If the system of equations 2x + 3y = 7 and (a + b)x + (2a – b)y = 21 has infinitely many solutions, then :

Question

Accepted Answer

To determine why Option 2 is the correct answer, we need to analyze the conditions under which the given system of equations has infinitely many solutions. The system of equations is:

1. $2x + 3y = 7$
2. $(a + b)x + (2a - b)y = 21$

For the system to have infinitely many solutions, the second equation must be a scalar multiple of the first equation. This means that the coefficients of $x$, $y$, and the constant term in the second equation must be proportional to those in the first equation.

Let's compare the coefficients:

- The coefficient of $x$ in the first equation is 2, and in the second equation, it is $a + b$.
- The coefficient of $y$ in the first equation is 3, and in the second equation, it is $2a - b$.
- The constant term in the first equation is 7, and in the second equation, it is 21.

For the equations to be proportional, the following conditions must hold:

$$
\frac{a + b}{2} = \frac{2a - b}{3} = \frac{21}{7}
$$

Simplifying $\frac{21}{7}$, we get 3. Therefore, we have:

1. $\frac{a + b}{2} = 3$
2. $\frac{2a - b}{3} = 3$

Solving these equations:

1. From $\frac{a + b}{2} = 3$, we get:
   $$
   a + b = 6
   $$

2. From $\frac{2a - b}{3} = 3$, we get:
   $$
   2a - b = 9
   $$

Now, solve these two equations simultaneously:

- Equation 1: $a + b = 6$
- Equation 2: $2a - b = 9$

Add the two equations to eliminate $b$:

$$
(a + b) + (2a - b) = 6 + 9
$$
$$
3a = 15
$$
$$
a = 5
$$

Substitute $a = 5$ back into Equation 1:

$$
5 + b = 6
$$
$$
b = 1
$$

Thus, the values $a = 5$ and $b = 1$ satisfy both conditions, confirming that Option 2 is correct.

Now, let's explain why the other options are incorrect:

1. **Option 1: $a = 1, b = 5$**
   - Substituting into the equations: $a + b = 1 + 5 = 6$ (correct), but $2a - b = 2(1) - 5 = -3$, which does not equal 9. Hence, this option is incorrect.

2. **Option 3: $a = -1, b = 5$**
   - Substituting into the equations: $a + b = -1 + 5 = 4$, which does not equal 6. Hence, this option is incorrect.

3. **Option 4: $a = 5, b = -1$**
   - Substituting into the equations: $a + b = 5 - 1 = 4$, which does not equal 6. Hence, this option is incorrect.

Therefore, Option 2 is the only correct choice, as it satisfies the conditions for the system of equations to have infinitely many solutions.