[Algebra 9] System of two linear equations in two variables

❄ SYSTEM OF TWO LINEAR EQUATIONS IN TWO VARIABLES

A system of two linear equations in two variables in the form of:

$\begin{cases}ax + by = c\\a'x + b'y = c'\end{cases}$

In order to solve it, we use the substitution method or the algebraic addition method.

Example: Solve the following system of equations $\begin{cases}2x + y = 1\\3x + 4y = -1\end{cases}$ (I)

Way 1: Using the substitution method

(I) $ \Leftrightarrow $ $\begin{cases}y = 1 - 2x\\3x + 4(1 - 2x) = -1\end{cases}$
$ \Leftrightarrow $ $\begin{cases}y = 1 - 2x\\-5x = -5\end{cases}$
$ \Leftrightarrow $ $\begin{cases}y = 1 - 2.1\\x = 1\end{cases}$
$ \Leftrightarrow $ $\begin{cases}y = -1\\x = 1\end{cases}$
Thus, the given system has a solution $\begin{cases}x = 1\\y = -1\end{cases}$

Way 2: Using the algebraic addition method

(I) $ \Leftrightarrow $ $\begin{cases}8x + 4y = 4\\3x + 4y = -1\end{cases}$
$ \Leftrightarrow $ $\begin{cases}5x = 5\\3x +4y = -1\end{cases}$
$ \Leftrightarrow $ $\begin{cases}x = 1\\3.1 + 4y = -1\end{cases}$
$ \Leftrightarrow $ $\begin{cases}x = 1\\y = -1\end{cases}$
Thus, the given system has a solution $\begin{cases}x = 1\\y = -1\end{cases}$