Foundational Mathematics for Corequisite Courses: Succeeding at College Mathematics

We have seen that we can solve systems of linear equations using substitution, but that one of the challenges of this method was the computational problem of manipulating fractions. The method of elimination is another approach to solving systems of equations that mostly sidesteps the issue (at least at this level), which often makes it a more efficient approach.

The challenge is to find useful equations so that adding the equations together gives a helpful result. Here are two examples to compare:

First, notice that the addition in columns is simply an organizational tool. By putting like terms together, it simplifies the process of combining them and avoids errors. Second, notice that the final equation on the right only has one variable even though the two initial equations had two variables. The goal of the method of elimination is to eliminate one of the variables. This allows you to solve for the remaining variable.

There’s no real "trick" to figuring out the numbers that you need to get the terms to cancel, but there is a general heuristic. Once you’ve identified the variable you want to eliminate, find the least common multiple of the coefficients, and multiply the equations by the appropriate value to make match, and then make sure that the signs are opposite each other. While some instructors suggest that you can "subtract" the equations, this ends up causing a lot of computational errors, and so it’s usually better to use addition.

PDF Version of these Worksheets

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Algebra is a skill, which means it requires practice to become proficient. But it will take more than rote repetition to get there. Deliberate practice is the thoughtful repetition of a task. For each of these sections, you will be given a list of specific skills or ideas to focus on as you practice thinking through the problems.

This section created a new toolbox of ideas to solve the types of equations that you already knew how to solve. In that sense, this section is extraneous. Why learn a new way to do something you already know how to do? But in another sense, this is a critical section because it teaches you that sometimes the way you’ve done things in the past isn’t necessarily the way you want to do things in the future.

The method of substitution works for a wide range of problems, and it follows a very straightforward pattern: Solve for one of the variables and plug it into the other equation. The method of elimination requires some intuition and intellectual flexibility. You have to actively assess the situation and determine what choice of multiplication values will cause the elimination to happen. But if you choose wisely, the calculations are much simpler. In fact, with a bit of practice you can solve some of these problems mentally (as long as the numbers aren’t too big). You would find that to be extremely difficult if you were thinking about using the method of substitution.

So consider this section to be one about simply expanding your toolbox of techniques. Instead of being confined to just one way of looking at these problems, you now have two. You have the option of looking at the equations to decide whether substitution or elimination makes more sense, and you have the freedom to pick the method that suits you the best. As you get further in your mathematical studies, you will find more and more situations where there are multiple approaches to solve a problem, and that you’ll be making more active decisions about the methods and techniques that you apply.

We will first wrap our minds around the geometric understanding of these equations. It turns out that these formulas form planes and that planes are the natural shape to generalize the "linear-ness" of lines in two variables. When we think about three simultaneous systems of equations in three variables, we’re looking for a point that lies on all three planes. Since the number of dimensions has increased, it makes sense that the number of possible arrangements would also increase. However, we still have the three same basic classifications that we had before. There will either be zero solutions, one solution, or infinitely many solutions.

In each of the systems above, there are no solutions. This may seem unusual at first, since there are clearly some points of intersection. However, we are working with simultaneous systems of equations, which means that we’re looking for points that lie on all three planes, not just pairs of planes. Notice that we cannot use the simple heuristic of just thinking about whether the planes are parallel, because the diagram on the right shows that it’s possible to have no solutions even though none of the planes are parallel to each other.

A key observation about these diagrams is that when there are intersections between planes, it forms a line. This is always true of the intersection of two planes, which is an idea we’ll come back to when we look at the algebra.

This is an example of a system with one solution. Although everything is drawn to be at right angles with each other, there is a lot of flexibility in terms of the relative positions of the planes. You should be able to visualize each pair of planes forming a line, and that each of those lines cross each other in exactly one place.

Lastly, we have these configurations to get infinitely many points of intersection. The diagram on the left has all three planes overlapping each other. The middle diagram has two planes overlapping each other with the third plane cutting across them. The last diagram has all three planes intersecting along the same line. Even though these all have infinitely many solutions, there is a distinction between the first one and the last two. Intuitively, what we’re seeing are different numbers of dimensions in the overlap. The intersection of the first diagram is a two-dimensional plane, whereas the intersection of the last two are a one-dimensional line.

When it comes to solving these systems algebraically, it’s just the methodical application of either substitution or elimination to reduce the number of variables. Conceptually, we’re going to find the intersection of planes to get lines, then find the intersection of lines to get points. When working through the process, it’s much more about keeping the work organized so that you don’t get lost.