Certainly! Let's analyze each of the given equations to determine if they have one solution, no solutions, or infinitely many solutions.
### Equation [tex]\(a\)[/tex]: [tex]\(5x - 2 = 5x + 9\)[/tex]
1. Simplify both sides:
[tex]\[
5x - 2 = 5x + 9
\][/tex]
2. Subtract [tex]\(5x\)[/tex] from both sides:
[tex]\[
5x - 5x - 2 = 5x - 5x + 9 \implies -2 = 9
\][/tex]
3. The resulting statement [tex]\(-2 = 9\)[/tex] is a contradiction because it is not true.
Thus, Equation [tex]\(a\)[/tex] has no solution since there is no value of [tex]\(x\)[/tex] that can satisfy this equation.
### Equation [tex]\(b\)[/tex]: [tex]\(6x + 8 = 8 + 6x\)[/tex]
1. Simplify both sides:
[tex]\[
6x + 8 = 8 + 6x
\][/tex]
2. Subtract [tex]\(6x\)[/tex] from both sides:
[tex]\[
6x - 6x + 8 = 8 + 6x - 6x \implies 8 = 8
\][/tex]
3. The resulting statement [tex]\(8 = 8\)[/tex] is always true, which means the original equation does not depend on [tex]\(x\)[/tex].
Thus, Equation [tex]\(b\)[/tex] has infinitely many solutions since any value of [tex]\(x\)[/tex] will satisfy this equation.
### Equation [tex]\(c\)[/tex]: [tex]\(3x + 4 = 2x + 8\)[/tex]
1. Simplify both sides:
[tex]\[
3x + 4 = 2x + 8
\][/tex]
2. Subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[
3x - 2x + 4 = 2x - 2x + 8 \implies x + 4 = 8
\][/tex]
3. Subtract 4 from both sides:
[tex]\[
x + 4 - 4 = 8 - 4 \implies x = 4
\][/tex]
Thus, Equation [tex]\(c\)[/tex] has one solution, which is [tex]\(x = 4\)[/tex].
### Summary
- Equation [tex]\(a\)[/tex]: No solution
- Equation [tex]\(b\)[/tex]: Infinitely many solutions
- Equation [tex]\(c\)[/tex]: One solution (specifically [tex]\(x = 4\)[/tex])
