Question
Aanya wrote the four equations that are shown below.
A
6 x minus 8 = 4 (x minus 2) + 2 x
B
3 (x minus 1) + 2 x = 3 (x minus 1) + 2
C
7 x + 2 minus x = 6 (x + 2)
D
4 (x + 3) + x = 5 (x + 1) + 7
She examined them, without solving them, to determine which has exactly one solution. Which of Aanya’s equations has exactly one solution?
6 x minus 8 = 4 (x minus 2) + 2 x
3 (x minus 1) + 2 x = 3 (x minus 1) + 2
7 x + 2 minus x = 6 (x + 2)
4 (x + 3) + x = 5 (x + 1) + 7
A
6 x minus 8 = 4 (x minus 2) + 2 x
B
3 (x minus 1) + 2 x = 3 (x minus 1) + 2
C
7 x + 2 minus x = 6 (x + 2)
D
4 (x + 3) + x = 5 (x + 1) + 7
She examined them, without solving them, to determine which has exactly one solution. Which of Aanya’s equations has exactly one solution?
6 x minus 8 = 4 (x minus 2) + 2 x
3 (x minus 1) + 2 x = 3 (x minus 1) + 2
7 x + 2 minus x = 6 (x + 2)
4 (x + 3) + x = 5 (x + 1) + 7
Answers
GPT-4o mini
To determine which of Aanya’s equations has exactly one solution, we need to analyze each equation for the possibility of having one, no, or infinitely many solutions.
### Equation A:
\[ 6x - 8 = 4(x - 2) + 2x \]
Simplifying the right side:
\[ 6x - 8 = 4x - 8 + 2x \]
Combining like terms on the right:
\[ 6x - 8 = 6x - 8 \]
This simplifies to a true statement, which indicates that there are **infinitely many solutions**.
### Equation B:
\[ 3(x - 1) + 2x = 3(x - 1) + 2 \]
Simplifying both sides:
\[ 3x - 3 + 2x = 3x - 3 + 2 \]
Combining like terms:
\[ 5x - 3 = 3x - 3 + 2 \]
Which simplifies to:
\[ 5x - 3 = 3x - 1 \]
Subtracting \(3x\) from both sides:
\[ 2x - 3 = -1 \]
Adding 3 to both sides:
\[ 2x = 2 \]
Dividing by 2:
\[ x = 1 \]
This equation has **exactly one solution**.
### Equation C:
\[ 7x + 2 - x = 6(x + 2) \]
Simplifying the left side:
\[ 6x + 2 = 6x + 12 \]
Subtracting \(6x\) from both sides:
\[ 2 = 12 \]
This is a false statement, indicating that there is **no solution**.
### Equation D:
\[ 4(x + 3) + x = 5(x + 1) + 7 \]
Expanding both sides:
\[ 4x + 12 + x = 5x + 5 + 7 \]
Combining like terms:
\[ 5x + 12 = 5x + 12 \]
This also simplifies to a true statement, indicating there are **infinitely many solutions**.
### Conclusion:
The only equation that has **exactly one solution** is Equation B:
\[ 3(x - 1) + 2x = 3(x - 1) + 2 \]
### Equation A:
\[ 6x - 8 = 4(x - 2) + 2x \]
Simplifying the right side:
\[ 6x - 8 = 4x - 8 + 2x \]
Combining like terms on the right:
\[ 6x - 8 = 6x - 8 \]
This simplifies to a true statement, which indicates that there are **infinitely many solutions**.
### Equation B:
\[ 3(x - 1) + 2x = 3(x - 1) + 2 \]
Simplifying both sides:
\[ 3x - 3 + 2x = 3x - 3 + 2 \]
Combining like terms:
\[ 5x - 3 = 3x - 3 + 2 \]
Which simplifies to:
\[ 5x - 3 = 3x - 1 \]
Subtracting \(3x\) from both sides:
\[ 2x - 3 = -1 \]
Adding 3 to both sides:
\[ 2x = 2 \]
Dividing by 2:
\[ x = 1 \]
This equation has **exactly one solution**.
### Equation C:
\[ 7x + 2 - x = 6(x + 2) \]
Simplifying the left side:
\[ 6x + 2 = 6x + 12 \]
Subtracting \(6x\) from both sides:
\[ 2 = 12 \]
This is a false statement, indicating that there is **no solution**.
### Equation D:
\[ 4(x + 3) + x = 5(x + 1) + 7 \]
Expanding both sides:
\[ 4x + 12 + x = 5x + 5 + 7 \]
Combining like terms:
\[ 5x + 12 = 5x + 12 \]
This also simplifies to a true statement, indicating there are **infinitely many solutions**.
### Conclusion:
The only equation that has **exactly one solution** is Equation B:
\[ 3(x - 1) + 2x = 3(x - 1) + 2 \]