Question

its 2 30 AM need to wake up in 3 hours, please help me finish this last question fast
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

if we multiply the first equation by -3, we get:
\[9x - 21y = 48\]

Answer 2

Now if we sum that equation with the second one, what do you get?

Answer 3

x-16y=64?

Answer 4

hint:
\[9x - 21y + \left( { - 9x + 5y} \right) = 48 + 16\]
please simplify

Answer 5

but wouldnt that be x-16y=16?

Answer 6

no, since the next step is:
\[9x - 21y - 9x + 5y = 48 + 16\]
please simplify

Answer 7

oh you put addition at first,
x-25y=64

Answer 8

hint:
what is 9 -9 =...

Answer 9

0

Answer 10

ok! so the coefficient of the term with x is zero.
Now what is -21 + 5= ...?

Answer 11

-16

Answer 12

ok! so the coefficient of the term with y is -16

Answer 13

then we can write:
\[0x - 16y = 64\]
and finally:
\[ - 16y = 64\]

Answer 14

am I right?

Answer 15

yeah

Answer 16

ok! so the requested system of part A, can be this:
\[\left\{ \begin{gathered}
- 3x + 7y = - 16 \hfill \\
- 16y = 64 \hfill \\
\end{gathered} \right.\]

Answer 17

thats it for part A?

Answer 18

yes!

Answer 19

perfect, now what do we do for part B?

Answer 20

we can consider the last system, namely the equivalent system, and we can solve the second equation for y, namely:
\[ - 16y = 64\]
what is y?

Answer 21

hint:
divide both sides by -16, what do you get?

Answer 22

y=-4

Answer 23

ok!

Answer 24

now substitute that value of y into the first equation of the equivalent system, namely:
\[ - 3x + 7y = - 16\]
what equation do you get?

Answer 25

-3x+7-4=16

Answer 26

not exactly, here is the right step:
\[ - 3x + 7 \times \left( { - 4} \right) = - 16\]
please simplify

Answer 27

what is:
\[7 \times \left( { - 4} \right) = ...?\]

Answer 28

-3x-21=-16?

Answer 29

hint:
\[7 \times \left( { - 4} \right) = - 28\]
am I right?

Answer 30

oops lol, yeah you're right, i accidentally multiplied by -3

Answer 31

sorry its real late over here, cant think properly

Answer 32

ok! So we have:
\[ - 3x - 28 = - 16\]

Answer 33

now if we add 28 to both sides, we get:
\[ - 3x - 28 + 28 = - 16 + 28\]
please simplify

Answer 34

-3x=12

Answer 35

ok! then if we divide both sides by -3, we can write:
\[\frac{{ - 3x}}{{ - 3}} = \frac{{12}}{{ - 3}}\]
please simplify

Answer 36

-4

Answer 37

x=-4

Answer 38

perfect! So the solution of the equivalent system is:
\[\left\{ \begin{gathered}
x = - 4 \hfill \\
y = - 4 \hfill \\
\end{gathered} \right.\]

Answer 39

now, please substitute that solution into the first equation of the original system, namely:
\[ - 3x + 7y = - 16\]
what do you get?

Answer 40

-3-4+7-4=-16

Answer 41

oops

Answer 42

hint:
\[ - 3 \times \left( { - 4} \right) + 7 \times \left( { - 4} \right) = - 16\]

Answer 43

gotta multiply

Answer 44

yes!

Answer 45

yeah i meant that

Answer 46

what is (-3) * (-4)=...
and 7*(-4)=...?

Answer 47

12
-28

Answer 48

ok! so we can write:
\[12 - 28 = - 16\]

Answer 49

now what is 12-28=...?

Answer 50

-16

Answer 51

ok! So we can write:
\[ - 16 = - 16\]

Answer 52

ok thanks a lot! appreciate every thing, you taught me better than any teacher has in school!

Answer 53

reassuming the solution of the equivalent system makes the first equation an identity

Answer 54

now we have to do the same procedure for the second equation of the original system. If the solution of the equivalent system makes an identity the second equation of the original system then we can state that the two systems, namely the original system and the equivalent system have the same solution

Answer 55

i think we only have to do it with one equation, but i might be wrong

Answer 56

so if we substitute the solution of the equivalent system:
\[\left\{ \begin{gathered}
x = - 4 \hfill \\
y = - 4 \hfill \\
\end{gathered} \right.\]
into the second equation of the original system:
\[ - 9x + 5y = 16\]
what do you get?

Answer 57

-9(-4)+5(-4)=16

Answer 58

ok!
\[ - 9 \times \left( { - 4} \right) + 5 \times \left( { - 4} \right) = 16\]

Answer 59

now please simplify:
what is (-9)*(-4)=...?
and
5*(-4)=...?

Answer 60

36
-20

Answer 61

so we can write:
\[36 - 20 = 16\]
now what is 36-20=...?

Answer 62

16

Answer 63

16=16

Answer 64

ok! so we got an identity again!

Answer 65

then we can state that both systems have the same solution, which is:
\[\left\{ \begin{gathered}
x = - 4 \hfill \\
y = - 4 \hfill \\
\end{gathered} \right.\]

Answer 66

ok, thanks for all the help! greatly appreciate it

Answer 67

:)