If the system of equations
$2x+3y=7$,
$(p+q)x+(2p-q)y=21$
has infinitely many solutions, then:

Question

If the system of equations 
$2x+3y=7$, 
$(p+q)x+(2p-q)y=21$ 
has infinitely many solutions, then:

Accepted Answer

To determine why Option 1 ($p=5, q=1$) is the correct answer, we need to analyze the conditions under which the given system of equations has infinitely many solutions. A system of linear equations has infinitely many solutions if the equations are dependent, meaning one equation is a scalar multiple of the other.

The given system of equations is:

1. $2x + 3y = 7$
2. $(p+q)x + (2p-q)y = 21$

For these equations to be dependent, the coefficients of $x$ and $y$ in the second equation must be proportional to those in the first equation, and the constant terms must also be proportional. Specifically, we need:

$$
\frac{p+q}{2} = \frac{2p-q}{3} = \frac{21}{7}
$$

Calculating the constant term ratio:

$$
\frac{21}{7} = 3
$$

Now, let's set up the proportionality conditions:

1. $\frac{p+q}{2} = 3$
2. $\frac{2p-q}{3} = 3$

Solving these equations:

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

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

Now, we solve the system of equations:

- $p + q = 6$
- $2p - q = 9$

Adding these two equations to eliminate $q$:

$$
(p + q) + (2p - q) = 6 + 9
$$
$$
3p = 15
$$
$$
p = 5
$$

Substituting $p = 5$ back into $p + q = 6$:

$$
5 + q = 6
$$
$$
q = 1
$$

Thus, the values $p = 5$ and $q = 1$ satisfy both conditions, confirming that Option 1 is correct.

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

- **Option 2: $p=1, q=5$**
  - Substituting into $p + q = 6$ gives $1 + 5 = 6$, which is correct, but substituting into $2p - q = 9$ gives $2(1) - 5 = 2 - 5 = -3$, which is not equal to 9. Hence, this option does not satisfy both conditions.

- **Option 3: $p=2, q=2$**
  - Substituting into $p + q = 6$ gives $2 + 2 = 4$, which is not equal to 6. Therefore, this option is incorrect.

- **Option 4: $p=0, q=3$**
  - Substituting into $p + q = 6$ gives $0 + 3 = 3$, which is not equal to 6. Thus, this option is incorrect.

In conclusion, only Option 1 ($p=5, q=1$) satisfies both proportionality conditions, confirming it as the correct answer.