Video: Solving Simultaneous Equations by Elimination

Video Transcript

By first multiplying the equations to make the coefficients of either 𝑥 or 𝑦 equal, use the elimination method to solve the given simultaneous equations to find 𝑥 and 𝑦: two 𝑥 plus three 𝑦 equals four; three 𝑥 plus four 𝑦 equals eight.

The elimination method looks something like this. It’s when your variable terms can cancel each other out if you add them together. So in this example that I have drawn, two 𝑥 minus two 𝑥 would be zero and the 𝑥 term would drop out. Now back to our problem, we actually don’t have any terms that are equal. Two 𝑥 and three 𝑦 can’t be added together; three 𝑦 and four 𝑥 cannot be added together. What we want to do is multiply both of these equations by numbers so that the coefficients of one of the variables, either 𝑥 or 𝑦, are equal in both of these equations.

For example, if I multiplied the equation on the left by three, then I would multiply three times two 𝑥, which would give me six 𝑥. I can then multiply the equation on the right by two; in this case we’d be multiplying two times three 𝑥, which would give us six 𝑥.

And now, we will have two equal variables that we could work with. So let’s go with three on the left side. Three times two 𝑥 equals six, three times three 𝑦 equals nine 𝑦, three times four equals 12.

And our equation on the right, we’ll multiply by two. Two times three 𝑥 equals six 𝑥, two times four 𝑦 equals eight 𝑦, two times eight equals 16.

Now we’ll take our first equation and subtract our second equation from the first. Here, we’ll take six 𝑥 from six 𝑥, which would equal zero. After that, we need to subtract eight 𝑦 from nine 𝑦. So we say nine 𝑦 minus eight 𝑦 equals one 𝑦. And finally, 12 minus 16 equals negative four. This tells us that our 𝑦-value is negative four.

But what we want to do now is to use one of the equations and plug in negative four for 𝑦. And our equation would look like this: two 𝑥 plus three times negative four equals four. We multiply three times negative four to equal negative 12. Bring down our two 𝑥 and our four. To isolate 𝑥, we’ll need to add 12 to both sides of our equation. On the left side, we’re left with two 𝑥. On the right side, four plus 12 equals 16.

Our final step to solve for 𝑥 would be to divide both sides of the equation by two. Two 𝑥 divided by two equals 𝑥; 16 divided by two equals eight. This tells us that our 𝑥-value equals eight and our 𝑦-value equals negative four.