### Video Transcript

If π¦ is three times as large as π₯ and π¦ minus π₯ equals eight, what is the value of π¦?

First, letβs take the phrase π¦ is three times as large as π₯ and turn it into an equation. π¦ is three times as large as π₯. We can represent that as π¦ equals three π₯. And now we have two equations π¦ equals three π₯ and π¦ minus π₯ equals eight. We want both of these equations to be true, so we can set them equal to each other.

Weβll take our second equation and get it into the format of π¦ equals something. To do that, weβll add π₯ to both sides. And our second equation will then say π¦ equals eight plus π₯. Weβre interested in the case when π¦ equals three π₯ and π¦ equals eight plus π₯ are both true. So, we set them equal to each other. Three π₯ equals eight plus π₯.

And now solving for π₯, we subtract π₯ from both sides. Three π₯ minus π₯ equals two. And π₯ minus π₯ cancels out. To get rid of two π₯, weβll divide both sides by two, which leaves us on the left with π₯. And eight divided by two equals four. These two statements are true when π₯ equals four. And we need to plug that π₯-value in to find our π¦ since our question has asked what the value of π¦ is.

π¦ equals three π₯, and π₯ equals four. Three times four is 12. Our π¦ equals 12. We can check our answer here by plugging in what we found for π₯ and π¦ into the equation π¦ minus π₯ equals eight. 12 minus four is equal to eight. And 12 is three times larger than four. π¦ equals 12.

Letβs consider a second way to solve this problem. Weβll still need our two equations. Notice how with our second equation the first time we solved for π¦. This time weβre going to solve our second equation for π₯. To solve for π₯, weβll need to subtract π¦ from both sides, which leaves us with negative π₯ equals eight minus π¦. To get rid of that negative in front of our π₯, weβll multiply the whole equation by negative one.

Negative one times negative π₯ equals π₯. Negative one times eight equals negative eight. Negative one times negative π¦ equals positive π¦. π₯ equals negative eight plus π¦. Or we can rewrite that to say π₯ equals π¦ minus eight. This time we know what π₯ equals. So, we take what we found for π₯ and we plug it into our equation one.

We know that π₯ equals π¦ minus eight. So, when we plug that back in, our new equation says π¦ equals three times π¦ minus eight. Distribute the three, and you get three π¦. Three times negative eight is negative 24. π¦ equals three π¦ minus 24. To solve for π¦, weβll subtract three π¦ from both sides. One π¦ minus three π¦ equals negative two π¦. And negative two π¦ equals negative 24. Now weβll divide both sides by negative two. π¦ equals 12. Negative 24 divided by negative two equals positive 12.