Given the system of equations:

(I) $3x + 2y = 7$

(II) $2x + 5y = 12$

Determine the correct relationship between $x$ and $y$ from the options below.

Question

Given the system of equations:

(I) $3x + 2y = 7$

(II) $2x + 5y = 12$

Determine the correct relationship between $x$ and $y$ from the options below.

Accepted Answer

**Solution:**

1. We are given two equations:
   - (I) $3x + 2y = 7$
   - (II) $2x + 5y = 12$

2. Solve equation (I) for $x$:
   $$3x = 7 - 2y \implies x = \frac{7 - 2y}{3}$$

3. Substitute this value of $x$ into equation (II):
   $$2 	imes \frac{7 - 2y}{3} + 5y = 12$$

4. Multiply both sides by 3 to eliminate the denominator:
   $$2(7 - 2y) + 15y = 36$$

5. Simplify:
   $$14 - 4y + 15y = 36$$
   $$14 + 11y = 36$$

6. Solve for $y$:
   $$11y = 36 - 14 = 22 \implies y = 2$$

7. Substitute $y = 2$ back into $x = \frac{7 - 2y}{3}$:
   $$x = \frac{7 - 2 	imes 2}{3} = \frac{7 - 4}{3} = \frac{3}{3} = 1$$

8. Now compare $x$ and $y$:
   $$x = 1, \quad y = 2$$

Since $1 < 2$, we conclude:

**Answer:** $x < y$

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