Elimination was used to solve a system of equations. One of the intermediate steps led to the equation 9x=27 . Which of the following systems could have led to this equation?
4x+3y=24
−5x−3y=3
7x−2y=15
x+y=6
9x+2y=21
−9x−2y=21
10x−y=15
x+y=−12
7x−2y=15
x+y=6
Multiply second equation by two and then add the equations.
Well, let's see here. We're looking for a system of equations that, when solved using elimination, would lead to the equation 9x=27.
Looking at the first option, 4x+3y=24 and −5x−3y=3, if we add these equations together to eliminate the y terms, we get -x = 27. So that doesn't match.
Moving on to the second option, 7x−2y=15 and x+y=6. If we multiply the second equation by 2 and add it to the first equation, we get 9x=27. Well, hot dog! We have a winner!
Let's keep going just for kicks. In the third option, 9x+2y=21 and −9x−2y=21, if we add these equations together, we get 0 = 42. Nope, doesn't work.
Finally, in the fourth option, 10x−y=15 and x+y=−12, no matter how we manipulate these equations, we won't be getting a 9x=27 situation.
So, the second option, 7x−2y=15 and x+y=6, is the only system that could have led to the equation 9x=27. Looks like we have a solution! Who's the mathlete now?
The system of equations that could have led to the equation 9x = 27 is:
9x + 2y = 21
-9x - 2y = 21
To use elimination to solve a system of equations, the goal is to eliminate one variable by adding or subtracting the equations. Let's consider each system of equations and see if any of them could have led to the equation 9x=27.
Option 1:
4x + 3y = 24
-5x - 3y = 3
In this system, adding the two equations together will lead to the elimination of the y variable:
(4x + 3y) + (-5x - 3y) = 24 + 3
- x = 27
So, this system could not have led to the equation 9x = 27.
Option 2:
7x - 2y = 15
x + y = 6
In this system, if we multiply the second equation by 2, we can eliminate the y variable by adding the equations together:
7x - 2y + 2x + 2y = 15 + 12
9x = 27
So, this system could have led to the equation 9x = 27.
Option 3:
9x + 2y = 21
-9x - 2y = 21
In this system, if we add the two equations together, the y variable will be eliminated:
(9x + 2y) + (-9x - 2y) = 21 + 21
0 = 42
So, this system could not have led to the equation 9x = 27.
Option 4:
10x - y = 15
x + y = -12
In this system, if we add the two equations together, the y variable will be eliminated:
(10x - y) + (x + y) = 15 + (-12)
11x = 3
So, this system could not have led to the equation 9x = 27.
In summary, the only system that could have led to the equation 9x = 27 is:
7x - 2y = 15
x + y = 6