Question

Solve the equation. State whether the equation is a​ contradiction, an​ identity, or a conditional equation.
16 a minus 9 left parenthesis a plus 1 right parenthesis equals 5 left parenthesis a minus 2 right parenthesis16a−9(a+1)=5(a−2)

Answer 1

To solve the equation \( 16a - 9(a + 1) = 5(a - 2) \), we will first simplify both sides.

1. Expand both sides:

\[
16a - 9a - 9 = 5a - 10
\]

This simplifies to:

\[
7a - 9 = 5a - 10
\]

2. Next, we can isolate \( a \) by moving the terms with \( a \) to one side and the constant terms to the other side:

\[
7a - 5a = -10 + 9
\]

Simplifying further:

\[
2a = -1
\]

3. To solve for \( a \), divide both sides by \( 2 \):

\[
a = -\frac{1}{2}
\]

Now we have found a solution \( a = -\frac{1}{2} \).

### Determine the type of equation:
- A contradiction would have no solutions (e.g., something like \( 0 = 5 \)).
- An identity would be true for all values of the variable (e.g., something like \( 0 = 0 \)).
- A conditional equation has exactly one solution or is true for specific values of the variable.

Since we found a specific value for \( a \), which is \( a = -\frac{1}{2} \), the original equation is a **conditional equation**.