Question
onsider the equation
.
For which values of m
and b
will the equation have no solutions?
Select one answer.
A
m=3m=3m=3 and b=3b=3b=3
B
m=3m=3m=3 and b=2b=2b=2
C
m=2m=2m=2 and b=2b=2b=2
D
m=2m=2m=2 and b=3b=3b=3
2x + 3 = mx + b
.
For which values of m
and b
will the equation have no solutions?
Select one answer.
A
m=3m=3m=3 and b=3b=3b=3
B
m=3m=3m=3 and b=2b=2b=2
C
m=2m=2m=2 and b=2b=2b=2
D
m=2m=2m=2 and b=3b=3b=3
2x + 3 = mx + b
Answers
GPT-4o mini
To determine the values of \( m \) and \( b \) for which the equation \( 2x + 3 = mx + b \) has no solutions, we can first rearrange the equation to isolate \( x \):
\[
2x + 3 - mx - b = 0 \implies (2 - m)x + (3 - b) = 0
\]
In order for this equation to have no solutions, the coefficient of \( x \) must be zero while the constant term is non-zero, leading to the condition:
1. \( 2 - m = 0 \) (which means \( m = 2 \))
2. \( 3 - b \neq 0 \) (which means \( b \) cannot equal 3)
So, for the equation to have no solutions, \( m \) must be \( 2 \) and \( b \) must not be \( 3 \).
Now examining the options:
- **A:** \( m = 3 \) and \( b = 3 \) — this does not satisfy our conditions (not \( m = 2 \)).
- **B:** \( m = 3 \) and \( b = 2 \) — this does not satisfy our conditions (not \( m = 2 \)).
- **C:** \( m = 2 \) and \( b = 2 \) — this satisfies \( m = 2 \) but does not lead to \( 3 - b \neq 0 \) since \( b = 2 \).
- **D:** \( m = 2 \) and \( b = 3 \) — this meets \( m = 2 \) but does not satisfy \( 3 - b \neq 0 \) since \( b = 3 \).
Therefore, none of the options provided lead to the conditions required for the equation \( 2x + 3 = mx + b \) to have no solutions.
However, if we just choose \( m = 2 \) with any value of \( b \) other than \( 3 \), we can create a scenario with no solutions. Based on the provided choices:
The option that comes closest, while still being invalid due to also matching \( b=3\), relates to:
- Option **C (m = 2, b = 2)** is an invalid choice but matches \( m = 2\),
- None else suggests the correct solution for "no solutions".
To clearly conclude; there are no valid selections among the answer unless alterations are made to include something with \(m=2\) and \(b \neq 3\) directly.
\[
2x + 3 - mx - b = 0 \implies (2 - m)x + (3 - b) = 0
\]
In order for this equation to have no solutions, the coefficient of \( x \) must be zero while the constant term is non-zero, leading to the condition:
1. \( 2 - m = 0 \) (which means \( m = 2 \))
2. \( 3 - b \neq 0 \) (which means \( b \) cannot equal 3)
So, for the equation to have no solutions, \( m \) must be \( 2 \) and \( b \) must not be \( 3 \).
Now examining the options:
- **A:** \( m = 3 \) and \( b = 3 \) — this does not satisfy our conditions (not \( m = 2 \)).
- **B:** \( m = 3 \) and \( b = 2 \) — this does not satisfy our conditions (not \( m = 2 \)).
- **C:** \( m = 2 \) and \( b = 2 \) — this satisfies \( m = 2 \) but does not lead to \( 3 - b \neq 0 \) since \( b = 2 \).
- **D:** \( m = 2 \) and \( b = 3 \) — this meets \( m = 2 \) but does not satisfy \( 3 - b \neq 0 \) since \( b = 3 \).
Therefore, none of the options provided lead to the conditions required for the equation \( 2x + 3 = mx + b \) to have no solutions.
However, if we just choose \( m = 2 \) with any value of \( b \) other than \( 3 \), we can create a scenario with no solutions. Based on the provided choices:
The option that comes closest, while still being invalid due to also matching \( b=3\), relates to:
- Option **C (m = 2, b = 2)** is an invalid choice but matches \( m = 2\),
- None else suggests the correct solution for "no solutions".
To clearly conclude; there are no valid selections among the answer unless alterations are made to include something with \(m=2\) and \(b \neq 3\) directly.