What Are Systems of Linear Equations?

Systems of linear equations are two (or sometimes more) linear equations that normally intersect at a single point. This is essentially just two lines on a single graph, and where they intersect is the solution to the system. There are two main ways of solving the system (here, “solving” means finding the point where the two lines intersect)- graphically, and algebraically.

Solving Systems Graphically

Solving a system of linear equations graphically is very straightforward; you graph the two lines on a cordinate plane (also called a grid) and then simply look to see where they intersect. Solving a system of linear equations graphically, however, can take a large amount of time and a lot of paper (especially if you have to draw your own grid), making it an unsuitable solving method for most problems, like those on a test. You will most likely have to solve some systems graphically for tests, though, so you should be prepared.

Solving Systems Algebraically

As you know, you are actually trying to find a value of x and a value of y that, when inserted into the formulas, makes both true. TO solve them algebraically, we must first eliminate one of the two variables, since it is impossible to solve for two variables definatley. To do this, we either add or subtract one equation to/from the other, such that one of the variables cancels out with the other. Many times, though, this is not possible because of the coefficients- then you must multiply one wholeequation by a number to make some of the coefficients the same. Then, you do your addition/subtraction and solve for the remaining varialbe. To find the last value, substitute the variable for which you have the value of into its place in one of the equations and solve for the last variable. SOund complicated? It’s not really once you get used to it.

An example

Systems of Linear Equations Graph

Look at the graph above. As you can see, the graph features two lines- the red line with a slope of 1 and a y intersect of 0, and the blue line with a slope of 2 and a y interesect of -1. Written in slope-intercept form, these two lines are y=x and y=2x-1. As we can easily see by looking at the graph, these two lines intersect at the point (1,1). How would we solve this algebraically, though?

First, we must isolate just one variable- either x or y- by adding or subtracting one equation to/from the other.



Now we have our x value. By substituting it into either of the equations (y=x obviously being the easiest) we can find the value of y, and hence, our solution.


Of course, that is a quite simplified example, but the principal is the same. Take a look at another, more realistic example…



…and they get much more complicated than that. Just remember your rules and you won’t fail.

Two Special Cases

There are two special cases, however, which you will definatley need to know. They pop up fairly often on tests, when test makers try to trick you

Parallel Lines

As you should remember, parallel lines will never meet- they each stay the same distance apart from each other to infitiny in each direction. Of course, because of this they will never meet. How are you to find the intersect point?

As far as answers are concerned, you will usually put “parallel” or “undefined” (or something along those lines) as your teacher specifies. Algebraically, you can tell if they are parallel when you eliminate all the varialbes and get two numerical values that are NOT equal.

The Same Line

Like parallel lines, when the formulas are the same (usually disguised, though), you will get an answer of infinity or “same line”, again depending on the teacher. This occurs algebraically when you get down to the subtraction or addition and end up with nothing left- either 0=0 or two numbers that are equal to each other. Graphically, this occurs when your two lines are the same.

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