Question

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.

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

Answer 1

I would suggest to multiply the 1st equation times 3, and then subtract the equations .
Alternatively, you can multiply the first equation times -3, and add the equations. Makes no difference.

Answer 2

I really don't know how to do this, I'm just trying to finish up for the segment. :(

Answer 3

lets do a multiplication times -3, and then adding, okay>

Answer 4

So, when you multiply \(\large\color{black}{ -3x + 7y = -16 }\) times -3, you are multiplying EVERY THING in the equation times -3.

So can you tell me:

\(\large\color{black}{ ( -3x)\times (-3)=? }\)
\(\large\color{black}{ ( 7y)~~~\times (-3)=? }\)
\(\large\color{black}{ ( -16)\times (-3)=? }\)

Answer 5

9 -21 and 48?

Answer 6

you are forgetting the Ys and Xs for the first 2. For the last one, 48 is correct.

Answer 7

But what are the values for x & y?

Answer 8

when you multiply \(\large\color{black}{ ( -3y)\times (-3) }\) you get \(\large\color{black}{ 9y }\), not just \(\large\color{black}{ 9 }\) correct?

Answer 9

I mean 3x, not 3y.

Answer 10

Oh okay.

Answer 11

So what would I put for Part A? Sorry to rush its timed though :/

Answer 12

for part A, you need to just add the equations, and this will be an equation 1 of your new system.
and the second equation of your new system, is a multiply of any of the initial two equations.

Answer 13

But how would I type it up? I'm really confused I'm sorry.

Answer 14

okay, can you add your 2 equations?

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

Answer 15

Is the first one 36?

Answer 16

I added the stuff from before..

Answer 17

add the equations, please. I don't know what just a plain 36 means.

Answer 18

x + y = 36??? I really don't know.

Answer 19

-3x+(-9x)=?
(7y)+(5y)=?
(-16)+(16)=?

Answer 20

-12x 12y and 0

Answer 21

so you have:
-12x + 12y = 0 as your first equation.

Answer 22

then choose any equation

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

and multiply it by any number (besides zero).

I would just multiply the 1st equation times -2.

Answer 23

Wait what?

Answer 24

multiply any of your (initial) equations by any number.
This is what they want, they want the second new equation to be a multiple of one of the first two.

Answer 25

Like multiply the 1st equation times -2.
-3x + 7y = -16
times(-2),

(-3x)(-2) + (7y)(-2) = (-16)(-2)

do you see what I am doing?
what do you get?

Answer 26

32

Answer 27

im sorry to say this but can you please just give me the answer for this one? I really can't figure this out and I'm running out of time and I still have to work on other questions.

Answer 28

no you are multiplying everything as I said;
(-3x)(-2) + (7y)(-2) = (-16)(-2)


you just need to simplify. And sorry i lost connection.

Answer 29

-5x + -14?

Answer 30

-14y *

Answer 31

not exactly:
(-3x)(-2) + (7y)(-2) = (-16)(-2)

6x + (-14y) = 32

6x -14y = 32 \(\normalsize\color{blue}{ \rm correct ? }\)

Answer 32

so there is your system that you need for A.
-12x + 12y = 0
6x -14y = 32

Answer 33

Oh oops! I added -3x and -2

Answer 34

yes lol...

Answer 35

So that's what I put for part a?

Answer 36

yes.

Answer 37

And what do I put for part b?

Answer 38

This is the system you had initially.
-3x + 7y = -16
-9x + 5y = 16

This is the system that you created.
-12x + 12y = 0
6x -14y = 32

you are to show that these two systems have the same solution, by solving each one of them.

Answer 39

So how would I lay that out?

Answer 40

you just need to solve each system. let's start from the
-3x + 7y = -16
-9x + 5y = 16

okay?

Answer 41

Wasn't it -12x - 12y = 0?

Answer 42

this is just the addition of the 2 equations of your intial system that you needed for the 1st equation of part A.

Answer 43

You need to solve for x and y.
-3x + 7y = -16
-9x + 5y = 16

multiply the 1st equation times -3. (we have already done that, I think)

Answer 44

Yeah it was -48 right?

Answer 45

you are multiplying every term by -3.
the -3x times -3,
7y times -3,
and -16 times -3.

Answer 46

\(\large\color{black}{ (-3x)(-3) + (7y)(-3) = (-16)(-3) }\)

Answer 47

9x -2y & 48

Answer 48

-21y*

Answer 49

when we did it before (in this same question)
we got.
\(\large\color{black}{ 9x-21y=48 }\)

you are correct.

Answer 50

you had:

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


multiplied the first equation times -3 and now have:
9x -21y = 48
-9x + 5y = 16


now, please add the equations for me.

9x -21y = 48
-9x + 5y = 16
--------------

Answer 51

64?

Answer 52

9x + (-9x) =?

-21y + 5y= ?

and 64 (for the sum of 48 and 16) is correct.

Answer 53

0 & -16y

Answer 54

yes. so when you add you get.

-16y = 64

(and as you can notice, the Xs cancel)

can you solve for y?

Answer 55

-4?

Answer 56

yes, y=-4.

Answer 57

now, plug in -4 instead of y into any of the equations of the original system

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

Answer 58

if you plug -4 for y into both of the equations and get the same answer (though), this will not only find the x, but also verify and check.

Answer 59

Yeah I got the same answer

Answer 60

so that's what I put?

Answer 61

you got the same answer for both, what did you get for x?