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Video: Solving a System of Two Linear Equations Simultaneously

Kathryn Kingham

Solve the simultaneous equations 𝑥 + 4𝑦 = 17 and 2𝑥 + 7𝑦 = 5.


Video Transcript

Solve the simultaneous equations π‘₯ plus four 𝑦 equals 17 and two π‘₯ 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 π‘₯ and two π‘₯; 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 π‘₯. How would I make my first equation also have two π‘₯ in it? If I multiply the whole first equation by two, we’ll end up with two π‘₯. We distribute our two to the π‘₯. Two times four 𝑦 equals eight 𝑦. Two times 17 equals 34. And now both our first and second equation have two π‘₯ in it.

Here’s what we do now. We take our first equation, two π‘₯ plus eight 𝑦 equals 34, and we subtract our second equation, two π‘₯ plus seven 𝑦 equals five, from the first equation. For the first term, two π‘₯ minus two π‘₯ 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 π‘₯. I can choose any of the equations we’ve been working with to solve for 𝑦, but I’m gonna choose π‘₯ plus four 𝑦 equals 17. Here, I plug in 29 for 𝑦, and we can solve for π‘₯. Four times 29 equals 116. Bring down the rest of our equation. To get π‘₯ by itself we’ll subtract 116 from both sides of the equation leaving us with π‘₯ on the left and negative 99 on the right.

Our final answer is then: π‘₯ equals 99; 𝑦 equals 29.