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 a,b.

Options

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

Answer

To solve the problem, we will substitute the given solution (-3, 4) into both equations to find the values of constants \( a \) 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 \( a \times b \)
Now that we have the values of \( a \) and \( b \):
\[ a = 2, \quad b = 5 \]
We can find \( a \times b \):
\[ a \times b = 2 \times 5 = 10 \]

### Final Answer
The values of \( a \) and \( b \) are:
\[ a = 2, \quad b = 5 \]
Thus, \( a \times b = 10 \).

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