Question
Justify why 6x−5=3(x+2)+3x−7
has one solution, infinitely many solutions, or no solution.(1 point)
Responses
It has infinitely many solutions because the solution is in the form a=a.
It has infinitely many solutions because the solution is in the form
It has one solution because the solution is in the form x=a.
The solution is x=0.
It has one solution because the solution is in the form The solution is x is equal to 0 point
It has one solution because the solution is in the form x=a.
The solution is x=−1.
It has one solution because the solution is in the form The solution is x is equal to negative 1 point
It has no solution because the solution is in the form a=b.
has one solution, infinitely many solutions, or no solution.(1 point)
Responses
It has infinitely many solutions because the solution is in the form a=a.
It has infinitely many solutions because the solution is in the form
It has one solution because the solution is in the form x=a.
The solution is x=0.
It has one solution because the solution is in the form The solution is x is equal to 0 point
It has one solution because the solution is in the form x=a.
The solution is x=−1.
It has one solution because the solution is in the form The solution is x is equal to negative 1 point
It has no solution because the solution is in the form a=b.
Answers
GPT-4o mini
To determine whether the equation \( 6x - 5 = 3(x + 2) + 3x - 7 \) has one solution, infinitely many solutions, or no solution, we need to simplify the equation.
Starting with the given equation:
\[
6x - 5 = 3(x + 2) + 3x - 7
\]
1. Distribute on the right side:
\[
6x - 5 = 3x + 6 + 3x - 7
\]
2. Combine like terms on the right side:
\[
6x - 5 = 6x - 1
\]
3. Now subtract \( 6x \) from both sides:
\[
6x - 6x - 5 = 6x - 6x - 1
\]
This simplifies to:
\[
-5 = -1
\]
Since \( -5 \neq -1 \), we have reached a contradiction. This indicates that the equation has no valid solutions.
Thus, the correct justification for the number of solutions is:
**It has no solution because the solution is in the form \( a \neq b \).**
Starting with the given equation:
\[
6x - 5 = 3(x + 2) + 3x - 7
\]
1. Distribute on the right side:
\[
6x - 5 = 3x + 6 + 3x - 7
\]
2. Combine like terms on the right side:
\[
6x - 5 = 6x - 1
\]
3. Now subtract \( 6x \) from both sides:
\[
6x - 6x - 5 = 6x - 6x - 1
\]
This simplifies to:
\[
-5 = -1
\]
Since \( -5 \neq -1 \), we have reached a contradiction. This indicates that the equation has no valid solutions.
Thus, the correct justification for the number of solutions is:
**It has no solution because the solution is in the form \( a \neq b \).**