Question

Math 8B Unit 4 Study Guide
12. Which of the following equations has exactly one solution?
4x - 3 = 4x + 3
B. -4x + 3 = 4x + 7
C.
-4x + 3 = 4x + 7
-7x + 3 = -7x + 3

Answer 1

To determine which equation has exactly one solution, let's analyze each option:

**A. \(4x - 3 = 4x + 3\)**

Subtract \(4x\) from both sides:
\[
4x - 3 - 4x = 4x + 3 - 4x
\]
This simplifies to:
\[
-3 = 3
\]
This is a false statement, indicating that there are **no solutions**.

---

**B. \(-4x + 3 = 4x + 7\)**

Add \(4x\) to both sides:
\[
-4x + 4x + 3 = 4x + 4x + 7
\]
This simplifies to:
\[
3 = 8x + 7
\]
Now, subtract \(7\) from both sides:
\[
3 - 7 = 8x
\]
This simplifies to:
\[
-4 = 8x
\]
Now, divide both sides by \(8\):
\[
x = -\frac{1}{2}
\]
This equation has **exactly one solution**.

---

**C. \(-4x + 3 = 4x + 7\) and \(-7x + 3 = -7x + 3\)**

The first part is already analyzed above:
\[
-4x + 3 = 4x + 7 \quad \text{(equation B)}
\] which has exactly one solution \(x = -\frac{1}{2}\).

Now analyze the second part:
\[
-7x + 3 = -7x + 3
\]
This simplifies to:
\[
0 = 0
\]
This is a true statement and indicates that there are **infinitely many solutions**.

---

In conclusion, the equation that has exactly one solution is **B. \(-4x + 3 = 4x + 7\)**.