Animated Solution for Mathematics - Matrices and Determinants: If the system of linear equations $2x+2y+3z=a$, $3x-y+5z=b$, $x-3y+2z=c$ where $a,b,c$ are non-zero real numbers, has more than one solution, then :

Imagine you are standing in a vast, three-dimensional space. Before you, there are three flat, infinite planes, which we call $P_1,P_2,$ and $P_3$.

These planes are defined by the equations:

The problem states that this system has more than one solution. In the world of linear algebra, this is a massive hint. If three planes intersect at more than one point, they cannot just meet at a single coordinate; they must be locked together along an entire common line. This implies there are infinitely many solutions.

Uncovering Linear Dependency

When a system has infinitely many solutions, it tells us something profound: the equations are not telling us three completely different stories. They are redundant. One of these planes is essentially a linear combination of the others.

In JEE problems, we often reach for the determinant $\Delta =0$ to prove this, but there is a more elegant, intuitive path. Let us play detective and inspect the coefficients of our planes.

Look at $P_1$ and $P_3$. If we add their left-hand sides, we observe: - For the $x$-terms: $2x+x=3x$ - For the $y$-terms: $2y+(-3y)=-y$ - For the $z$-terms: $3z+2z=5z$

When we combine these, we get $3x-y+5z$. This is a beautiful moment of clarity—this expression is exactly the left-hand side of our second plane, $P_2$.

Because the left-hand sides of $P_1$ and $P_3$ add up perfectly to the left-hand side of $P_2$, the system is only consistent if the right-hand sides follow the exact same rule. If the left sides are locked in this relationship, the constants $a,b,$ and $c$ must also be locked.

Therefore, we must have the condition:

This is the core of the problem. The planes are not just floating randomly; they are constrained by this specific relationship.

To match the standard format of such problems, we rearrange this equation:

By visualizing the geometry and trusting the arithmetic pattern, we have bypassed the need for heavy matrix calculations and arrived at the solution with confidence. Remember, in JEE, the most complex-looking systems often hide the simplest patterns. Keep your eyes open, trust your intuition, and always look for the underlying symmetry.