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Section 7.1 : Linear Systems with Two Variables

3. Use the Method of Substitution to find the solution to the following system or to determine if the system is inconsistent or dependent.

\[\begin{align*}3x + 9y & = - 6\\ - 4x - 12y & = 8\end{align*}\]

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Start Solution

Before we get started with the solution process for this system we need to make it clear that there is no “one correct solution path”. There are lots of solution paths that we can take to find the solution to this system. All are correct and all will end up with the same solution to the system (provided the work has been done correctly of course…).

Okay, let’s get started on the solution to this system.

The Method of Substitution tells us that we first need to solve one of the equations for one of the variables. The equation we solve and the variable we solve for technically doesn’t matter as noted above.

In this case both equations seem equally “easy” to deal with and so let’s solve the second equation for \(x\) since that is a combination we didn’t use in the first couple of problems.

\[\begin{align*} - 4x - 12y & = 8\\ - 4x & = 12y + 8\hspace{0.25in} \Rightarrow \hspace{0.25in}x = - 3y - 2\end{align*}\] Show Step 2

We now take the equation for \(x\) we found above and substitute this into the other equation (the first equation in this case). Doing this gives,

\[\begin{align*}3x + 9y & = - 6\\ 3\left( { - 3y - 2} \right) + 9y & = - 6\end{align*}\] Show Step 3

We can now solve the equation we found in the previous step for \(y\). Doing this gives,

\[\begin{align*}3\left( { - 3y - 2} \right) + 9y & = - 6\\ - 9y - 6 + 9y & = - 6\\ - 6 & = - 6\end{align*}\] Show Step 4

Now, the result from the previous step is true for any value of \(y\) or \(x\) and so we know that the system is dependent and there will be an infinite number of solutions to the system. We can write the “solution” to this system as follows,

\[\begin{array}{*{20}{c}}{x = - 3t - 2}\\{y = t}\end{array}\hspace{0.25in}t{\mbox{ is any number}}\]