Question

MEDAL AND FANS :)
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

-4x+16y=12 would this equation count for part a?

Answer 2

@phi

Answer 3

how did you get your new equation?

Answer 4

i got -4 because it is a multiple of -16 and 16 is the sum of one of the equations i just plugged them in

Answer 5

@phi are you still there?

Answer 6

call the first equation A and the second equation B

they want you to
replacing one equation with the sum of that equation and a multiple of the other.

in other words, if you pick A to replace, they want you to find a new equation (call it C) equal to
A + nB where n is a number.

if you pick n=1 that makes it easy: the new equation C= A+B
can you do that ?

Answer 7

yeah thanks is that all for part a?

Answer 8

what did you get?

Answer 9

umm hold on

Answer 10

-4+20=16

Answer 11

the equations have variables x and y
they (usually) never go away.

here is how to add the two equations:
-3x + 7y = -16
-9x + 5y = 16

you add the left hand sides and the right hand sides:
-3x + 7y + -9x + 5y = -16+16

Answer 12

you can simplify that. For example notice you have 7 y's and 5 more y's
how many y's do you have on the left hand side?

Answer 13

12

Answer 14

you write 12y (remember we are counting how many y's)
we also have -3x - 9x
can you simplify that ?

Answer 15

-12x

Answer 16

so you now have
-12x + 12y = -16+16
simplify the right side: -16+16 is 0.
we get
-12x + 12y = 0

notice if we start with
-3x + 7y = -16
-9x + 5y = 16
we could just "add the columns"
-12x + 12y = 0

Answer 17

thank you so much :) can you also help with part b

Answer 18

part B uses the same idea of adding equations.

we have to solve the problem two ways. The first way is with the original equations
-3x + 7y = -16
-9x + 5y = 16

try this: multiply the first equation by -3. that means multiply both sides and *all terms* each by -3. can you do that ?

Answer 19

yeah 9x+-21y=-48

Answer 20

-3*-16 is not -48

Answer 21

oh positive sorry

Answer 22

yes. when we do these there are painful if you make a simple mistake. they take a lot of work

Answer 23

yes!!!!!!

Answer 24

so we now have these equations (the first multiplied by -3, the 2nd as it was)
9x + -21y= 48
-9x + 5y = 16

add the two equations. what do you get ?

Answer 25

-16y=64

Answer 26

is that all @phi

Answer 27

notice that the 9x + -9x added to zero, and the x "went away"
(that is why I picked -3 to multiply the first equation ... I wanted a +9x)

to solve
-16y=64
divide both sides by -16
what do you get?

Answer 28

4 yay thank you

Answer 29

you seem to have trouble with the minus signs. make a note: be careful of the signs

Answer 30

-16x/-16 = 64/-16

-16/-16 is 1
x = 64/-16

x= -4

Answer 31

leave it up to me to make a small mistake but thank you so so much you are a lifesavor

Answer 32

except that should be y= -4 !

we use that in either of the equations to find x
for example
-3x + 7y = -16
replace y with -4
-3x + 7*-4 = -16

can you solve for x ?

Answer 33

-6.3.....

Answer 34

is that right @phi

Answer 35

or is it -4?? @phi

Answer 36

-3x + 7*-4 = -16
7 * -4 = -28

-3x -28 = -16
add +28 to both sides
-3x -28 + 28 = -16 + 28
-3x = 12

divide both sides by -3
x = 12/-3 = -4

yes, x= -4,

so the solution is x=-4, y=-4

Answer 37

thank you so much *hugs you with death grip* sorry for bugging you you saved my life

Answer 38

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


the new system (as you may remember) is
-12x + 12y = 0
-9x + 5y = 16


one way to show this system has the same solution, is to use x=-4 and y=-4
and show that *both* equations are true.

can you do that ?

Answer 39

yeah thats all you help me so much thank you again