Write down the euqation of the line which makes an intercepts of 2a on the x-axis and 3a on the y-axis. Given that the line passes through the point (14, -9), find the numerical value of a.

Answer

To solve the problem step by step, we will derive the equation of the line with the given intercepts and then find the value of \( a \) using the point through which the line passes.

### Step 1: Write the equation of the line using intercepts
The line has an x-intercept of \( 2a \) and a y-intercept of \( 3a \). The general form of the equation of a line with x-intercept \( a \) and y-intercept \( b \) is given by:

\[ \frac{x}{a} + \frac{y}{b} = 1 \]

For our case, substituting \( a = 2a \) and \( b = 3a \):

\[ \frac{x}{2a} + \frac{y}{3a} = 1 \]

### Step 2: Simplify the equation
To simplify the equation, we can multiply through by \( 6a \) (the least common multiple of the denominators):

\[ 6a \left( \frac{x}{2a} + \frac{y}{3a} \right) = 6a \cdot 1 \]

This simplifies to:

\[ 3x + 2y = 6a \]

### Step 3: Use the point (14, -9) to find \( a \)
Since the line passes through the point \( (14, -9) \), we can substitute \( x = 14 \) and \( y = -9 \) into the equation \( 3x + 2y = 6a \):

\[ 3(14) + 2(-9) = 6a \]

Calculating the left side:

\[ 42 - 18 = 6a \]

This simplifies to:

\[ 24 = 6a \]

### Step 4: Solve for \( a \)
Now, divide both sides by 6 to find \( a \):

\[ a = \frac{24}{6} = 4 \]

### Final Answer
The numerical value of \( a \) is \( 4 \).

---