Answer :
Let's solve each equation one by one to determine whether they have no solution, one solution, or infinitely many solutions.
### Equation 1:
[tex]\[ -1.7v + 2.8 = 1.4v - 3.1v + 2.8 \][/tex]
1. Combine like terms on the right-hand side:
[tex]\[ -1.7v + 2.8 = (1.4v - 3.1v) + 2.8 \][/tex]
[tex]\[ -1.7v + 2.8 = -1.7v + 2.8 \][/tex]
2. Subtract [tex]\(-1.7v + 2.8\)[/tex] from both sides:
[tex]\[ 0 = 0 \][/tex]
This equation simplifies to an identity [tex]\(0 = 0\)[/tex], which means it has infinitely many solutions.
### Equation 2:
[tex]\[ 4a - 3 + 2a = 7a - 2 \][/tex]
1. Combine like terms on the left-hand side:
[tex]\[ (4a + 2a) - 3 = 7a - 2 \][/tex]
[tex]\[ 6a - 3 = 7a - 2 \][/tex]
2. Subtract [tex]\(6a\)[/tex] from both sides:
[tex]\[ -3 = a - 2 \][/tex]
3. Add 2 to both sides:
[tex]\[ -1 = a \][/tex]
This equation has one solution, [tex]\(a = -1\)[/tex].
### Equation 3:
[tex]\[ \frac{1}{5}f - \frac{2}{3} = -\frac{1}{5}f + \frac{2}{3} \][/tex]
1. Combine like terms by adding [tex]\(\frac{1}{5}f\)[/tex] to both sides:
[tex]\[ \frac{1}{5}f + \frac{1}{5}f - \frac{2}{3} = \frac{2}{3} \][/tex]
[tex]\[ \frac{2}{5}f - \frac{-2}{3} = \frac{2}{3} \][/tex]
2. Add [tex]\(\frac{2}{3}\)[/tex] to both sides:
[tex]\[ \frac{2}{5}f = \frac{2}{3} + \frac{2}{3} \][/tex]
[tex]\[ \frac{2}{5} + \frac{2}{6} != \frac{1}{5} \][/tex]
This equation has no solution.
### Equation 4:
[tex]\[ 2y - 3 = 5 + 2(y - 1) \][/tex]
1. Distribute the 2 on the right-hand side:
[tex]\[ 2y - 3 = 5 + 2y - 2 \][/tex]
Combine like terms on the right-hand side:
[tex]\[ 2y - 3 = 2y + 3 \][/tex]
2. Subtract [tex]\(2y\)[/tex] from both sides:
[tex]\[ -3 = 3 \][/tex]
This is a contradiction, so the equation has no solution.
### Equation 5:
[tex]\[ -3(n + 4) + n = -2(n + 6) \][/tex]
1. Distribute the constants:
[tex]\[ -3n - 12 + n = -2n - 12 \][/tex]
2. Combine like terms on both sides:
[tex]\[ -2n - 12 = -2n - 12\][/tex]
Since both sides are equal, this equation has infinitely many solutions.
### Summary:
- Equation 1: Infinitely many solutions
- Equation 2: One solution [tex]\((a = -1)\)[/tex]
- Equation 3: No solution
- Equation 4: No solution
- Equation 5: Infinitely many solutions
### Equation 1:
[tex]\[ -1.7v + 2.8 = 1.4v - 3.1v + 2.8 \][/tex]
1. Combine like terms on the right-hand side:
[tex]\[ -1.7v + 2.8 = (1.4v - 3.1v) + 2.8 \][/tex]
[tex]\[ -1.7v + 2.8 = -1.7v + 2.8 \][/tex]
2. Subtract [tex]\(-1.7v + 2.8\)[/tex] from both sides:
[tex]\[ 0 = 0 \][/tex]
This equation simplifies to an identity [tex]\(0 = 0\)[/tex], which means it has infinitely many solutions.
### Equation 2:
[tex]\[ 4a - 3 + 2a = 7a - 2 \][/tex]
1. Combine like terms on the left-hand side:
[tex]\[ (4a + 2a) - 3 = 7a - 2 \][/tex]
[tex]\[ 6a - 3 = 7a - 2 \][/tex]
2. Subtract [tex]\(6a\)[/tex] from both sides:
[tex]\[ -3 = a - 2 \][/tex]
3. Add 2 to both sides:
[tex]\[ -1 = a \][/tex]
This equation has one solution, [tex]\(a = -1\)[/tex].
### Equation 3:
[tex]\[ \frac{1}{5}f - \frac{2}{3} = -\frac{1}{5}f + \frac{2}{3} \][/tex]
1. Combine like terms by adding [tex]\(\frac{1}{5}f\)[/tex] to both sides:
[tex]\[ \frac{1}{5}f + \frac{1}{5}f - \frac{2}{3} = \frac{2}{3} \][/tex]
[tex]\[ \frac{2}{5}f - \frac{-2}{3} = \frac{2}{3} \][/tex]
2. Add [tex]\(\frac{2}{3}\)[/tex] to both sides:
[tex]\[ \frac{2}{5}f = \frac{2}{3} + \frac{2}{3} \][/tex]
[tex]\[ \frac{2}{5} + \frac{2}{6} != \frac{1}{5} \][/tex]
This equation has no solution.
### Equation 4:
[tex]\[ 2y - 3 = 5 + 2(y - 1) \][/tex]
1. Distribute the 2 on the right-hand side:
[tex]\[ 2y - 3 = 5 + 2y - 2 \][/tex]
Combine like terms on the right-hand side:
[tex]\[ 2y - 3 = 2y + 3 \][/tex]
2. Subtract [tex]\(2y\)[/tex] from both sides:
[tex]\[ -3 = 3 \][/tex]
This is a contradiction, so the equation has no solution.
### Equation 5:
[tex]\[ -3(n + 4) + n = -2(n + 6) \][/tex]
1. Distribute the constants:
[tex]\[ -3n - 12 + n = -2n - 12 \][/tex]
2. Combine like terms on both sides:
[tex]\[ -2n - 12 = -2n - 12\][/tex]
Since both sides are equal, this equation has infinitely many solutions.
### Summary:
- Equation 1: Infinitely many solutions
- Equation 2: One solution [tex]\((a = -1)\)[/tex]
- Equation 3: No solution
- Equation 4: No solution
- Equation 5: Infinitely many solutions
