Question

Justify why 6x−5=3(x+2)+3x−7 has one solution, infinitely many solutions, or no solution.(1 point) Responses It has no solution because the solution is in the form a=b. It has no solution 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 infinitely many solutions because the solution is in the form a=a.

Answer 1

To determine if the equation \(6x - 5 = 3(x + 2) + 3x - 7\) has one solution, infinitely many solutions, or no solution, we can solve the equation step by step.

Starting with the original equation:

\[
6x - 5 = 3(x + 2) + 3x - 7
\]

First, we simplify the right side:

\[
3(x + 2) = 3x + 6
\]
So,
\[
3(x + 2) + 3x - 7 = 3x + 6 + 3x - 7 = 6x - 1
\]

Now, we rewrite the equation:

\[
6x - 5 = 6x - 1
\]

Next, we can subtract \(6x\) from both sides:

\[
-5 = -1
\]

This statement is not true; hence, the equation has no solutions. It leads to a contradiction, indicating that the two expressions can never be equal.

Thus, the correct justification is:

**It has no solution because the solution is in the form \(a = b\) (where \( -5 \neq -1\)).**