SOLUTION: Hi! I have been trying to use old, retired ACT exams to prepare to take the actual test, and in checking my work to find what I need to improve on I've run into a problem that I

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Question 1158969: Hi!
I have been trying to use old, retired ACT exams to prepare to take the actual test, and in checking my work to find what I need to improve on I've run into a problem that I have solved but don't quite understand.
"The solution of the system of equations below is the set of all (x,y)such that 2x-3y=6. What is the value of k?
18x-27y=54
6x+ky=-2k
A.-9
B.-1
C.3
D.6
E.9
I solved by substituting each of the options given for k in the bottom equation, and then using the elimination method for solving systems of equations. However, it was very time consuming and I did not understand how the equation in the paragraph fit in with the other two equations. The answer was A. I did not understand why the answer that made the two equations come out to 0 was correct. I did not even use the equation in the paragraph to solve, and I only got the answer correct when fixing my work later, not when taking the test (I simulated an actual test as much as I could by timing myself). Is there a faster, more accurate method? Perhaps using the calculator? I do own a graphing calculator, but it is not a TI-84.
I hope this makes sense. Thank you for taking the time to help me!
Anna McLeran
Found 3 solutions by josgarithmetic, MathLover1, MathTherapy:
Answer by josgarithmetic(39617)   (Show Source): You can put this solution on YOUR website!
-------------------------------
"The solution of the system of equations below is the set of all (x,y)such that 2x-3y=6. What is the value of k?
18x-27y=54
6x+ky=-2k
------------------------------

The first equation of the system


---------the same equation refered in the descriptive part before your listing of the equations of the system.

Your equation using k shows a coefficent on x being 6. The way to get the simplified equation to have this same coefficient is multiply 2x-3y=6 by 3. This gives you:
.

Compare the corresponding parts.

This shows that .
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

.......both sides multiply by
---------------------


--------------------------subtract

........group
...factor out common



=>
->
=>->
find





so, check your system:


-> true


->true


Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

Hi!
I have been trying to use old, retired ACT exams to prepare to take the actual test, and in checking my work to find what I need to improve on I've run into a problem that I have solved but don't quite understand.
"The solution of the system of equations below is the set of all (x,y)such that 2x-3y=6. What is the value of k?
18x-27y=54
6x+ky=-2k
A.-9
B.-1
C.3
D.6
E.9
I solved by substituting each of the options given for k in the bottom equation, and then using the elimination method for solving systems of equations. However, it was very time consuming and I did not understand how the equation in the paragraph fit in with the other two equations. The answer was A. I did not understand why the answer that made the two equations come out to 0 was correct. I did not even use the equation in the paragraph to solve, and I only got the answer correct when fixing my work later, not when taking the test (I simulated an actual test as much as I could by timing myself). Is there a faster, more accurate method? Perhaps using the calculator? I do own a graphing calculator, but it is not a TI-84.
I hope this makes sense. Thank you for taking the time to help me!
Anna McLeran
18x - 27y = 54 ----- eq (i)

6x + ky = - 2k ----- eq (ii)

18x + 3ky = - 6k --- Multiplying eq (ii) by 3 ----- eq (iii)

18x - 27y =   54 --- eq (i)

Looking at eqs (i) & (iii), and EQUATING terms, you'll see that 3ky = - 27y

Therefore, 

You DON'T have to go any further, but just for the fun of it, when you EQUATE the right-side. you'll also see that - 6k = 54. 

Then, we get:  

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