Let's analyze each equation one by one and match them to their solutions:
1. Equation: [tex]\(-4 + 6x = 2(3x - 3)\)[/tex]
Simplify the right-hand side:
[tex]\[
2(3x - 3) = 6x - 6
\][/tex]
So the equation becomes:
[tex]\[
-4 + 6x = 6x - 6
\][/tex]
Subtract [tex]\(6x\)[/tex] from both sides:
[tex]\[
-4 = -6
\][/tex]
This statement is false. Therefore, this equation has no solution.
2. Equation: [tex]\(3(5x + 2) = 5(3x - 4)\)[/tex]
Simplify both sides:
[tex]\[
3(5x + 2) = 15x + 6
\][/tex]
[tex]\[
5(3x - 4) = 15x - 20
\][/tex]
So the equation becomes:
[tex]\[
15x + 6 = 15x - 20
\][/tex]
Subtract [tex]\(15x\)[/tex] from both sides:
[tex]\[
6 = -20
\][/tex]
This statement is also false. Therefore, this equation also has no solution.
3. Equation: [tex]\(8 - 2x = 2x - 8\)[/tex]
Move all terms involving [tex]\(x\)[/tex] to one side:
[tex]\[
8 - 2x - 2x = -8
\][/tex]
[tex]\[
8 - 4x = -8
\][/tex]
Add 8 to both sides:
[tex]\[
8 + 8 - 4x = 0
\][/tex]
[tex]\[
16 - 4x = 0
\][/tex]
Move the constant term to the other side:
[tex]\[
-4x = -16
\][/tex]
Divide both sides by [tex]\(-4\)[/tex]:
[tex]\[
x = 4
\][/tex]
Therefore, this equation has the solution [tex]\(x = 4\)[/tex].
4. Equation: [tex]\(-3x + 3 = -3(1 + x)\)[/tex]
Expand the right-hand side:
[tex]\[
-3x + 3 = -3 - 3x
\][/tex]
Move all terms involving [tex]\(x\)[/tex] to one side:
[tex]\[
-3x + 3x + 3 = -3
\][/tex]
[tex]\[
3 = -3
\][/tex]
This statement is false. Therefore, this equation has no solution.
So, the final matching is:
- [tex]\(-4 + 6x = 2(3x - 3)\)[/tex] matches no solution
- [tex]\(3(5x + 2) = 5(3x - 4)\)[/tex] matches no solution
- [tex]\(8 - 2x = 2x - 8\)[/tex] matches x = 4
- [tex]\(-3x + 3 = -3(1 + x)\)[/tex] matches no solution
