Video Transcript
If 𝑥, 𝑦 is a solution to the system 𝑦 equals negative eight 𝑥 plus 13 and 𝑦 equals 𝑥 squared minus 10𝑥 plus 13, which of the following is a possible value of 𝑥 plus 𝑦? a) Negative one, b) negative three, c) zero, or d) two.
If 𝑥, 𝑦 is a solution to the system, then 𝑥, 𝑦 is the place where these two lines intersect. It’s a coordinate that makes both of these statements true at the same time. We can find this solution by setting these equations equal to each other. If 𝑦 equals negative eight 𝑥 plus 13 and 𝑦 equals 𝑥 squared minus 10𝑥 plus 13, then we wanna know for what 𝑥-value is negative eight 𝑥 plus 13 equal to 𝑥 squared minus 10𝑥 plus 13.
First I notice that I have a positive 13 on either side of the equation. And those cancel out. So we have negative eight 𝑥 equals 𝑥 squared minus 10𝑥. And then, we wanna get all the 𝑥s on the same side of the equation. So I add eight 𝑥 to both sides. Negative eight 𝑥 plus eight 𝑥 equals zero. And negative 10𝑥 plus eight 𝑥 equals negative two 𝑥. We have two factors of 𝑥. If we pull out one factor of 𝑥, we’ll be left with 𝑥 minus two. So we have zero equals 𝑥 times 𝑥 minus two. We take each of these factors and set them equal to zero. 𝑥 equals zero. And 𝑥 minus two equals zero. Add two to both sides and we get 𝑥 equals two. This is telling us that there are two places where these two lines intersect. They intersect when 𝑥 equals zero. And they intersect when 𝑥 equals two.
Before we can find the possible values of 𝑥 plus 𝑦, we now need to solve for 𝑦. Using this first equation, 𝑦 equals negative eight 𝑥 plus 13, we need to solve when 𝑥 equals zero and when 𝑥 equals two. For the first equation, we plug in zero for 𝑥. Negative eight times zero equals zero. That term cancels out. And 𝑦 equals 13. This means that when 𝑥 equals zero, 𝑦 equals 13. This is one possible solution for 𝑥, 𝑦.
Let’s do the same thing when 𝑥 equals two. 𝑦 equals negative eight times two plus 13. Negative eight times two equals negative 16. Negative 16 plus 13 equals negative three. When 𝑥 equals two, 𝑦 equals negative three. We have a second solution for 𝑥, 𝑦 at two, negative three. What about the value of 𝑥 plus 𝑦? It could be zero plus 13. Or it could be two plus negative three. Zero plus 13 equals 13. And two plus negative three equals negative one.
Only one of these options appears in our answer choice, option a). One possible value of 𝑥 plus 𝑦 is negative one.
