`gx-hy=78`
`4x+3y=13`
In the system of equations above, g and h are constants. If the system has infinity many solutions, what is the value of gh?

Options

A. `-432`
B. `-6`
C. `6`
D. `432`

Answer

To solve the system of equations given by \( gx - hy = 78 \) and \( 4x + 3y = 13 \) for the constants \( g \) and \( h \) such that the system has infinitely many solutions, we can follow these steps:

### Step 1: Understand the condition for infinitely many solutions
For a system of linear equations to have infinitely many solutions, the two equations must be equivalent. This means that one equation can be obtained from the other by multiplying by a constant.

### Step 2: Multiply the second equation
We start with the second equation:
\[ 4x + 3y = 13 \]

To make the right-hand side equal to 78 (as in the first equation), we can multiply the entire equation by 6:
\[ 6(4x + 3y) = 6(13) \]
This gives us:
\[ 24x + 18y = 78 \]

### Step 3: Set the equations equal
Now we have two equations:
1. \( gx - hy = 78 \)
2. \( 24x + 18y = 78 \)

Since these two equations must be equivalent for there to be infinitely many solutions, we can equate the coefficients of \( x \) and \( y \) from both equations.

### Step 4: Equate coefficients
From the equations:
- Coefficient of \( x \): \( g = 24 \)
- Coefficient of \( y \): \( -h = 18 \) (which implies \( h = -18 \))

### Step 5: Calculate \( gh \)
Now that we have the values of \( g \) and \( h \):
- \( g = 24 \)
- \( h = -18 \)

We can find \( gh \):
\[ gh = 24 \times (-18) = -432 \]

### Final Answer
Thus, the value of \( gh \) is:
\[ \boxed{-432} \]