This is one of the easiest topics in Quant and also one of the foundation topics. Apart from direct questions on linear equations, many questions on other topics also involve solving linear equations from time to time. Be careful of silly mistakes in this topic as that is how students generally lose marks here. Also lookout for the tricky special equations mentioned at the bottom of the formulae list. Generally when the number of equations is less than the number of variables the answer cannot be determined EXCEPT in a few cases where either additional conditions are specified or special combination of variables is asked.

Equations with three variables: Let the equations be $$ a_{1}x+b_{1}y+c_{1}z$$=$$d_{1}$$, $$ a_{2}x+b_{2}y+c_{2}z$$=$$d_{2}$$ and $$ a_{3}x+b_{3}y+c_{3}z$$=$$d_{3}$$. Here we define the following matrices: $$$D=\begin{bmatrix} a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} & c_{2}\\ a_{3} & b_{3} & c_{3}\end{bmatrix}$$$, $$$D_{x}=\begin{bmatrix} d_{1} & b_{1} & c_{1}\\ d_{2} & b_{2} & c_{2}\\ d_{3} & b_{3} & c_{3}\end{bmatrix}$$$, $$$D_{y}=\begin{bmatrix} a_{1} & d_{1} & c_{1}\\ a_{2} & d_{2} & c_{2}\\ a_{3} & d_{3} & c_{3}\end{bmatrix}$$$, $$$D_{z}=\begin{bmatrix} a_{1} & b_{1} & d_{1}\\ a_{2} & b_{2} & d_{2}\\ a_{3} & b_{3} & d_{3}\end{bmatrix}$$$

• Find the determinant of $$D$$ = $$a_{1}(b_{2}c_{3}-c_{2}b_{3})$$ + $$b_{1}(c_{2}a_{3}-a_{2}c_{3})$$ + $$c_{1}(a_{2}b_{3}$$-$$b_{2}a_{3})$$
• Similarly find the determinant of Dx, Dy and Dz
• If $$ \textrm{determinant of D}\neq 0$$, there exists a unique solution
• If $$ \textrm{determinant of D} = 0$$ and at least one of the determinants of Dx, Dy or Dz are non-zero then the system of equations has no solution
• If $$ \textrm{determinant of D} = 0$$ and all three determinants Dx, Dy and Dz are zero then there are infinitely many solutions

• If the system of equations has n variables with n-1 equations then the solution is indeterminate
• If the system of equations has n variables with n-1 equations with some additional conditions like the variables are integers then the solution may be determinate
• If the system of equations has n variables with n-1 equations then some combination of variables may be determinable.
  
• For Example, if ax+by+cz=d and mx+ny+pz=q if a,b, and c are in Arithmetic progression and m,n and p are in AP then the sum x+y+z is determinable