Question

Solve each equation. Check your solution.

$2\left(4r+6\right)=\frac{2}{3}\left(12r+18\right)$

Short answer

The given equation is an identity. Any value of r is a solution.

Step-by-step solution

Step 1. Write the given equation.

The given equation is “$2\left(4r+6\right)=\frac{2}{3}\left(12r+18\right)$”.

Step 2. Solve the given equation.

Solving,

$2\left(4r+6\right)=\frac{2}{3}\left(12r+18\right)$ (Given equation)

$8r+12=\frac{2}{3}\times 12r+\frac{2}{3}\times 18$ (Distributive property)

$8r+12=8r+12$ (Simplify)

The expressions on both sides are identical.

This implies that the given equation is an identity. So, it is true for all values of r.

Step 3. Check the obtained solution.

Put $r=1$in the given equation $2\left(4r+6\right)=\frac{2}{3}\left(12r+18\right)$to get,

$\begin{array}{l}2\left(4+6\right)=\frac{2}{3}\left(12+18\right)\\ 2\left(10\right)=\frac{2}{3}\left(30\right)\\ 20=2\left(10\right)\\ 20=20\end{array}$

Since the values on the left-hand side and the right-hand side are equal, it can be said that $r=1$is a solution of the given equation.

Put $r=0$in the given equation $2\left(4r+6\right)=\frac{2}{3}\left(12r+18\right)$to get,

$\begin{array}{l}2\left(4\left(0\right)+6\right)=\frac{2}{3}\left(12\left(0\right)+18\right)\\ 2\left(0+6\right)=\frac{2}{3}\left(0+18\right)\\ 2\left(6\right)=\frac{2}{3}\left(18\right)\\ 12=2\left(6\right)\\ 12=12\end{array}$

Since the values on the left-hand side and the right-hand side are equal, it can be said that $r=0$is a solution of the given equation.

Thus, it can be seen that a linear equation has at least two different solutions. This verifies that the given equation is an identity.