Question

[6.03] What is the value of the x variable in the solution to the following system of equations?
7x - 2y = 21
4x + y = 57

Answer 1

What methods of solving systems of linear equations have you been taught/ do you want to use on this one?

Answer 2

elimination?

Answer 3

That works, but if I may, this particular problems seems to be *begging* to be solved by substitution, are you fine with that? :)

Answer 4

*problem, not problems, jeez

Answer 5

yes

Answer 6

Ok, look at the second equation:
4x + y = 57
Whenever you have a variable already alone (ie it has no number beside it)
that usually makes life easier for you.

In this case, try to isolate y (in other words, put it on one side of the equation, and all the rest on the other side)

Go ahead :D

Answer 7

y=0

Answer 8

No. I meant like this: (I'm going to take you through this step by step, ok? )
4x + y = 57

Now add (-4x) to both sides

4x + y - 4x = 57 - 4x
4x - 4x + y = 57 - 4x
y = 57 - 4x

Get me so far? :)

Answer 9

-16+-4y=-228 and yes its fine the step by step

Answer 10

If I may ask, where did you get that equation?

Answer 11

from the 4x+y=57

Answer 12

Oh, so you multiplied everything by -4?
I see no reason to do that, perhaps you should try multiplying by 2, instead

(by the way, this is in no way involved with substitution method, this is elimination method, but if this is what you're comfortable with, then go right ahead :) )

Answer 13

2(4x+y=57)2
8x+2y=114
-8 -8
2y=106
2/106= 53

Answer 14

Ok, I see you have trouble with elimination method, would you like me to explain it to you, step by step?
(Don't worry, I'm more than willing to do so :D )

Answer 15

yes that would be fine and thank you so much

Answer 16

I'll start from the VERY basics, ok?
For instance, if we have
a = b

and

c = d

it follows that
a + c = b + d,
right?
if you don't get it, I can explain further, if you wish

Answer 17

no i dont get it im sorry

Answer 18

Ok, well,
a = b, right?
since they're equal, we can add c to both sides and still have it equal, like so:
a + c = b + c
but c = d, so in any expression that has c, we can replace it with d, so...
a + c = b + d (we replaced the second c with d, since they're equal anyway)

Get it now?

Answer 19

yes

Answer 20

ok, so we have established that if
a = b
and
c = d
then

a + c = b + d

right?

Well, your systems of linear equations isn't THAT much different
you could say your a, b, c, and d just got fancier :)

Let me show you what I mean:

Answer 21

7x - 2y = 21
4x + y = 57

your "a" is 7x - 2y
your "b" is 21
your "c" is 4x + y
your "d" is 57

Do you get me so far?

Answer 22

okay i get that part

Answer 23

so, if we apply the same rules:
7x - 2y = 21
4x + y = 57

we can say that
7x - 2y + 4x + y = 21 + 57

right?

Answer 24

yes

Answer 25

We can simplify it to
11x - y = 78
But this does not really help us

The trick to solving by elimination is properly *manipulating* the equations
Ready for me to show you?

Answer 26

yes

Answer 27

ok, back to our original pair of equations
7x - 2y = 21
4x + y = 57

What we want to do is to alter them a bit, so that when we add them up like we just did, one of the variables will cancel out, leaving us with a linear equation in one variable.
Any idea how to do this?

Answer 28

no

Answer 29

Ok, look at them again
7x - 2y = 21
4x + y = 57

Let's look at 7x for now
What would cancel out 7x?
-7x would...
but there's no immediate way to turn 4x into -7x, is there?

Perhaps the -2y is easier
What would cancel out -2y?

Answer 30

7 would cancel out to be just x and no there is no way 4 can turn to a -7x

Answer 31

You're about right in your second statement, the first one is rather vague, but what would cancel out -2y?

Answer 32

i need to turn this test in half hour and i need to check the other questions too

Answer 33

Ok, I'll demonstrate it for you, but try to learn it ok?

Answer 34

okay

Answer 35

7x - 2y = 21
4x + y = 57

We take the second equation, and multiply everything by 2

2(4x + y) = 2(57)
8x + 2y = 114

We replace the second equation with this one

7x - 2y = 21
8x + 2y = 114

Now we add up all those on the left side, and all those on the right

7x - 2y + 8x + 2y = 21 + 114
15x = 135 (the y got cancelled out!)
x = 9 (divided both sides by 15)

So now that we know that x = 9, then we substitute it for the x in the first equation
7x - 2y = 21
7(9) - 2y = 21
63 - 2y = 21
- 2y = 21 - 63
- 2y = -42
y = 21 (dividing both sides by -2)

So x=9 and y=21
To be safe, let's try it with the second equation as well

4x + y = 57
4(9) + 21 = 57
36 + 21 = 57
57 = 57 (it works! )

Thus, we're sure that
x = 9
and
y = 21

END

Hope that helps
And remember to study the process I went through to get to the answer

Answer 36

one more question and that would be all [6.02] Use the Substitution Method to solve the following system of equations.
4x + y = 11
x + 2y = 8

(3, 2)

(2, 3)

(-2, -3)

(-2, 3)

Answer 37

4x + y = 11
x + 2y = 8

Take the first equation and add -4x to both sides
4x + y - 4x = 11 - 4x
y = 11 - 4x

We have obtained a value for y in terms of x. Now we substitute that value for the y in the second equation:
x + 2y = 8
x + 2(11 - 4x) = 8
x + 22 - 8x = 8
-7x = 8 - 22
-7x = -14 (divide both sides by -7)
x = 2

Now we substitute 2 for x in any of the two given equations, in the first:
4x + y = 11
4(2) + y = 11
8 + y = 11
y = 11 - 8
y = 3

We have x = 2 and y = 3. Let's verify with the second equation:
x + 2y = 8
2 + 2(3) = 8
2 + 6 = 8
8 = 8 (it works!)

Thus, we're sure that
x = 2
and
y = 3

END

PS: Please go through the process involved, it's very important :)