To determine which equation has exactly one solution, let's solve each equation one at a time.
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Equation: \(3x + 8 = -3x - 8\)
Adding \(3x\) to both sides: \[ 6x + 8 = -8 \] Subtracting 8 from both sides: \[ 6x = -16 \] Dividing by 6: \[ x = -\frac{8}{3} \] This equation has exactly one solution: \(x = -\frac{8}{3}\). -
Equation: \(3x - 8 = 3x - 8\)
If we subtract \(3x\) from both sides: \[ -8 = -8 \] This is a true statement but does not isolate \(x\).
This equation has infinite solutions. -
Equation: \(3x - 8 = 3x + 8\)
Subtracting \(3x\) from both sides: \[ -8 = 8 \] This is a false statement and represents no solution.
This equation has no solutions. -
Equation: \(8x + 3 = -8x + 3\)
Adding \(8x\) to both sides: \[ 16x + 3 = 3 \] Subtracting 3 from both sides: \[ 16x = 0 \] Dividing by 16: \[ x = 0 \] This equation has exactly one solution: \(x = 0\).
In conclusion, the equations that have exactly one solution are:
- \(3x + 8 = -3x - 8\) with the solution \(x = -\frac{8}{3}\)
- \(8x + 3 = -8x + 3\) with the solution \(x = 0\)
Therefore, the equations having exactly one solution are:
- 3x + 8 = -3x - 8 and 8x + 3 = -8x + 3.