Match the linear equations given in Column-I with their solutions given in Column-II and select the correct option.
`{:(,"Column-I",,"Column-II"),((P),5x=-2y+7,(a),(0,0)),((Q),4x-6y=0,(b),(2,0)),((R),3y=(5)/(3)x+7,(c),(3,-4)),((S),2x-y=4,(d),(3,4)):}`

Options

A. `(P)to(b),(Q)to(c),(R)to(a),(S)to(d)`
B. `(P)to(c),(Q)to(b),(R)to(d),(S)to(a)`
C. `(P)to(c),(Q)to(a),(R)to(d),(S)to(b)`
D. `(P)to(d),(Q)to(b),(R)to(a),(S)to(c)`

Answer

To solve the problem of matching the linear equations from Column-I with their corresponding solutions from Column-II, we will evaluate each equation with the given points in Column-II.

### Step 1: Analyze the equations and points

We have the following equations in Column-I:
1. \( P: 5x = -2y + 7 \)
2. \( Q: 4x - 6y = 0 \)
3. \( R: 3y = \frac{5}{3}x + 7 \)
4. \( S: 2x - y = 4 \)

And the points in Column-II:
- \( a: (0,0) \)
- \( b: (2,0) \)
- \( c: (3,-4) \)
- \( d: (3,4) \)

### Step 2: Substitute the points into the equations

#### For Equation P: \( 5x = -2y + 7 \)
1. Check point \( a: (0,0) \)
\[ 5(0) = -2(0) + 7 \implies 0 = 7 \quad \text{(False)} \]
2. Check point \( b: (2,0) \)
\[ 5(2) = -2(0) + 7 \implies 10 = 7 \quad \text{(False)} \]
3. Check point \( c: (3,-4) \)
\[ 5(3) = -2(-4) + 7 \implies 15 = 8 + 7 \implies 15 = 15 \quad \text{(True)} \]
4. Check point \( d: (3,4) \)
\[ 5(3) = -2(4) + 7 \implies 15 = -8 + 7 \implies 15 = -1 \quad \text{(False)} \]

**Match for P: (3,-4) → c**

#### For Equation Q: \( 4x - 6y = 0 \)
1. Check point \( a: (0,0) \)
\[ 4(0) - 6(0) = 0 \implies 0 = 0 \quad \text{(True)} \]
2. Check point \( b: (2,0) \)
\[ 4(2) - 6(0) = 0 \implies 8 = 0 \quad \text{(False)} \]
3. Check point \( c: (3,-4) \)
\[ 4(3) - 6(-4) = 0 \implies 12 + 24 = 0 \quad \text{(False)} \]
4. Check point \( d: (3,4) \)
\[ 4(3) - 6(4) = 0 \implies 12 - 24 = 0 \quad \text{(False)} \]

**Match for Q: (0,0) → a**

#### For Equation R: \( 3y = \frac{5}{3}x + 7 \)
1. Check point \( a: (0,0) \)
\[ 3(0) = \frac{5}{3}(0) + 7 \implies 0 = 7 \quad \text{(False)} \]
2. Check point \( b: (2,0) \)
\[ 3(0) = \frac{5}{3}(2) + 7 \implies 0 = \frac{10}{3} + 7 \quad \text{(False)} \]
3. Check point \( c: (3,-4) \)
\[ 3(-4) = \frac{5}{3}(3) + 7 \implies -12 = 5 + 7 \implies -12 = 12 \quad \text{(False)} \]
4. Check point \( d: (3,4) \)
\[ 3(4) = \frac{5}{3}(3) + 7 \implies 12 = 5 + 7 \implies 12 = 12 \quad \text{(True)} \]

**Match for R: (3,4) → d**

#### For Equation S: \( 2x - y = 4 \)
1. Check point \( a: (0,0) \)
\[ 2(0) - 0 = 4 \implies 0 = 4 \quad \text{(False)} \]
2. Check point \( b: (2,0) \)
\[ 2(2) - 0 = 4 \implies 4 = 4 \quad \text{(True)} \]
3. Check point \( c: (3,-4) \)
\[ 2(3) - (-4) = 4 \implies 6 + 4 = 4 \quad \text{(False)} \]
4. Check point \( d: (3,4) \)
\[ 2(3) - 4 = 4 \implies 6 - 4 = 4 \quad \text{(False)} \]

**Match for S: (2,0) → b**

### Final Matches
- \( P \) matches with \( c \) (3,-4)
- \( Q \) matches with \( a \) (0,0)
- \( R \) matches with \( d \) (3,4)
- \( S \) matches with \( b \) (2,0)

### Summary of Matches
- \( P \) → \( c \)
- \( Q \) → \( a \)
- \( R \) → \( d \)
- \( S \) → \( b \)