I'm learning about equivalent systems of equations, !

OK to solve this, we have to solve each system presented through elimination or substitution and find which one is equivalent to that of the teacher's!

First let's solve for the teacher's:
-2x+5y=10
-3x+9y=6
Solve by substitution (I think elimination might be easier to do for this one, but I don't really remember 100% sorry!)
Isolate the x (or y) variable in the first equation
-2x+5y=10
-2x=10-5y

Substitute x into the next equation and solve for y
-3(10-5y/2)+9y=6
3*10-5y/2+9y=6
(multiply both sides by 2)
3(10-5y)+18y=12
30-15y+18y=12
30+3y=12
3y=-18
y=-6

Substitute in x
x= -10-5(-6)/2
x=-20

TEACHER'S ANSWER (-20,-6)

GOKU
x-3y=-2
-2x+5y=-7
Do the same as above
Solve for x
x-3y=-2
x=3y-2
Plug in
-2(3y-2)+5y=-7
4-6y+5y=-7
4-y=-7
-y=-11
y=11

x=(3(11)-2)
x=31
GOKU'S ANSWER (31, 11)

SELINA:
-5x+14y=16
-3x+9y=12
One last time!! :)

-5x+14y=16
-5x=16-14y
x=(16-14y)/-5

-3(-(16-14y/5)+9y=12
3*16-14y/5+9y=12
3*16-14y+45y=60 
48-42y+45y=60
48+3y=60
3y=12
y=4

x=-(16-14(4))/5
x=8
SELINA'S ANSWER
(8,4)

So neither Goku or Selina got the same answer as the teacher

Here I go

The teacher's system is
-2x + 5y = 10
-3x + 9y = 6

To find the system that equals to the teachers solution, we must solve all three of them. (Yeah good luck)

Let's start solving the teacher's system

We need to solve -2x + 5y = 10 for x
-2x + 5y = 10
-2x = 10 - 5y
Divide both sides by -2
-2x/-2 = (10-5y)/-2
x = 5/2 y - 5
Now we need to substitute 5/2 y -5 for x in -3x +9y = 6
We gonna replace 5/2 y -5 by x
-3x + 9y = 6
-3(5/2 y - 5) + 9y = 6
Simplify both sides (I used a calculator)
3/2 y + 15 = 6
3/2 y = 6 - 15
3/2 y = -9
To find y, we need to divide both sides by 3/2
3/2 y/3/2 = -9/3/2  (I used a calculator)
y = -6
It easier to find x since we already find y
So, let's find x by substitute -6 for y in x=5/2 y - 5
What does that mean? We gonna replace the value of y which is -6 everywhere we find y (I hope it makes sense)
x = 5/2 y - 5
x = 5/2 (-6) - 5 (use a calculator)
x = -20

Therefor,
x = -20 and y = -6
This is the teacher's solution so we gonna solve the others and see what they have :)


Selina's system

-5x + 14y = 16
-3x + 9y = 12

So, we need to solve -5x + 14y = 16 for x
-5x + 14y = 16
-5x = 16 - 14y
Divide both sides by -5
-5x/-5 = (16 - 14y)/-5
x = 14/5 y + -16/5     (It looks messy here)
To continue, we need to substitute 14/5 y + -16/5 for x in -3x + 9y = 12
-3x + 9y = 12
-3( 14/5 y + -16/5) + 9y = 12
Simplify both sides
3/5 y + 48/5 = 12
3/5  y = 12 - 48/5
3/5 y = 12/ 5
To find y, we just need to divide both sides by 3/5
3/5 y/3/5 = 12/5/3/5  (cancel 5, then divide 12 by 3 = 4)
y = 4
We gonna use the value of y to find x
We gonna do that by substitute 4 for y in x = 14/5 y + -16/5
x = 14/5 y + -16/5
We gonna replace y by 4
x = 14/5 (4) + -16/5 (used a calculator)
x = 8
Therefore,
x = 8 and y = 4
So, Selina system doesn't equal to the teacher's system.

Goku's system

again, I used the same steps to solve all of them.

x - 3y = -2
-2x + 5y = -7

We need to solve x - 3y = -2 for x
x - 3y = -2
x = -2 + 3y
I prefer to let the unknown value leads, so it will be
x = 3y - 2  (it still the same thing :) )
Substitute 3y - 2 for x in -2x + 5y = -7
-2x + 5y = -7
-2(3y - 2) + 5y = -7
-6y + 4 + 5y = -7
-y + 4 = -7
-y = -7 - 4
-y = -11
We cannot leave the y with a negative sign, so we must divide both sides by -1
-y/-1 = -11/-1
y = 11
To find x, we need to substitute 11 for y in x = 3y - 2
x = 3y - 2
x = (3)(11)-2
x = 33-2
x = 31
Therefore,
x = 31 and y = 11
Goku's answer doesn't equal to the teacher's answer.



Answer : Neither Goku or Selina

I hope I helped!