The object of a number game is to combine four numbers...

The object of a number game is to combine four numbers, using addition, subtraction, multiplication, and/or division, to get the number 24. For example, the numbers 2, 5, 5, 4 can be combined as 2(5+5)+4=24. For the algebra edition of the game and the game card shown to the right, the object is to find single-digit positive integer values x and y so the found numbers x+y, 3x+2y, 8, and 9 can be combined to make 24. Use this information to answer parts (a) and (b). A game card displays the expressions x + y and 3x + 2y, and the numbers 8 and 9. (a) Using the game card, write a system of equations that, when solved, can be used to make 24 from the game card. What is the solution to this system and how can it be used to make 24 on the game card? What is a possible system? A. x+y=1 3x+2y=2 B. x+y=5 3x+2y=9 C. x+y=4 3x+2y=5 D. x+y=3 3x+2y=8 What is the solution to the system?

Transcript

00:01 Okay, so for this question, we are first asked to find a system of equations that we can basically use to represent different numbers to combine with the numbers 8 and 9 to get 24 at the end.

00:20 So basically the first part of this question, which it says you already got right.

00:25 So i'll just go through it really quickly.

00:26 But the first part of this question is basically what two other numbers can we combine in some way with 8 and 9 to get 24? and there are a lot of different numbers that we could choose.

00:42 But out of the ones we are given in our answer choices, which are either 1 and 2 for a, 5 and 9 for b, 4 and 5 for c or 3 and 8 for d, out of all of those, it's going to have to be one and two.

01:02 If you're curious, the way that we can combine these is we can do 9 times 8 is 72 divided by 1 plus 2, which is 3, and then 72 divided by 3 is 24.

01:18 So because we're able to set up an equation like this using the numbers 1 and 2 in addition to 8 and 9, that means that our first system that they give us, which is x plus y equals one, and 3x plus 2y equals two, is a possible system for x and y to be equal to.

01:53 So like i said, it already says that you chose that correctly.

01:56 So i hope that wasn't too quick explaining how that is the correct answer for that.

02:01 The second part, which it looks like is the part that you're struggling with, asks what is the solution to this specific system of equations that you chose.

02:10 And in order to do this, we're going to need to multiply our first equation by negative three.

02:22 You could do, you could do positive three.

02:24 I'm going to do negative three just because it's going to make our lives a little bit easier in the next step.

02:29 But the goal of this is to get the same number as our bottom equation.

02:39 So we could either multiply by two, positive two, three, negative three, anything to get our numbers matching in either our x values or y values.

02:48 So i'm going to go ahead and multiply by negative three.

02:51 And when i do that, we're going to get negative 3x minus 3y equals negative 3.

03:01 So now we have 3x and we have negative 3x, which is great, because if we add these two equations together, now 3x plus negative 3x is going to cancel...

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