Showing $x_1x_3 + 2x_2 + x_1 = 4$ is not a linear equation

Question

Prove that $x_1x_3 + 2x_2 + x_1 = 4$ is not a linear equation.

I do understand that in order to be a linear equation, it must fall inside the graph linearly. This equation does show a linear equation but I don't get it why it is not a linear equation.

I am sorry, I am really new to linear algebra.

Answer 1

Hint $\#1$:

Typically, an equation which is "linear" will meet the condition that two members of your set of variables are not multiplied together.

For example, consider $f(x) = x^2.$ This is not linear, because we have a term that is $x\cdot x$. (Of course this also means that if you have a variable raised to a power which is not $0$ or $1$, you can immediately conclude the equation is nonlinear.)

Similarly, $f(x,y) = xy$ is not linear, because here we have two variables, $x,y$, being multiplied together.

In your case, since you're in a linear algebra course, your variables often will be $x_i$ for some integer $i$. With this in mind, what can you conclude about your equation?

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Hint $\#2$:

Look at the first term of your equation. If any term in the equation is nonlinear, the equation is nonlinear. (Assuming of course the term is not somehow negated - for example, $x^2 - x^2$ is still linear since it's $0$, but that's a fringe concern and not relevant here.)