Question

How many solutions can be found for the linear equation?

3(x/2+ 5) - 6 =( 9x+18)/3

Responses
A no solutionsno solutions
B one solutionone solution
C two solutionstwo solutions
D infinitely many solutions

Answer 1

To determine how many solutions there are for the given linear equation, let's start by simplifying and solving the equation step by step.

The equation given is:

\[
3\left(\frac{x}{2} + 5\right) - 6 = \frac{9x + 18}{3}
\]

First, we simplify each side.

**Step 1: Simplifying the left side**

Distribute the 3 on the left:

\[
3 \cdot \frac{x}{2} + 3 \cdot 5 - 6 = \frac{3x}{2} + 15 - 6 = \frac{3x}{2} + 9
\]

So the left side simplifies to:

\[
\frac{3x}{2} + 9
\]

**Step 2: Simplifying the right side**

Now simplify the right side:

\[
\frac{9x + 18}{3} = \frac{9x}{3} + \frac{18}{3} = 3x + 6
\]

**Step 3: Setting the simplified expressions equal**

Now we have the equation:

\[
\frac{3x}{2} + 9 = 3x + 6
\]

**Step 4: Eliminating the fraction**

To eliminate the fraction, multiply the entire equation by 2:

\[
2\left(\frac{3x}{2} + 9\right) = 2(3x + 6)
\]

This simplifies to:

\[
3x + 18 = 6x + 12
\]

**Step 5: Rearranging to isolate x**

Next, rearranging the terms gives:

\[
18 - 12 = 6x - 3x
\]

Which simplifies to:

\[
6 = 3x
\]

**Step 6: Solving for x**

Divide both sides by 3:

\[
x = 2
\]

**Conclusion**

Since we found one value for \(x\), the equation has exactly one solution.

Thus, the correct response is:

**B: one solution**.