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)

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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)

Mathematics
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if we multiply the first equation by -3, we get: \[9x - 21y = 48\]
Now if we sum that equation with the second one, what do you get?
x-16y=64?

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hint: \[9x - 21y + \left( { - 9x + 5y} \right) = 48 + 16\] please simplify
but wouldnt that be x-16y=16?
no, since the next step is: \[9x - 21y - 9x + 5y = 48 + 16\] please simplify
oh you put addition at first, x-25y=64
hint: what is 9 -9 =...
0
ok! so the coefficient of the term with x is zero. Now what is -21 + 5= ...?
-16
ok! so the coefficient of the term with y is -16
then we can write: \[0x - 16y = 64\] and finally: \[ - 16y = 64\]
am I right?
yeah
ok! so the requested system of part A, can be this: \[\left\{ \begin{gathered} - 3x + 7y = - 16 \hfill \\ - 16y = 64 \hfill \\ \end{gathered} \right.\]
thats it for part A?
yes!
perfect, now what do we do for part B?
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?
hint: divide both sides by -16, what do you get?
y=-4
ok!
now substitute that value of y into the first equation of the equivalent system, namely: \[ - 3x + 7y = - 16\] what equation do you get?
-3x+7-4=16
not exactly, here is the right step: \[ - 3x + 7 \times \left( { - 4} \right) = - 16\] please simplify
what is: \[7 \times \left( { - 4} \right) = ...?\]
-3x-21=-16?
hint: \[7 \times \left( { - 4} \right) = - 28\] am I right?
oops lol, yeah you're right, i accidentally multiplied by -3
sorry its real late over here, cant think properly
ok! So we have: \[ - 3x - 28 = - 16\]
now if we add 28 to both sides, we get: \[ - 3x - 28 + 28 = - 16 + 28\] please simplify
-3x=12
ok! then if we divide both sides by -3, we can write: \[\frac{{ - 3x}}{{ - 3}} = \frac{{12}}{{ - 3}}\] please simplify
-4
x=-4
perfect! So the solution of the equivalent system is: \[\left\{ \begin{gathered} x = - 4 \hfill \\ y = - 4 \hfill \\ \end{gathered} \right.\]
now, please substitute that solution into the first equation of the original system, namely: \[ - 3x + 7y = - 16\] what do you get?
-3-4+7-4=-16
oops
hint: \[ - 3 \times \left( { - 4} \right) + 7 \times \left( { - 4} \right) = - 16\]
gotta multiply
yes!
yeah i meant that
what is (-3) * (-4)=... and 7*(-4)=...?
12 -28
ok! so we can write: \[12 - 28 = - 16\]
now what is 12-28=...?
-16
ok! So we can write: \[ - 16 = - 16\]
ok thanks a lot! appreciate every thing, you taught me better than any teacher has in school!
reassuming the solution of the equivalent system makes the first equation an identity
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
i think we only have to do it with one equation, but i might be wrong
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?
-9(-4)+5(-4)=16
ok! \[ - 9 \times \left( { - 4} \right) + 5 \times \left( { - 4} \right) = 16\]
now please simplify: what is (-9)*(-4)=...? and 5*(-4)=...?
36 -20
so we can write: \[36 - 20 = 16\] now what is 36-20=...?
16
16=16
ok! so we got an identity again!
then we can state that both systems have the same solution, which is: \[\left\{ \begin{gathered} x = - 4 \hfill \\ y = - 4 \hfill \\ \end{gathered} \right.\]
ok, thanks for all the help! greatly appreciate it
:)

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