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.

2x = 3y + 4
4x = 3 - 5y

Accepted Answer

Hey everyone welcome back in this problem. We are asked to solve the following system of equations. Okay. And the system of equations were given is five X equals minus two. Y minus 37 10 X equals three y minus 137 K. And were also asked to determine if the system ends up having no solution or infinitely many solutions. Ok, Alright. So the first equation we have is five x equals -2, Y -37. And the second equation we have is 10 x equals three Y -137. All right, and it's a good idea to label your equations. It's just easier to refer to them later on. Okay. All right. So, we can solve this a number of ways. So we can use the substitution method or we can use the elimination method sometimes called the addition method. Now, in this case substitution, we're gonna have to do some dividing the numbers. Aren't the nicest for dividing. And what we see is this five X. Here, If we multiply this by two, it's going to be equal to the 10 X in the second equation. Okay, so that's a good candidate for using elimination or addition method. Okay, so let's go ahead and multiply equation one by two. Okay, and that's gonna give us 10 X. Is equal to minus or y minus 74. And then we're going to take we'll call this one star. Okay, we're gonna take equation one star and subtract equation to Okay, so we have 10 X equals minus four. Y minus 74. We're gonna subtract 10 X equals three Y minus 137. Okay. On the left hand side we're gonna have 10 minus 10 X zero On the right side -4 Y -3. Y is -7. Y. Okay and then we have minus 74 minus negative 137. That's going to be plus 100 and 37. Okay so we're gonna get plus 63 moving the seven Y to the left hand side seven Y is equal to 63 dividing gives us Y is equal to nine. So we have our Y value and that's great. Okay so we've done the elimination or addition method. It allows us to get rid of this X term and solve for y. Now we need to find the corresponding X value. Okay so we want this to be a solution for either equation. We're solving the system so we can sub this Y value into either equation. Okay so let's choose equation one, Subsequute y equals nine into equation one. Okay so we're gonna have five X. Is equal to minus two times Y. Which is nine minus 37. So five x is equal to - -37. Five, X is equal to minus 55 dividing by five gives us X equals minus 11. Okay so we have an X value and we have a y value. So our solution is going to be the set containing the point minus 9. That is our solution and that is going to correspond with. See, the solution set is the .119. 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. 2x = 3y + 4 4x = 3 - 5y