Solve the simultaneous equations 𝑥 plus four 𝑦 equals 17 and two 𝑥 plus
seven 𝑦 equals five.
We wanna solve this problem by elimination. But in elimination, you need the same amount of one of your variables. Here, we have one 𝑥 and two 𝑥; that won’t work. We have four 𝑦 and seven 𝑦. So it seems like the elimination wouldn’t work here. But, we can use multiplication as a tool to help us here.
My second equation has two 𝑥. How would I make my first equation also have two
𝑥 in it? If I multiply the whole first equation by two, we’ll end up with two 𝑥. We distribute our two to the 𝑥. Two times four 𝑦 equals eight 𝑦. Two times 17 equals 34. And now both our first and second equation have two 𝑥 in it.
Here’s what we do now. We take our first equation, two 𝑥 plus eight 𝑦 equals
34, and we subtract our second equation, two 𝑥 plus seven 𝑦 equals five, from the
first equation. For the first term, two 𝑥 minus two 𝑥 equals zero. Eight 𝑦 minus seven 𝑦 equals one 𝑦. And finally, 34 minus five equals 29. 𝑦 equals 29.
We can use this information, 𝑦 equals 29, to help us find 𝑥. I can choose any of the equations we’ve been working with to solve for 𝑦, but
I’m gonna choose 𝑥 plus four 𝑦 equals 17. Here, I plug in 29 for 𝑦, and we can solve for 𝑥. Four times 29 equals 116. Bring down the rest of our equation. To get 𝑥 by itself we’ll subtract 116 from both sides of the
equation leaving us with 𝑥 on the left and negative 99 on the right.
Our final answer is then: 𝑥 equals 99; 𝑦 equals 29.