Question

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

$23z+19=3\left(5z-9\right)+8z+46$

Short answer

The equation is an identity and the solution of this equation is "all real numbers".

Step-by-step solution

Step 1. Given information.

The given equation is$23z+19=3\left(5z-9\right)+8z+46$.

Step 2. Apply the distributive property.

Simplify the equation.

$$\begin{gathered} 23z+19=3\left(5z-9\right)+8z+46 \\ 23z+19=3\times 5z-3\times 9+8z+46 \\ 23z+19=15z-27+8z+46 \\ 23z+19=23z+19 \end{gathered}$$

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

Subtract $23z$from each side.

$$\begin{gathered} 23z+19-23z=23z+19-23z \\ 19=19 \end{gathered}$$

This is a true statement.

It means the equation is true for any value of the variable.

So, the equation is an identity.

Hence, the solution of this equation is "all real numbers".