# Use the Elimination Method to Solve Systems of Linear Equations

The main concept behind the **elimination method** is to create terms with opposite coefficients because they cancel each other when added. In the end, we should deal with a simple linear equation to solve, like a one-step equation in x or in y.

## Two Ideal Cases of the Elimination Method

I can summarize the “big” ideas about the elimination method when solving systems of linear equations using the illustrations below. Here I present two ideal cases that I want to achieve during the solving process. Take a look at them and hopefully, it makes sense. Otherwise, go directly to the six (6) worked examples to see how actual problems are being solved.

__Case 1__: By Adding the Two Equations, the Variable “X” is Eliminated

The coefficients of **variable x** are opposites.

__Case 2__: By Adding the Two Equations, the Variable “Y” is Eliminated

The coefficients of **variable y** are opposites.

### Examples of Solving Systems of Linear Equations by Elimination Method

**Example 1:** Solve the system of linear equations by elimination method .

I have observed that adding the x-column will not eliminate the variable x. However, if I add the y-column the variable y disappears. This happens because the coefficients of y are opposite of each other in terms of signs. Now, I will proceed with the second option.

After doing so, I end up with an easy equation.

I divide both sides by the coefficient of x which gives the answer of **x = 4**.

The next step is to find the corresponding value of y. This is easy to find since I already know what x is. I will pick any of the two original equation, which in this case, I chose the top equation. Then, I will plug in the value of x = 4 to get y. The process or procedure of solving for y should be similar below.

Here I get **y = −4**. The final answer in point notation is shown below.

Graphically, the solution looks like this.

**Example 2: **Solve the system of linear equations by elimination method .

This is quite interesting because no variables will cancel when added. What I want is to introduce a multiplier to one of the equations, or both, and then observe if I arrive at some coefficients that only differ in signs.

There are a

Multiplying the entire equation by any nonzero number **does not change** its original meaning. What will change is just its form. I call it equation “revision” or “modification”.

This is a one-step equation so I solve y by dividing both sides by its coefficient.

Great! I obtained the value **y = 2**. Next, I will solve x using back substitution using either of the original equations. For this, I will utilize the top equation because it is less complicated.

I obtained the value **x = −1**. I can now write the final answer as the ordered pair written below .

The graph below verifies that our solution is correct.

**Example 3:** Use the method of elimination or linear combination to solve .

There’s some twist on this problem because the coefficients of x variables are exactly the same, **both −2**. The only thing I need to fix here is to make one of them positive. Now, I decide to multiply the top equation by −1. It should also work just fine if I multiply the bottom by −1.

You should see that the plan works since adding the x-column results to the cancellation of x.

I solved the value of y by dividing both sides by −17 which results to **y = 3**. This time, I will back solve the value of x using the bottom equation because I know what y is.

After a few steps in solving the equation above, I arrive at** x = 2**. The final answer as an ordered pair is shown below .

Indeed, the two lines intersect at the point that we found in our calculations.

**Example 4:** Use the method of elimination or linear combination to solve .

This example follows along the line of example 3 where we have exactly the same coefficients. I see that variable y both have coefficients of 8. So, I will need to tweak it a bit in order to make their signs opposite. I now have two options on how to proceed. I can multiply the top equation by −1 or the bottom by −1 as well. For this exercise, I choose to do the latter.

Applying the −1 multiplier on the bottom equation and adding them together results to y going away.

Solve the simple equation that arises from it.

I got** x = 4** by dividing both sides by −9. The next obvious step is to solve for the other variable y using back substitution. Pick any of the original equations, plug x = 4, and you will get y in no time.

The answer is **y = −1**. The final answer in the ordered pair form is shown below .

The graphical solution looks like this.

**Example 5: **Use the method of elimination or linear combination to solve .

This type of problem requires us to simultaneously multiply both the top and bottom equations by some number in order to generate coefficients with opposite signs.

If I decide to eliminate x, I can multiply the top equation by −2 and the bottom by 9. By doing so, I should end up with x terms, 18x and −18x, respectively, which would cancel when added together.

For this exercise, I want to eliminate y. Therefore, I will multiply the top by 5 and the bottom by 3.

As predicted, I was able to get rid of y which leaves us with a simple equation to deal with.

You should arrive at ** x = 1**. Proceed on solving the other variable which is y. Plug in x = 1 then solve for y.

I got **y = −2**. Putting it together, our final answer is the order pair below** .**

The graphical representation of the two lines intersecting at the solved point is…

**Example 6: **Use the method of elimination or linear combination to solve .

This last example is very similar to the previous one. As it stands, no variables will be eliminated after adding the columns of x and y. However, I can eliminate the x variables by multiplying the first equation by 5 and the second by − 4, and then add them together. The rest is history!

I will end up solving a simple equation as shown.

I now have **y = 5** after dividing both sides by 8. I will then substitute this value of y to any of the original equations to solve for the corresponding x-value.

This yields an answer of **x = − 6**. The final answer should be \left( {x,y} \right) = \left( { - \,6,5} \right) .

This point is where the two lines intersect, as shown below.

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