## Question

The family of lines *x*(*a* + 2*b*) + *y*(*a *+ 3*b*) = *a *+ *b* passes through the point for all values of *a* and *b*. Find the point.

### Solution

(2, –1)

The given equation can be written as

*a*(*x* + *y* – 1) + *b*(2*x* + 3*y* – 1) = 0

which is equation of a line passing through the point of intersection of the lines *x* + *y* – 1 = 0 and 2*x* + 3*y* – 1 = 0. The point of intersection of these lines is (2, –1). Hence the given family of lines passing through the point (2, –1) for all values of *a* and *b*.

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