(8-2b\right)\\ 2b=3\left(8-2b\right)\end{array}$

Step 3. Apply distributive property.

Expand the right-hand side of the equation obtained in step 2 using the distributive property.

$\begin{array}{c}2b=3\times 8-3\times \left(2b\right)\\ =24-6b\end{array}$

Step 4. Apply the addition property of equality.

If $p=q$and $r=s$, then $p+r=q+s$.

Step 5. Description of step.

Add $6b$on each side of $2b=24-6b$and simplify.

$2b+6b=24-6b+6b$

$8b=24$

Step 6. Apply division property of equality.

If $p=q$and $r\ne 0$, then $\frac{p}{r}=\frac{q}{r}$.

Step 7. Description of step.

Divide each side of the equation obtained in steps 5 by 8 to find the value of b.

$\begin{array}{c}\frac{8b}{8}=\frac{24}{8}\\ b=3\end{array}$

Therefore, the value of b is 3.