Animated Solution for Mathematics - Matrices and Determinants: If the system of linear equations $x-4y+7z=g$, $3y-5z=h$, $-2x+5y-9z=k$ is consistent, then :

Imagine you are standing in a vast, three-dimensional space. You are surrounded by three massive, flat sheets—these are our planes, defined by the equations $P_1:x-4y+7z=g$, $P_2:3y-5z=h$, and $P_3:-2x+5y-9z=k$.

The problem asks us to find the condition under which this system is consistent. This means these three planes, in their infinite expanse, must share at least one common point.

The Determinant

Before we dive into the geometry, we need to check the 'heartbeat' of our system: the determinant of the coefficient matrix $A$. We extract the coefficients of $x,y,$ and $z$ to form the matrix:

We calculate the determinant $\Delta =|A|$ by expanding along the first column, which is the most efficient path due to the zero in the second row. The calculation unfolds as:

Infinite Solutions

When $\Delta =0$, the system is at a crossroads. It either has no solution (the planes never meet) or infinitely many solutions (the planes intersect along a common line).

Since the problem guarantees consistency, we know we are in the realm of infinite solutions. Geometrically, this means our three planes are not independent; they are locked in a dance where one is essentially a combination of the others.

Linear Dependency

To find the condition on $g,h,$ and $k$, we must uncover the 'linear dependency'—the hidden relationship between the equations. We assume there exist constants $a,b,$ and $c$ such that $a{P}_1+b{P}_2+c{P}_3=0$.

Let's break this down by coefficients:

For $x$: $a(1)+b(0)+c(-2)=0⟹a=2c$.

For $y$: $a(-4)+b(3)+c(5)=0$. Substituting $a=2c$, we get $-4(2c)+3b+5c=0$, which simplifies to $-8c+3b+5c=0$, or $3b=3c$, meaning $b=c$.

By setting $c=1$, we find $a=2$ and $b=1$. Our dependency is $2P_1+1P_2+1P_3=0$. We verify this with the $z$-coefficients: $2(7)+1(-5)+1(-9)=14-5-9=0$.

It works perfectly! For the system to be consistent, this same dependency must apply to the constants $g,h,$ and $k$ on the right-hand side:

And there it is! The elegance of linear algebra reveals that the relationship between the planes is mirrored in their constants. The final condition for consistency is $2g+h+k=0$.