Question

Classify the below equation as a conditional equation, an identity, or a contradiction, and then state the solution.

$30\left(2n-1\right)=5\left(10n+8\right)$

Short answer

The equation is a conditional equation and the solution of this equation is$n=7$.

Step-by-step solution

Step 1. Given information.

The given equation is$30\left(2n-1\right)=5\left(10n+8\right)$.

Step 2. Apply the distributive property to simplify the equation.

$$\begin{gathered} 30\times 2n-30=5\times 10n+5\times 8 \\ 60n-30=50n+40 \end{gathered}$$

Step 3. Collect all variables terms on one side of the equation.

Subtract $50n$from each side.

$$\begin{gathered} 60n-30-50n=50n+40-50n \\ 10n-30=40 \end{gathered}$$

Step 4. Collect all constant terms on one side of the equation and then simplify.

Add 30 to each side.

$$\begin{gathered} 10n-30+30=40+30 \\ 10n=70 \end{gathered}$$

Divide each side by $10$.

$$\begin{gathered} \frac{10n}{10}=\frac{70}{10} \\ n=7 \end{gathered}$$

From the above result, we can say that the given equation is true only for $n=7$.

So, the equation is a conditional equation.

Hence, the solution of the given equation is$n=7$.