Question

what makes the following have exactly 1 solution:

−3(8x+10)=−14x+2

x=

Answer 1

To determine whether the equation \(-3(8x + 10) = -14x + 2\) has exactly one solution, let's solve it step-by-step.

1. **Distribute on the left side:**
\[
-3(8x + 10) = -24x - 30
\]
So the equation becomes:
\[
-24x - 30 = -14x + 2
\]

2. **Rearrange the equation to isolate the variable \(x\):**
First, add \(24x\) to both sides:
\[
-30 = 10x + 2
\]
Now, subtract \(2\) from both sides:
\[
-30 - 2 = 10x
\]
That simplifies to:
\[
-32 = 10x
\]

3. **Solve for \(x\):**
Divide both sides by \(10\):
\[
x = -\frac{32}{10} = -\frac{16}{5}
\]

This process shows that the equation has exactly one solution, namely:
\[
x = -\frac{16}{5}
\]

### Summary of Conditions for Exactly One Solution:
An equation will have exactly one solution when it can be simplified to a form where the variable appears with a non-zero coefficient, ensuring that there's a unique \(x\) that satisfies the equation. In this case, we found one unique value of \(x\).

Final answer:
\[
x = -\frac{16}{5}
\]