# How to solve a system of equation with a 3 variable and a 2 variable system

\begin{cases}\begin{align}x-4y+3z=5\\y+2z=6\end{align}\end{cases} Given that system of equation, the question calls on to solve the system

I was very confused because with two variables on the bottom equation and three variables in this I had difficulties in solving for any of the variables and would like some help.

Currently I have subtracted the first equation with $4(y+2z)=24$ which cancels out the $y$ variable but that didn't help at all.

• If one equation has three variables and the other has two, simply put a zero in front of the missing variable in the second equation. That is, consider the system: $$\begin{cases}x-4y+3z=5\\0x+y+2z=6\end{cases}$$ Are you familiar with row operations and matrices? – Dave Jun 8 '17 at 0:22

## 2 Answers

First, this system has many solutions. You can check that both $(x,y,z)=(29,6,0)$ and $(-4,0,3)$ are solutions. So the solution is not of the form that $x$, $y$ and $z$ equal to some particular numbers.

If you let $z=t$, then $y=-2t+6$ and therefore,

$$x-4(-2t+6)+3t=0$$

and hence $x=11t+24$.

$(x,y,z)=(11t+24,-2t+6,t)$ is a solution for any $t\in \mathbb{R}$. This is the general solution.

• so does that mean that there are infinitely many solution sets to this system of equations? – John Rawls Jun 8 '17 at 0:36
• Yes. Each $t$ is corresponding to a solution. The solutions form a line in $\mathbb{R}^3$ – CY Aries Jun 8 '17 at 1:43

I noticed in a comment that the OP wasn't sure if there were an infinite number of solutions. How about looking at something really simple to get some insight.

Here is a linear system with 2 equation and 3 unknowns.

$x + y = 1$
$z = 1$

The first equation is a line on the x,y-axis. The second equation is telling you that in the $x,y,z$ space, the line is up one unit off of the x,y plane. But it is a line.

How do you describe a line in 3-space? Well, if you want to see a line, you would like to see a 1:1 correspondence with the real number line. There is no unique way to do this, but this process is called 'parameterization'.

You can google this parameterization of a line in 3 space.