If `3x-by+2=0` and `9x+3y+a=0` represent the same straight line, find the values of 'a' and 'b'.

Answer

To find the values of 'a' and 'b' such that the lines represented by the equations \(3x - by + 2 = 0\) and \(9x + 3y + a = 0\) are the same, we can follow these steps:

### Step 1: Rewrite the equations in slope-intercept form

The first equation is:
\[ 3x - by + 2 = 0 \]
Rearranging gives:
\[ by = 3x + 2 \implies y = \frac{3}{b}x + \frac{2}{b} \]

The second equation is:
\[ 9x + 3y + a = 0 \]
Rearranging gives:
\[ 3y = -9x - a \implies y = -3x - \frac{a}{3} \]

### Step 2: Set the slopes equal to each other

Since the two lines are the same, their slopes must be equal. From the first equation, the slope is \(\frac{3}{b}\), and from the second equation, the slope is \(-3\). Therefore, we set the slopes equal:
\[ \frac{3}{b} = -3 \]

### Step 3: Solve for 'b'

To solve for \(b\), we cross-multiply:
\[ 3 = -3b \implies b = -1 \]

### Step 4: Set the y-intercepts equal

Next, we need to set the y-intercepts equal. The y-intercept from the first equation is \(\frac{2}{b}\) and from the second equation is \(-\frac{a}{3}\). Setting these equal gives:
\[ \frac{2}{b} = -\frac{a}{3} \]

### Step 5: Substitute the value of 'b'

Substituting \(b = -1\) into the equation gives:
\[ \frac{2}{-1} = -\frac{a}{3} \implies -2 = -\frac{a}{3} \]

### Step 6: Solve for 'a'

Multiplying both sides by -3 gives:
\[ 6 = a \]

### Final values

Thus, we find:
\[ b = -1 \quad \text{and} \quad a = 6 \]

### Summary of the solution:
- The value of \(b\) is \(-1\).
- The value of \(a\) is \(6\).

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