There are linear algebraic solutions to second-order and third-order linear equations in textbooks.

But not in the form of "linear equations". However, from the content of the textbook, in fact, the equation has been solved by Cramer's law (using second-order and third-order determinants).

But this part of the content is not tested in the college entrance examination. So generally speaking, high schools don't take it seriously or even talk about it. So the average student has no impression of it.