Animated Solution for Mathematics - Straight Lines: The number of integral values of m so that the abscissa of point of intersection of lines $3x+4y=9$ and $y=mx+1$ is also an integer, is :

Imagine you are standing on a coordinate plane. You have a fixed line, $3x+4y=9$, sitting perfectly still, like a rigid steel beam.

Then, you have a second line, $y=mx+1$, which is dynamic. It is pivoting around the point $(0,1)$ on the y-axis.

As you change the slope $m$, this line sweeps across the plane, intersecting our fixed beam at different locations. The goal is to find the specific integer slopes $m$ that force the intersection point $(x,y)$ to have integer coordinates.

To find where these two entities collide, we must solve their equations simultaneously: $L_1:3x+4y=9$ $L_2:y=mx+1$

The most elegant way to bridge these two is the method of substitution. We take the definition of $y$ from the second line and inject it into the first.

By substituting $y=mx+1$ into $3x+4y=9$, we obtain:

Distributing the $4$ carefully, we get:

Grouping the $x$ terms yields:

Finally, we isolate our abscissa:

This equation is the key to the entire problem. It dictates exactly where the intersection happens for any given slope $m$.

The problem demands that $x$ must be an integer. For $x$ to be an integer, the denominator $(3+4m)$ must be a divisor of the numerator, $5$.

Since $5$ is a prime number, its divisors are limited to the set $\{1,-1,5,-5\}$. We must test each divisor to see if it yields an integer value for $m$.

We test the cases systematically:

Case 1: $3+4m=1$ This is not an integer. We reject it.

Case 2: $3+4m=-1$ This is an integer. We accept it.

Case 3: $3+4m=5$ This is not an integer. We reject it.

Case 4: $3+4m=-5$ This is an integer. We accept it.

We have navigated the geometry, mastered the algebra, and applied the number theory. We found that only two slopes, $m=-1$ and $m=-2$, satisfy the condition.

The journey from a simple line intersection to a discrete set of integer solutions is what makes mathematics so thrilling. Always look for the 'integer' constraints hidden in the algebra to conquer the JEE Advanced.