*We can see the two graphs intercept at the point \((4,2)\). Push ENTER one more time, and you will get the point of intersection on the bottom! Substitution is the favorite way to solve for many students!*

So, again, now we have three equations and three unknowns (variables).

We’ll learn later how to put these in our calculator to easily solve using matrices (see the Matrices and Solving Systems with Matrices section), but for now we need to first use two of the equations to eliminate one of the variables, and then use two other equations to eliminate the same variable: Now this gets more difficult to solve, but remember that in “real life”, there are computers to do all this work!

(You can also use the WINDOW button to change the minimum and maximum values of your \(x\) and \(y\) values.) TRACE” (CALC), and then either push 5, or move cursor down to intersect. The reason it’s most useful is that usually in real life we don’t have one variable in terms of another (in other words, a “\(y=\)” situation).

The main purpose of the linear combination method is to add or subtract the equations so that one variable is eliminated.

It’s easier to put in \(j\) and \(d\) so we can remember what they stand for when we get the answers.

There are several ways to solve systems; we’ll talk about graphing first.

The easiest way for the second equation would be the intercept method; when we put for the “\(d\)” intercept.

We can do this for the first equation too, or just solve for “\(d\)”.

When equations have no solutions, they are called inconsistent equations, since we can never get a solution.

Here are graphs of inconsistent and dependent equations that were created on the graphing calculator: Let’s get a little more complicated with systems; in real life, we rarely just have two unknowns with two equations.

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