Solving Problems With Linear Systems
There are several ways to solve systems; we’ll talk about graphing first.
Remember that when you graph a line, you see all the different coordinates (or \(x/y\) combinations) that make the equation work.
“Systems of equations” just means that we are dealing with more than one equation and variable.
So far, we’ve basically just played around with the equation for a line, which is \(y=mx b\).
Let's look at an example: Find the ordered pair for which Thus, the solution to the system is .What we want to know is how many pairs of jeans we want to buy (let’s say “\(j\)”) and how many dresses we want to buy (let’s say “\(d\)”).Always write down what your variables will be: equations as shown below.The third method for solving a system of linear equations is to graph them in the plane and observe where they intersect.We'll go back to our same example to illustrate this.We can do this for the first equation too, or just solve for “\(d\)”.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!We’ll need to put these equations into the \(y=mx b\) (\(d=mj b\)) format, by solving for the \(d\) (which is like the \(y\)): First of all, to graph, we had to either solve for the “\(y\)” value (“\(d\)” in our case) like we did above, or use the cover-up, or intercept method.The easiest way for the second equation would be the intercept method; when we put for the “\(d\)” intercept.Now, you can always do “guess and check” to see what would work, but you might as well use algebra!It’s much better to learn the algebra way, because even though this problem is fairly simple to solve, the algebra way will let you solve any algebra problem – even the really complicated ones.