In Exercises 31–42, solve by the method of your choice. Identify ...

Question

In Exercises 31–42, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.

x + 3y = 2
3x + 9y = 6

Accepted Answer

Hey everyone welcome back in this problem. We are asked to solve the system of equations. Okay. And the system of equations were given is six X plus seven Y equals 15, 24 X plus 28 Y is equal to 60. Okay, so writing down our equations six X plus seven Y equals 15. An equation to will be 24 X plus 28 Y equals 60. Okay. We just labeled our equations as a good practice so that when you're using them later on, you can keep track of what's going on. All right now, we haven't been told how to solve this so we can solve whichever way we'd like. We could use substitution method. We can use the elimination or addition method. Sometimes it's called. Okay, so we could use either of those. Looking at our equations. The substitution method will have to do some dividing the numbers. Aren't the nicest. That will work out fine. But let's go ahead and try the elimination method instead. What we see is that this 24 X. Here, we can turn this six X into 24 X. By multiplying the equation by four. Okay, that's gonna allow us to eliminate X. So if we take equation one, we multiply it by four, We're going to get 24x. Okay, we can't just multiply that first term we have to multiply the entire equation so that we keep the same solutions. We're gonna have plus 28 y. is equal to 60. Okay, Now let's go ahead and subtract the second equation 24 x Plus 28 Y equals 60. What you'll notice is that these equations are actually the same. So when we do the subtraction on the left hand side, we're going to have zero plus zero. On the right hand side, 60 minus 60 is zero. Okay, so we get zero equals zero. Well, this is always true. Okay, this is a true statement. No matter what X and Y value you choose to put into this equation zero equals zero is true. That means that we're gonna have infinitely many solutions. Ok so we have infinitely many solutions. Now, how do we write the solution set if there are infinitely many solutions? Well, we know that the solutions are going to be along this equation there along this line. Given by the equations in the question. These are multiples of each other. So any multiple of this equation will satisfy will will be the solution. Okay, so looking at the problems or the answers that we've given, we have six X plus 17. Y equals 15. Well, that's not our equation. We have six X plus seven. Y. Okay, so it's not going to be this one. But this 1 24 X plus 28 Y equals 60. That is our equation K. We can write our solution set like that. So we have our solution set is any point X. Y. That satisfies this equation 24 X plus 28 Y equals 60. Okay, our solution is B It's gonna be the set of the points that satisfied 24 X plus 28 Y equals 60. Okay, that's it for this one. See you in the next video.

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In Exercises 31–42, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. x + 3y = 2 3x + 9y = 6