Question Video: Solving Equations Involving Rational Functions Mathematics

Video Transcript

Solve two 𝑥 plus one over three 𝑥 plus four is equal to three over five.

To solve this equation, we’ll want to get rid of the denominators. To get rid of the five in the denominator on the right-hand side, we’ll multiply the whole right-hand side by five over one. And to keep our statement balanced, we’ll have to multiply the left-hand side by five over one. On the right side, they cancel out. And we just have three. On the left-hand side, we have five times two 𝑥 plus one over three 𝑥 plus four.

And we’ll now need to get rid of the three 𝑥 plus four in the denominator, the same way we got rid of the five in the denominator, by multiplying three 𝑥 plus four over one on both sides of the equation. On the left side they cancel out. And remaining is five times two 𝑥 plus one. And on the right side, we have three times three 𝑥 plus four. We need to distribute the five. Five times two 𝑥 equals 10𝑥. Five times one equals five. And on the right side, three times three 𝑥 equals nine 𝑥. And three times four equals 12.

When you see the instruction “solve,” we want to know for what 𝑥-value is this statement true. And that means we need to isolate our 𝑥-variable. We need to get 𝑥 by itself. And to get 𝑥 by itself, we’ll need to get 𝑥 on the same side of the equation. We have nine 𝑥 on the right side. If we subtract nine 𝑥 from both sides of the equation, 10𝑥 minus nine 𝑥 equals one 𝑥, which we usually write as 𝑥. And nine 𝑥 minus nine 𝑥 equals zero.

We now can say that 𝑥 plus five equals 12. And we’ll subtract five from both sides. 12 minus five equals seven. 𝑥 equals seven. Before we say for certain that 𝑥 equals seven, let’s plug it back in to check. We’ll have two times seven plus one over three times seven plus four. And we want to know is that equal to three over five. Two times seven equals 14 plus one equals 15. Three times seven equals 21 plus four equals 25. Does 15 over 25 reduce to three over five? Both 15 and 25 are divisible by five. 15 divided by five equals three. 25 divided by five equals five. When 𝑥 equals seven, the statement is true.