Solving Linear Equations, with 3 unknowns

Question

$$6x-8y=24$$

$$\frac{-2}{3}x+ \frac{8}{9}y = m$$

I want to solve for all three of the variables. I did plot this into my calculator and the answer is $$\frac{-8}{3}$$ I want to know the process in solving it.

Answer 1

Since you has three unknowns, you need three equations to be able to solve for all three variables (your system of linear equations is underdetermined).

However, what you can do is solve for $m$.

Multiplying the second equation through by $-9$ we get:

$$ 6x - 8y = -9m$$

Aha! This looks a lot like our first equation: we already know $6x - 8y = 24$. Putting these together gives $-9m = 24$ and hence $$m = -\frac{8}{3}$$

However, without another equation we cannot determine $x$ and $y$ (for example, both $x = 0, y=-3$ and $x = 4, y=0$ satisfy your system of equations. In fact, there will be infinitely many values of $x$ and $y$ satisfying the two equations).

Answer 2

From $\frac{-2}{3}x+ \frac{8}{9}y = m$ we get $-6x+8y=9m$. If we add this to the first equation we get $24+9m=0$, hence $m=\frac{-8}{3}.$

It remains only one equation: $ 6x-8y=24$. There are infinitely many pairs $(x,y)$ which are solutions of this equation.