If 𝑦 is three times as large as 𝑥 and 𝑦 minus 𝑥 equals eight, what is the value of 𝑦?
First, let’s take the phrase 𝑦 is three times as large as 𝑥 and turn it into an equation. 𝑦 is three times as large as 𝑥. We can represent that as 𝑦 equals three 𝑥. And now we have two equations 𝑦 equals three 𝑥 and 𝑦 minus 𝑥 equals eight. We want both of these equations to be true, so we can set them equal to each other.
We’ll take our second equation and get it into the format of 𝑦 equals something. To do that, we’ll add 𝑥 to both sides. And our second equation will then say 𝑦 equals eight plus 𝑥. We’re interested in the case when 𝑦 equals three 𝑥 and 𝑦 equals eight plus 𝑥 are both true. So, we set them equal to each other. Three 𝑥 equals eight plus 𝑥.
And now solving for 𝑥, we subtract 𝑥 from both sides. Three 𝑥 minus 𝑥 equals two. And 𝑥 minus 𝑥 cancels out. To get rid of two 𝑥, we’ll divide both sides by two, which leaves us on the left with 𝑥. And eight divided by two equals four. These two statements are true when 𝑥 equals four. And we need to plug that 𝑥-value in to find our 𝑦 since our question has asked what the value of 𝑦 is.
𝑦 equals three 𝑥, and 𝑥 equals four. Three times four is 12. Our 𝑦 equals 12. We can check our answer here by plugging in what we found for 𝑥 and 𝑦 into the equation 𝑦 minus 𝑥 equals eight. 12 minus four is equal to eight. And 12 is three times larger than four. 𝑦 equals 12.
Let’s consider a second way to solve this problem. We’ll still need our two equations. Notice how with our second equation the first time we solved for 𝑦. This time we’re going to solve our second equation for 𝑥. To solve for 𝑥, we’ll need to subtract 𝑦 from both sides, which leaves us with negative 𝑥 equals eight minus 𝑦. To get rid of that negative in front of our 𝑥, we’ll multiply the whole equation by negative one.
Negative one times negative 𝑥 equals 𝑥. Negative one times eight equals negative eight. Negative one times negative 𝑦 equals positive 𝑦. 𝑥 equals negative eight plus 𝑦. Or we can rewrite that to say 𝑥 equals 𝑦 minus eight. This time we know what 𝑥 equals. So, we take what we found for 𝑥 and we plug it into our equation one.
We know that 𝑥 equals 𝑦 minus eight. So, when we plug that back in, our new equation says 𝑦 equals three times 𝑦 minus eight. Distribute the three, and you get three 𝑦. Three times negative eight is negative 24. 𝑦 equals three 𝑦 minus 24. To solve for 𝑦, we’ll subtract three 𝑦 from both sides. One 𝑦 minus three 𝑦 equals negative two 𝑦. And negative two 𝑦 equals negative 24. Now we’ll divide both sides by negative two. 𝑦 equals 12. Negative 24 divided by negative two equals positive 12.