Animated Solution for Mathematics - Matrices and Determinants: If the following system of linear equations $2x+y+z=5$, $x-y+z=3$, $x+y+az=b$ has no solution, then :

Imagine you are standing in a vast, empty three-dimensional room. Each of the equations given, $2x+y+z=5$, $x-y+z=3$, and $x+y+az=b$, represents a flat, infinite sheet of paper—a plane—floating in that room.

When we ask for a solution to this system, we are asking: "Where do these three planes meet?" If they meet at a single point, we have a unique solution; if they meet along a line, we have infinite solutions.

We are hunting for the "No Solution" scenario. Geometrically, this means the planes are arranged like the sides of a triangular tent, where they might intersect in pairs, but the three planes never share a single common point.

Cramer's Rule

To solve this, we turn to the elegant machinery of Cramer's Rule. The behavior of the system is dictated by the determinant of the coefficient matrix, denoted as $D$.

If $D eq 0$, the system is well-behaved and has a unique solution. When $D=0$, the system becomes "singular"—it either collapses into infinite solutions or becomes inconsistent with no solution at all.

We construct the determinant $D$ using the coefficients of $x,y,$ and $z$:

Expanding along the first row, we calculate:

Simplifying the terms:

For the system to have no solution, we must have $D=0$. Setting $1-3a=0$ gives us $a=\frac{1}{3}$.

Setting $D=0$ is necessary, but not sufficient to guarantee "no solution." It could still be a system with infinite solutions. To ensure there is truly no solution, we must check the numerator determinants, specifically $D_3$.

We construct $D_3$ by replacing the third column of the coefficient matrix with the constants $5,3,$ and $b$:

Expanding along the first row:

Simplifying the expression:

For the system to have no solution, the determinant $D_3$ must be non-zero. If $D_3$ were zero, the planes would be dependent, leading to infinite solutions.

Setting $7 - 3b eq 0$ leads us to $3b eq 7$, or $b eq \frac{7}{3}$.

The exact conditions for the system to have no solution are $a=\frac{1}{3}$ and $b eq \frac{7}{3}$.