Question

MEDAL AND FAN!
A system of equations is shown below.

-3x + 7y = -16
-9x + 5y = 16

Part A: Create an equivalent system of equations by replacing one equation with the sum of that equation and a multiple of the other. Show the steps to do this. (6 points)

Part B: Show that the equivalent system has the same solution as the original system of equations. (4 points)

Answer 1

Replace (2) by adding (2) to two times (1):
2 times (1): -6x + 14y = -32
Add (2): -9x + 5y = 16
--------------------
-15x + 19y = -16

So the second pair of equations are:
-3x + 7y = -16 ---- (1)
-15x + 19y = -16 ----- (3)

Answer 2

@SolomonZelman can you help me with part B please?

Answer 3

@Michele_Laino Can you please help me?

Answer 4

@Secret-Ninja can you help me with part B?

Answer 5

ok! please now you have to solve the second part

Answer 6

how do I do that?

Answer 7

another solution, can be this:
I multiply your first equation by (-3) and then I add it to the second equation, namely:
(-3) times (1)--->9x-21y=48
add (2)-------> -9x+5y=16
------------------
-16y=64
SO the second pair of equations are:
-3x+7y=-16
-16y=64

Answer 8

Wait so my part A wasn't right?

Answer 9

no, no, it's right!

Answer 10

Oh :o so what did u just do right there? Was that a different solution to part A?

Answer 11

what I want you to understand is that the purpose of manipulating the equations of a system is to obtain the simplified equations

Answer 12

Okays I understand how manipulating equations of a system can obtain simplified equations but how do I apply that to part b?

Answer 13

for example, both results, namely mine and yours are correct, even if, by my answer we have the second equation already simplified, and we can solve it rapidly

Answer 14

solution of second system is:
x=-4, y=-4
do you agree?

Answer 15

No the second system is -3x+7y=-16 and -15x+19y=-16

Answer 16

where did you get x=-4 and y=-4?

Answer 17

please, I refer to my second system!

Answer 18

Oh! Your second system

Answer 19

:D I get it! Since you simplified 64 and -16

Answer 20

that's right!

Answer 21

now you have to solve your first system, and then discover that the resultant solution is equal to that of the second system

Answer 22

I know I need to Solve (1) and (2) for x,y and
Solve (1) and (3) for x,y

Answer 23

I solve your first system:
from first equation, we have:
\[x=\frac{ 7y +16}{ 3}\]
inserting that expression, into your second equation, we have:
\[-9*\frac{ 7y+16 }{ 3 }+5y=16\]
I simplify:
\[-16y=64\]
from which
y=-4
Now I substitute y=-4 into the expression for x, namely:
\[x=\frac{ 7y+16 }{ 3 }=\frac{ 7*(-4)+16 }{ 3 }=\frac{ -12 }{ 3 }=-4\]
then solution of your first system is
x=-4, y=-4
which is equal to solution of my second system.
In other words
your first system and my second system are equivalent

Answer 24

Thank you @Michele_Laino For all your help ^_^

Answer 25

Thank you @Sandybottoms1432