Equivalent expressions | Worked example

- [Instructor] Which of the following expressions are equivalent to three x minus 20 for all values of x, and it says choose all answers that apply and they give us three choices. So like always, pause this video and see if you can work through it on your own before we work through it together. All right, now the way that I would approach this is I would try to manipulate these expressions, and see if I can make it equivalent to this expression. Another way that you could do it that might involve a little bit less algebra is to just try out values for x, and see what three x minus 20 would be equal to, and try out these values. Now, that might be a little bit more spotty, depends on what values you pick. So let me do it the first way, and then we could do it the second way if you're interested. It's not as robust. So the first way is let's just rewrite this and try to write it in this form. So I would first try to distribute the three over the x minus seven, so that's going to be equal to three times x, which is three x, minus three times seven, which is minus 21. So all I did is I distributed the three on the x minus seven, and that's three x minus 21, and then we have plus one. Now, what is negative 21 plus one? Well, that's going to be equal to negative 20, so this is going to be equal to three x minus 20, right, which is exactly what we originally had. And since we did it algebraically, we know that it works for all values of x, so we like this choice. Now let's look at this one. So once again, we would probably want to start by distributing the three, so three times x plus one, then minus 23, this is going to be the same thing as three times x plus three times one, so plus three, and then we're going to subtract 23, minus 23. Now, what is three minus 23? Well, once again, we have to make sure that we're doing arithmetic with our negative numbers appropriately, but this is going to be negative 20. So this is once again going to be three x minus 20. Looks correct again, so I like that choice. Now let's try this one. So here, we have seven and then we have three times this expression, so we can distribute the three here. So this is going to be equal to seven plus, and then three times x, I'm just distributing the three now, minus three times nine, so minus three times nine is 27, and so let's see, I can rearrange these, so I could rewrite it as, this is the same thing as three x minus 27 plus seven. Now, what's negative 27 plus seven? Well, that once again is going to be negative 20, so this whole thing is going to be equal to three x minus 20. So actually, all of these answers apply. Now, another way that you could've done it as I mentioned that might be, that actually would be less robust, is to just try out values. So you said, okay, let me just see what happens when x is equal to one. So if x equal to one, then this one right over here would be three times one minus 20, which would be negative 17, and then over here, you would have three times one minus seven, so that would be negative six, and then three times negative six is negative 18, plus one would be negative 17, so that would actually look pretty good for just the case of x equals one, but once again, that wouldn't necessarily make you feel good for all x's, and you would be able to do the same thing with choice b and c, and you would get back to negative 17, but once again, this method is not as robust as just being able to algebraically manipulate these choices and see if you can make them equivalent to three x minus 20.