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.

3x - 2y = − 5
4x + y = 8

Accepted Answer

Hey everyone welcome back in this problem, we are asked to solve the system of equations. Okay. And to determine if it has no solution or infinitely many solutions. If we're in either of those cases. Okay. And the system we're given is four x minus Y equals minus 13 and X plus two Y equals minus 12. Okay, so let's go ahead and label those equations so that it's easier to refer to them later. So equation one is going to be four, x minus Y equals minus 13. An equation two is going to be 11, X plus two, Y equals minus 12. All right now we haven't been told what method to use. So we could use whichever method we like we have the substitution method. Okay? We also have um the elimination or addition method that it's sometimes called. So let's look at our equation. Um Either of these would be okay, so let's go ahead and try the addition method. Okay? We see that if we multiply equation one by two then we have minus two Y. And we can eliminate the two Y from each equation. Okay, so let's take equation one and multiply it by two. Okay, we can't just multiply the single term by two. We need to multiply the entire equation by two so that it remains consistent and it has the same solutions. So we're gonna have eight X minus two. Y. Is equal to minus 26. Okay, and again we're using the addition method. So we're gonna add 11 X plus two. Y equals minus 12. Now, if we do this edition we have eight X. Plus 11 X. Well that's 19 X. Okay we have minus two Y plus two Y. That's zero. Okay that's how we designed it. We multiplied the first equation by two. So that we would get this zero. Okay now we've eliminated Y. And we can solve for X. On the right hand side we have minus 38. Okay, dividing by 19. We get X is equal to -2. Alright great. So we found an X. Value but we want to find this solution. Sorry of the system. We can't just have an X value. We need a corresponding y value. Okay so let's go just up here so we can see the rest of our work and we're gonna sub X equals minus two. Okay we're gonna stab it into either of these equations. Okay this point has to satisfy both equations. So you can choose either equation one or equation to to sub into let's choose equation one. We're gonna have four X. So four times minus two -Y equals -13. So we have minus eight minus y equals minus 13. You can move the minus Y to the right hand side and the minus 13 to the left hand side and we're gonna get y is equal to five. And so this is our corresponding y value. So now we can write the solution set as the point that we found -2 and five. So that is a solution to the system that's going to correspond with Answer C. Thanks everyone for watching. 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. 3x - 2y = − 5 4x + y = 8