College Preparatory Mathematics

Section 4.3 Addition/Elimination

When solving systems we have found that graphing is very limited when solving equations. We then considered a second method known as substituion. This is probably the most used idea in solving systems in various areas of algebra. How- ever, substitution can get ugly if we don’t have a lone variable. This leads us to our second method for solving systems of equations. This method is known as either Elimination or Addition. We will set up the process in the following examples, then define the five step process we can use to solve by elimination.

In the previous example one variable had opposites in front of it, \(- 4y\) and \(4y\text{.}\) Adding these together eliminated the y completely. This allowed us to solve for the \(x\text{.}\) This is the idea behind the addition method. However, generally we won’t have opposites in front of one of the variables. In this case we will manipulate the equations to get the opposites we want by multiplying one or both equations (on both sides!). This is shown in the next example.

When we looked at the \(x\) terms, \(- 6x\) and \(2x\) we decided to multiply the \(2x\) by \(3\) to get the opposites we were looking for. What we are looking for with our opposites is the least common multiple (LCM) of the coefficients. We also could have solved the above problem by looking at the terms with \(y\text{,}\) \(5y\) and \(3y\text{.}\) The LCM of \(3\) and \(5\) is \(15\text{.}\) So we would want to multiply both equations, the \(5y\) by \(3\text{,}\) and the \(3y\) by \(- 5\) to get opposites, \(15y\) and \(- 15y\text{.}\) This illustrates an important point, some problems we will have to multiply both equations by a constant (on both sides) to get the opposites we want.

It is important for each problem as we get started that all variables and constants are lined up before we start multiplying and adding equations. This is illustrated in the next example which includes the five steps we will go through to solve a problem using elimination.

World View Note: The famous mathematical text, The Nine Chapters on the Mathematical Art, which was printed around \(179\) AD in China describes a for- mula very similar to Gaussian elimination which is very similar to the addition method.

Just as with graphing and substution, it is possible to have no solution or infinite solutions with elimination. Just as with substitution, if the variables all disappear from our problem, a true statment will indicate infinite solutions and a false stat- ment will indicate no solution.

We have covered three different methods that can be used to solve a system of two equations with two variables. While all three can be used to solve any system, graphing works great for small integer solutions. Substitution works great when we have a lone variable, and addition works great when the other two methods fail. As each method has its own strengths, it is important you are familiar with all three methods.