How to solve 3 linear equations in 3 variables using cross multiplication method? I have no problem in solving these equations using substituting. However, how do I solve these using cross multiplication? I can solve two equations with 3 variables but have no idea if I just one more equation. One way- I believe – is to solve two equations at a time but is there any better way?
By cross multiplication, I mean this.
Best Answer
A system of 3 equations with 3 unknowns $x,y,z$ can be written as
$$x\vec{U}+y\vec{V}+z\vec{W}=\vec{R}\tag{1}$$
with $\vec{U},\vec{V},\vec{W},\vec{R} \in \mathbb{R}^3$ (think to column vectors).
If you compute for example
$$\vec{V_1}:=\vec{V} \times \vec{W}, $$ it suffices to take the dot product of (1) with $\vec{V_1}$ to obtain
$$x\vec{U}.\vec{V_1}+y\underbrace{\vec{V}.\vec{V_1}}_{= 0}+z\underbrace{\vec{W}.\vec{V_1}}_{= 0}=\vec{R}.\vec{V_1}\tag{2}$$
whence :
It remains to do the same thing for unknowns $y$ and $z$.
Remark : In fact (3) is a quotient of 2 triple products ; thus we find back a way of establishing/writing Cramer's formulas as remarked by @ryan221b.