I'm studying High School Algebra and it had this question:
Solve the system by equations:
\begin{align*} x + y – z &= \,0 \\ 2x + 4y – 2z &= 6 \\ 3x +
6y – 3z &= \,9 \end{align*}
The solution was:
infinitely many solutions (x, 3, z) where x = z − 3; y = 3; z is any real number
I've spent hours on the problem. The textbook just gave a vague explanation and I can't seem to get how it works. Can someone please intuitively explain how this is?
Best Answer
Hint: Dividing the second equation by $2$ and the third by $3$ we get $$x+y-z=0$$ $$x+2y-z=3$$ $$x+2y-z=3$$ the second and the third equation are the same. Multiplying the first equation by $-1$ and adding to the second we get $y=3$. Plgging this into the first and second equation we get $$x-z=-3$$ $$x-z=-3$$ so we obtain the solutions $$x,3,x+3$$