In the equation shown below, a and b are unknown constants.
`3ax+ 4y= -2 and 2x+ by= 14`
If `(-3, 4)` is the solution of the given equations, find the value of ab.

Options

A. 10
B. 6
C. 12
D. 15

Answer

To solve the problem step by step, we will substitute the given solution point \((-3, 4)\) into both equations to find the values of \(a\) and \(b\).

### Step 1: Substitute \((-3, 4)\) into the first equation

The first equation is:
\[ 3ax + 4y = -2 \]
Substituting \(x = -3\) and \(y = 4\):
\[ 3a(-3) + 4(4) = -2 \]
This simplifies to:
\[ -9a + 16 = -2 \]

### Step 2: Solve for \(a\)

Now, we will isolate \(a\):
\[ -9a + 16 = -2 \]
Subtract 16 from both sides:
\[ -9a = -2 - 16 \]
\[ -9a = -18 \]
Now, divide both sides by -9:
\[ a = \frac{-18}{-9} = 2 \]

### Step 3: Substitute \((-3, 4)\) into the second equation

The second equation is:
\[ 2x + by = 14 \]
Substituting \(x = -3\) and \(y = 4\):
\[ 2(-3) + b(4) = 14 \]
This simplifies to:
\[ -6 + 4b = 14 \]

### Step 4: Solve for \(b\)

Now, we will isolate \(b\):
\[ -6 + 4b = 14 \]
Add 6 to both sides:
\[ 4b = 14 + 6 \]
\[ 4b = 20 \]
Now, divide both sides by 4:
\[ b = \frac{20}{4} = 5 \]

### Step 5: Calculate \(ab\)

Now that we have \(a = 2\) and \(b = 5\), we can find \(ab\):
\[ ab = 2 \times 5 = 10 \]

### Final Answer

Thus, the value of \(ab\) is:
\[ \boxed{10} \]