Answer :
To determine which of the given equations have exactly one solution, we need to solve each equation for [tex]\( x \)[/tex]. Let's analyze each one step-by-step.
### Equation A: [tex]\( 58x + 52 = -78x - 78 \)[/tex]
1. Combine like terms:
[tex]\[ 58x + 78x + 52 + 78 = 0 \][/tex]
Simplifies to:
[tex]\[ 136x + 130 = 0 \][/tex]
2. Isolate [tex]\( x \)[/tex]:
[tex]\[ 136x = -130 \][/tex]
[tex]\[ x = \frac{-130}{136} = -\frac{65}{68} = -\frac{65}{68} \Rightarrow \text{exactly one solution.} \][/tex]
### Equation B: [tex]\( 58x + 52 = 78x - 78 \)[/tex]
1. Combine like terms:
[tex]\[ 58x - 78x + 52 + 78 = 0 \][/tex]
Simplifies to:
[tex]\[ -20x + 130 = 0 \][/tex]
2. Isolate [tex]\( x \)[/tex]:
[tex]\[ -20x = -130 \][/tex]
[tex]\[ x = \frac{130}{20} = 6.5 \Rightarrow \text{exactly one solution.} \][/tex]
### Equation C: [tex]\( -78x + 52 = -52x - 78 \)[/tex]
1. Combine like terms:
[tex]\[ -78x + 52x + 52 + 78 = 0 \][/tex]
Simplifies to:
[tex]\[ -26x + 130 = 0 \][/tex]
2. Isolate [tex]\( x \)[/tex]:
[tex]\[ -26x = -130 \][/tex]
[tex]\[ x = \frac{130}{26} = 5 \Rightarrow \text{exactly one solution.} \][/tex]
### Equation D: [tex]\( 52x + 52 = 52x - 78 \)[/tex]
1. Combine like terms:
[tex]\[ 52x - 52x + 52 + 78 = 0 \][/tex]
Simplifies to:
[tex]\[ 130 = 0 \, (\text{contradiction!}) \][/tex]
This equation is contradictory and thus has no solution.
### Summary:
- Equation (A) [tex]\( 58x + 52 = -78x - 78 \)[/tex] has exactly one solution.
- Equation (B) [tex]\( 58x + 52 = 78x - 78 \)[/tex] has exactly one solution.
- Equation (C) [tex]\( -78x + 52 = -52x - 78 \)[/tex] has exactly one solution.
- Equation (D) [tex]\( 52x + 52 = 52x - 78 \)[/tex] has no solution.
Thus, the correct choices are:
- (A) [tex]\( 58 x + 52 = -78 x - 78 \)[/tex]
- (B) [tex]\( 58 x + 52 = 78 x - 78 \)[/tex]
- (C) [tex]\( -78 x + 52 = -52 x - 78 \)[/tex]
### Equation A: [tex]\( 58x + 52 = -78x - 78 \)[/tex]
1. Combine like terms:
[tex]\[ 58x + 78x + 52 + 78 = 0 \][/tex]
Simplifies to:
[tex]\[ 136x + 130 = 0 \][/tex]
2. Isolate [tex]\( x \)[/tex]:
[tex]\[ 136x = -130 \][/tex]
[tex]\[ x = \frac{-130}{136} = -\frac{65}{68} = -\frac{65}{68} \Rightarrow \text{exactly one solution.} \][/tex]
### Equation B: [tex]\( 58x + 52 = 78x - 78 \)[/tex]
1. Combine like terms:
[tex]\[ 58x - 78x + 52 + 78 = 0 \][/tex]
Simplifies to:
[tex]\[ -20x + 130 = 0 \][/tex]
2. Isolate [tex]\( x \)[/tex]:
[tex]\[ -20x = -130 \][/tex]
[tex]\[ x = \frac{130}{20} = 6.5 \Rightarrow \text{exactly one solution.} \][/tex]
### Equation C: [tex]\( -78x + 52 = -52x - 78 \)[/tex]
1. Combine like terms:
[tex]\[ -78x + 52x + 52 + 78 = 0 \][/tex]
Simplifies to:
[tex]\[ -26x + 130 = 0 \][/tex]
2. Isolate [tex]\( x \)[/tex]:
[tex]\[ -26x = -130 \][/tex]
[tex]\[ x = \frac{130}{26} = 5 \Rightarrow \text{exactly one solution.} \][/tex]
### Equation D: [tex]\( 52x + 52 = 52x - 78 \)[/tex]
1. Combine like terms:
[tex]\[ 52x - 52x + 52 + 78 = 0 \][/tex]
Simplifies to:
[tex]\[ 130 = 0 \, (\text{contradiction!}) \][/tex]
This equation is contradictory and thus has no solution.
### Summary:
- Equation (A) [tex]\( 58x + 52 = -78x - 78 \)[/tex] has exactly one solution.
- Equation (B) [tex]\( 58x + 52 = 78x - 78 \)[/tex] has exactly one solution.
- Equation (C) [tex]\( -78x + 52 = -52x - 78 \)[/tex] has exactly one solution.
- Equation (D) [tex]\( 52x + 52 = 52x - 78 \)[/tex] has no solution.
Thus, the correct choices are:
- (A) [tex]\( 58 x + 52 = -78 x - 78 \)[/tex]
- (B) [tex]\( 58 x + 52 = 78 x - 78 \)[/tex]
- (C) [tex]\( -78 x + 52 = -52 x - 78 \)[/tex]
