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Video: Solving Equations Involving Rational Functions

Bethani Gasparine

Solve 𝑦 = 2𝑥 + 1/3𝑥 + 4 with an expression for 𝑥 in terms of 𝑦.


Video Transcript

Solve 𝑦 equals two π‘₯ plus one divided by three π‘₯ plus four with an expression for π‘₯ in terms of 𝑦.

Essentially, this means we need to take our equation and solve for π‘₯. So the first thing that we should do is we should get rid of our denominator. The best way to do that is to multiply both sides by your denominator of three π‘₯ plus four. Again, multiplying both sides by three π‘₯ plus four, we’ll eliminate our denominator on the right-hand side of the equation. So we have three π‘₯ plus four times 𝑦 is equal to two π‘₯ plus one. Let’s go ahead and use the distributive property on the left-hand side. So we will distribute 𝑦 to three π‘₯ and 𝑦 to four. When we say distribute, that means we’re actually multiplying. We can see that we’ve a three π‘₯𝑦 term, a four 𝑦 term, and a two π‘₯ term.

Since we essentially are solving for π‘₯, let’s go ahead and get all of the terms that have an π‘₯ in it on one side of the equation. So we will go ahead and subtract three π‘₯𝑦 to both sides of the equation. On the left-hand side, the three π‘₯𝑦s cancelled. And on the right, the three π‘₯𝑦 cannot be combined with either of the two terms, two π‘₯ or one. So we just go ahead and just add it to that side of the equation. Now when we say add, it is a negative, so it’s still a minus three π‘₯𝑦. Now let’s go ahead and move the plus one that’s on the right-hand side of the equation over to the left side. So let’s subtract one from both sides of the equation.

On the left-hand side, four 𝑦 and minus one cannot be combined. And on the right-hand side, the ones cancelled. Now we can see on the right-hand side, we’re getting closer to just having an π‘₯ on that side. Let’s go ahead and move this. So that way, we have some more room. So on the right-hand side, we can take out a factor of π‘₯ since both terms have an π‘₯. So that turns into π‘₯ times two minus three 𝑦. So in order to isolate π‘₯, we can divide both sides by two minus three 𝑦. On the right-hand side, the two minus three 𝑦s cancel. And on the left, we can leave it.

Four 𝑦 minus one divided by two minus three 𝑦 does not reduce. This means our final answer is: π‘₯ equals four 𝑦 minus one divided by two minus three 𝑦.