Linear equations

Last class we started with something that lead us to a linear equation. Sometimes, equations has only (and only one) solution.

Equations like $4x+2=19$ has only one solution, which we can derive doing our basic algebra steps.

But, something like this

\begin{equation*} 3x+2y=9 \end{equation*}

has two unknowns and, therefore, we can't solve for both at once ..let's say I want to solve for $x$

\begin{equation*} 3x=9-2y \end{equation*}

\begin{equation*} x=3-\frac{2}{3}y \end{equation*}

If I want to solve for $y$ I end up with:

\begin{equation*} \frac{3}{2} x+\frac{9}{2}=y \end{equation*}

that is the same thing but written in a different way .. but .. no unique answer could be obtained from that.

Let's look at the begining $3x=9-2y$

let's say I force $x$ to be 1 ...

\begin{equation*} 3\cdot 1=9-2y \end{equation*}

now, there is only one unknown and it's possible to solve for $y$ ... what this tells us? There are infinite solutions for this kind of equations.

Not a very surprising thig, but, what if I decide to plot those "infiinte" results whatever $x$ I chose, paired with the y that matches the equation?

Well... we get a line. Not very clever that linear equations has a line as a plot for its answers, but it's an easy way to remember them.

go to desmos and play with its graphic calculator.

Another good resource is here for you to explore more on this subject.

Let's practice: