Question

Q26 Determine whether the following equations have the same solution set.
7z + 5 = 2z + 30
3z - 10 = 5
c - 5c + 9c = 27

Answer 1

To determine if the equations have the same solution set, we need to solve each equation for \( z \) (or \( c \) in the last equation).

### Equation 1:
\[ 7z + 5 = 2z + 30 \]

1. Subtract \( 2z \) from both sides:
\[ 7z - 2z + 5 = 30 \]
\[ 5z + 5 = 30 \]

2. Subtract 5 from both sides:
\[ 5z = 25 \]

3. Divide by 5:
\[ z = 5 \]

### Equation 2:
\[ 3z - 10 = 5 \]

1. Add 10 to both sides:
\[ 3z = 15 \]

2. Divide by 3:
\[ z = 5 \]

### Equation 3:
\[ c - 5c + 9c = 27 \]

1. Combine like terms:
\[ (1 - 5 + 9)c = 27 \]
\[ 5c = 27 \]

2. Divide by 5:
\[ c = \frac{27}{5} \]

### Conclusion:
- The solution for the first equation is \( z = 5 \).
- The solution for the second equation is \( z = 5 \).
- The solution for the third equation is \( c = \frac{27}{5} \).

Since the first two equations have the same solution \( z = 5 \), but the third equation has a solution \( c = \frac{27}{5} \), the equations do **not** have the same solution set.

**Final Answer:** No, they do not have the same solution set.