Module 4: Systems of Linear Equations and Inequalities
4.3: Solving a 2×2 System of Linear Equations by Elimination
Section 4.3 Learning Objectives
4.3: Solving a 2×2 System of Linear Equations by Addition/Elimination
Solve systems of linear equations using elimination
Recognize when systems of linear equations have no solution or an infinite number of solutions
Solve a system of equations using the elimination method
The elimination method for solving systems of linear equations uses the addition property of equality. You can add the same value to each side of an equation to eliminate one of the variable terms. In this method, you may or may not need to multiply the terms in one equation by a number first. We will first look at examples where no multiplication is necessary to use the elimination method. Then you will see examples using multiplication after you are familiar with the idea of the elimination method.
It is easier to show rather than tell with this method, so let’s dive right into some examples.
If you add the two equations,
[latex]x–y=−6[/latex] and [latex]x+y=8[/latex] together, watch what happens.
Unfortunately not all systems work out this easily. How about a system like [latex]2x+y=12[/latex] and [latex]−3x+y=2[/latex]. If you add these two equations together, no variables are eliminated.
But you want to eliminate a variable. So let’s add the opposite of one of the equations to the other equation. This means multiply every term in one of the equations by [latex]-1 [/latex], so that the sign of every term is opposite.
You have eliminated the [latex]y[/latex]-variable, and the problem can now be solved. The complete solution is shown in the example after the video.
The following video describe a similar problem where you can eliminate one variable by adding the two equations together.
Caution! When you add the opposite of one entire equation to another, make sure to change the sign of EVERY term on both sides of the equation. This is a very common mistake to make.
First, note that we will find the elimination process easier if the equations are in the standard form, [latex]Ax+By=C[/latex]. So, we will start by subtracting [latex]2x[/latex] from both sides in the first equation.
[latex]3y=2x-1[/latex]
[latex]-2x+3y=-1[/latex]
Then we will distribute and add [latex]5y[/latex] to both sides in the second equation.
Notice the coefficients of each variable in each equation. If you add these two equations, the [latex]x [/latex]-term will be eliminated since [latex]−2x+2x=0[/latex]. Thus, we can add the equations and solve for [latex]y [/latex].
Notice the coefficients of each variable in each equation. You will need to add the opposite of one of the equations to eliminate the variable y, as [latex]2y+2y=4y[/latex], but [latex]2y+\left(−2y\right)=0[/latex].
Change one of the equations to its opposite (i.e. multiply by [latex]-1[/latex]), add, and solve for [latex]x[/latex]. Let’s multiply the second equation by [latex]-1[/latex].
Go ahead and check this last example—substitute [latex](2, 3)[/latex] into both equations. You get two true statements: [latex]14=14[/latex] and [latex]16=16[/latex].
Notice that you could have used the opposite of the first equation rather than the second equation and the result would be the same.
Recognize systems that have no solution or an infinite number of solutions
Just as with the substitution method, the elimination method will sometimes eliminate both variables, and you end up with either a true statement or a false statement. Recall that a false statement means that there is no solution.
This false statement leads us to conclude that the system has no solution.
Answer
No Solution
Graphing these lines shows that they are parallel lines and as such do not share any point in common, verifying that there is no solution.
If both variables are eliminated and you are left with a true statement, this indicates that there are an infinite number of ordered pairs that satisfy both of the equations. In fact, the equations are the same line.
This true statement leads us to conclude that there are an infinite number of solutions, given by all points on the line. Recall that we learned the proper way of expressing such an answer earlier in this chapter.
Answer
There are an infinite number of solutions, given by
If you add the equations above, or add the opposite of one of the equations, you will get an equation that still has two variables. So let’s now use the multiplication property of equality first. You can multiply both sides of one of the equations by a number that will allow you to eliminate the same variable in the other equation.
We do this with multiplication. Notice that the first equation contains the term [latex]4y [/latex], and the second equation contains the term y. If you multiply the second equation by [latex]−4 [/latex], when you add both equations the [latex]y [/latex]–variable terms will add up to [latex]0 [/latex].
The following example takes you through all the steps to find a solution to this system.
Caution! When you use multiplication to eliminate a variable, you must multiply EACH term in the equation by the number you choose. Forgetting to multiply every term is a common mistake.
There are other ways to solve this system. Instead of multiplying one equation in order to eliminate a variable when the equations were added, you could have multiplied both equations by different numbers.
Let’s eliminate the variable [latex]x [/latex]this time. We can achieve this by multiplying the first equation by 5 and the second equation by [latex]−3[/latex].
In order to use the elimination method, you have to create variables that have the same coefficient in absolute value, but opposite signs—then you can eliminate them by adding the equation. Multiply the top equation by 5.
These equations were multiplied by 5 and [latex]−3[/latex] respectively, because that gave you terms that would add up to 0. Be sure to multiply all of the terms of the equation.
In the following video, you will see an example of using the elimination method for solving a system of equations.
The next example reminds us that sometimes we must deal with fractions.
While we could try to jump into the elimination process immediately, you may find it easier to first clear out the fractions by multiplying each equation by its LCD.
So, coincidentally, we also obtain a fractional answer. We now plug this value in to find [latex]x[/latex]. For simplicity, we can plug into one of the two equations we obtained after eliminating the fractions.
We conclude that the solution to the system is the ordered pair [latex]\left(\frac{3}{2},-\frac{1}{3}\right)[/latex].
Answer
The solution is [latex]\left(\frac{3}{2},-\frac{1}{3}\right)[/latex].
It is possible to use the elimination method with multiplication and get a result that indicates no solutions or infinitely many solutions, just as we saw in simpler systems earlier. In the following example, you will see a system that has infinitely many solutions.
Does this kind of solution look familiar? This implied that there were an infinite number of solution. If we solve both of these equations for [latex]y[/latex], you will see that they are the same equation.
In the following video, the elimination method is used to solve a system of equations. Notice that one of the equations needs to be multiplied by a negative one first. Additionally, this system has an infinite number of solutions.
Summary
Combining equations is a powerful tool for solving a system of equations. Adding or subtracting two equations in order to eliminate a common variable is called the elimination (or addition) method. Once one variable is eliminated, it becomes much easier to solve for the other one.
Multiplication can be used to set up terms that can be eliminated in the equations before they are combined to aid in finding a solution to a system. When using the multiplication method, it is important to multiply all the terms on both sides of the equation—not just the one term you are trying to eliminate. Solving using the elimination method will yield one of three results: a single value for each variable within the system (indicating one solution), an untrue statement (indicating no solutions), or a true statement (indicating an infinite number of solutions).
Caution! When you add the opposite of one entire equation to another, make sure to change the sign of EVERY term on both sides of the equation. This is a very common mistake to make.
