Solve : `(3)/(x)-(2)/(y)=0` and `(2)/(x)+(5)/(y)=19` . Hence , find 'a' if `y=ax+3`

Answer

To solve the given simultaneous equations and find the value of 'a', we will follow these steps:

### Step 1: Rewrite the equations
We have the equations:
1. \(\frac{3}{x} - \frac{2}{y} = 0\)
2. \(\frac{2}{x} + \frac{5}{y} = 19\)

Let's introduce new variables:
- Let \(X = \frac{1}{x}\)
- Let \(Y = \frac{1}{y}\)

Now, we can rewrite the equations as:
1. \(3X - 2Y = 0\) (Equation 1)
2. \(2X + 5Y = 19\) (Equation 2)

### Step 2: Solve for one variable
From Equation 1, we can express \(Y\) in terms of \(X\):
\[ 3X = 2Y \implies Y = \frac{3}{2}X \quad \text{(Equation 3)} \]

### Step 3: Substitute into the second equation
Now, substitute Equation 3 into Equation 2:
\[ 2X + 5\left(\frac{3}{2}X\right) = 19 \]
This simplifies to:
\[ 2X + \frac{15}{2}X = 19 \]

### Step 4: Combine like terms
To combine the terms on the left, we can convert \(2X\) into a fraction:
\[ \frac{4}{2}X + \frac{15}{2}X = 19 \]
This gives:
\[ \frac{19}{2}X = 19 \]

### Step 5: Solve for \(X\)
Multiply both sides by 2 to eliminate the fraction:
\[ 19X = 38 \implies X = 2 \]

### Step 6: Find \(Y\)
Now substitute \(X = 2\) back into Equation 3 to find \(Y\):
\[ Y = \frac{3}{2}(2) = 3 \]

### Step 7: Find \(x\) and \(y\)
Recall that \(X = \frac{1}{x}\) and \(Y = \frac{1}{y}\):
\[ \frac{1}{x} = 2 \implies x = \frac{1}{2} \]
\[ \frac{1}{y} = 3 \implies y = \frac{1}{3} \]

### Step 8: Substitute into the equation for \(a\)
We are given the equation \(y = ax + 3\). Substitute \(x\) and \(y\):
\[ \frac{1}{3} = a\left(\frac{1}{2}\right) + 3 \]

### Step 9: Solve for \(a\)
Rearranging gives:
\[ a\left(\frac{1}{2}\right) = \frac{1}{3} - 3 \]
Convert 3 to a fraction:
\[ 3 = \frac{9}{3} \implies \frac{1}{3} - \frac{9}{3} = -\frac{8}{3} \]
Thus:
\[ a\left(\frac{1}{2}\right) = -\frac{8}{3} \]
Multiply both sides by 2:
\[ a = -\frac{16}{3} \]

### Final Answer
The value of \(a\) is:
\[ \boxed{-\frac{16}{3}} \]