Question

Here is a system of equations:3x - y = 17x + 4y = 10Explain how you can tell, without graphing, that there is only one solution to the system.

Answer 1

To solve the exercise, we can consider the following system of two linear equations with two unknown variables:

$\begin{cases}Ax+By=C \\ Dx+Ey=F\end{cases}$

If the pair (A,B) is not proportional to the pair (D,E), then there is only one solution. That is, there is no such number k that

$\begin{gathered} D=kA \\ \text{ and} \\ E=kB \end{gathered}$

In other words, we have to verify the following:

$\begin{gathered} D=kA\Rightarrow\frac{D}{A}=k \\ E=kB\Rightarrow\frac{E}{B}=k \\ \text{ Then} \\ \frac{D}{A}\ne\frac{E}{B} \end{gathered}$

In this case, we have:

$\begin{cases}3x-y=17 \\ x+4y=10\end{cases}$$\begin{gathered} \frac{1}{3}\ne\frac{4}{-1} \\ 0.3\ne-4 \end{gathered}$

Therefore, since the pair (3, -1) is not proportional to the pair (1,4), then there is only one solution for the given system of linear equations.