Question

HELP! WILL 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.

Answer 1

Part A:
You can start this way:
Multiply the first equation by 2 on both sides.
Now add it to the second equation.
What do you get?

Answer 2

-3x + 7y = -16
X2 X2

Answer 3

-6x + 7y = -32

Answer 4

Remember to distribute the 2 on the left side.

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

Answer 5

do i solve this or something?

Answer 6

No, not yet.
We are working on multiplying the first equation by 2 on both sides.
On the left side, you need to distribute the 2.
See below.

\(\color{red}{2 \times} (\color{green}{-3x} + \color{blue}{7y}) = 2 \times (-16)\)

\(\color{red}{2 \times } \color{green}{-3x} + \color{red}{2 \times } \color{blue}{7y} = -32\)

\(-6x + 14y = -32\)

Answer 7

So far, we have multiplied the first equation by 2 on both sides.

Now we add that new equation to the second original equation:

\(~~~~~~~~~~-6x + 14y = -32\)

\(~~~+~~~-9x + ~~5y = 16\)
----------------------------
\(~~~~~~~~~~-15x + 19y = -16\)

Answer 8

are we still on Part A?

Answer 9

i dont have a lot of time and i still have 2 more questions to do after this one

Answer 10

The new equation just above is the sum of one original equation and a multiple of the other original equation.

Now we take the new equation and one of the original equations, and that completes Part A.

-3x + 7y = -16
-15x + 19y = -16

Answer 11

That is the end of part A.
We have a new system that is equivalent to the original one.

Answer 12

okay so what should i type in for part a?

Answer 13

Everything we did above.

Answer 14

Skipping the explanations.

Answer 15

I'll show you.

Answer 16

can it be put into not that because i cant write like that in my assignment

Answer 17

Part A.
-3x + 7y = -16
-9x + 5y = 16

2(-3x + 7y) = 2(-16)
-6x + 14y = -32
+ -9x + 5y = 16
---------------------
-15x + 19y = -16

-3x + 7y = -16
-15x + 19y = -16

Answer 18

Ok. The response above is Part A.

Answer 19

Okay thank you can you help me with part b too?

Answer 20

Yes.
Now in Part B, we need to show both systems are equivalent to each other and both have the same solution.
We could solve both systems, but there is a faster way.
We solve only the original system.
Then we try the solution of the original system in the second system.
If the solution works, that shows the systems are equivalent.

Answer 21

Let's solve the original system.
We can use the addition method.
We need to add the two equations together and make sure one of the variables adds to zero.

Answer 22

Since the first term of the first equation is -3x, if we multiply it by -3, we get 9x.
9x added to the first term of the second equation, -9x, will add to zero. That is what we want, so we start by multiplying the first equation by -3 on both sides.

-3x + 7y = -16 ------ *3 on both sides ----- > -9x -21y = 48
-9x + 5y = 16 ------ copied just as it is ---> 9x + 5y = 16
----------------
-- sum of the two equations --> -16y = 64

-16y = 64
y = -4

Now that we know y = -4, we substitute -4 in the first equation for y. Then we find x.

-3x + 7y = -16
-3x + 7(-4) = -16
-3x -28 = -16
-3x = 12
x = -4

x is also -4.

The solution of the original system of equations is:
x = -4, y = -4

Answer 23

so that is our part b correct?

Answer 24

One more step. We need to check the solution we got for the first system in the second system.

Answer 25

ok

Answer 26

We can solve this system, but it's faster to just check the solution we already know.
The question now is:
Is the solution x = -4 and y = -4 also a solution to the new system of equations?
Let's test (-4, -4) in both equations of the second system.

-3x + 7y = -16
-3(-4) + 7(-4) =? -16
12 + (-28) =? -16
-16 = -16
The solution works on the first equation.

-15x + 19y = -16
-15(-4) + 19(-4) =? -16
60 + (-76) =? -16
60 - 76 =? -16
-16 = -16
The solution works on the second equation.

This just showed that the two systems are equivalent.

Part B is done.

Answer 27

okay i get it thanks so much!

Answer 28

can u help with another? @mathstudent55

Answer 29

i just posted it

Answer 30

You're welcome.
I'll look.