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Question 72045: about 5 hours ago i emailed in reference to a problem that was solved but i wrote back to ask how they got a particular part of the problem could i get a response i know you are busy and i understand. thanks
addition of linear equations
8x-4y=16
4x+5y=22
what i need to know is how they got multiply 2nd by 2 to get 3rd: 8x+10y=44 where did the 2 come from
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website!
addition of linear equations:
8x - 4y = 16
4x + 5y = 22
What i need to know is how they got multiply 2nd by 2 to get 3rd: 8x + 10y = 44 where
did the 2 come from?
.
What this tutor is doing is showing you how to do variable elimination. The goal of variable
elimination is to get one term in one of the equations to be the same size as the corresponding
term in the other equation. That way you can subtract (or add) the two equations together
and get a third equation that only contains one variable. {Equations with one variable
can be solved for that variable.}
.
Let's work this problem and maybe the explanation will become clearer. Back to the
original two equations:
.
8x - 4y = 16
4x + 5y = 22
.
If you add these two equations together vertically you get 12x + y = 38. That doesn't
help because this new equation still has two variables ... x and y. You need to find a
way to get an equation with one variable.
.
One way to do this is to look at one of the equations and do something to that equation
to make one of its terms equal to the size of the corresponding term above it in the other
equation. In this case your tutor looked at the bottom equation and decided to double
the 4x to make it the same size as the 8x term just above it. But if you are going to
double the 4x term you have to double all the terms in the second equation. So do that ...
multiply everything in the second equation by 2 (both sides of the equal sign) and your
new "doubled" equation becomes: 8x + 10y = 44. Again, the only reason the tutor did that
was to make the get the 4x term changed to 8x. Now use this new equation as a replacement
for the second equation above. Your two equations (the original first equation and this
new equation) are now:
.
8x - 4y = 16
8x +10y = 44
Look what happens now if we subtract the two equations. The 8x take away the 8x from the
new equation becomes 0. The -4y take away 10y becomes -14y. And on the other side of
the equal sign the 16 take away 44 becomes -28. So the resulting equation is:
.
-14y = -28
.
And when you divide both sides by -14 you find the answer is y = +2. You can now plug
+2 in for y in either of the two original equations and solve for x.
.
Something else I'll point out. The problem statement seems to imply that your are to
solve by addition. We solved by subtraction. If you want to solve by addition, you can
just multiply the second equation by -2 instead of multiplying it by +2 as we did. If
you do that your equation pair becomes:
.
+8x - 4y = 16
-8x -10y = -44
.
Now you just add the two equations together and you get the two 8x terms to cancel again
because one of them has a + sign and the other has a minus sign. And as we got before,
we again end up with -14y = -28 so that y = +2.
.
Now don't get locked up with always trying to get rid of the x terms. In this case it
was probably the easiest thing to do. But suppose you decided that you wanted to get
rid of the two y terms. Let's go back to the original equations and discuss that a bit.
.
8x - 4y = 16
4x + 5y = 22
.
How would we make the two y terms (one in the top equation and one in the bottom equation) equal?
.
Let's work it sort of the same way as you would least common denominators. Could we
make the y term in the top equation = -20y? Sure could. All we would have to do is to
multiply it by 5. But if we do that we have to multiply all the terms on both sides by
5 so that we're not upsetting the balance of the equation. If we multiply everything
in the top equation by 5 it becomes:
.
40x - 20y = 80.
.
Why did we do that multiplication? Because what happens now if we multiply the bottom equation
by 4 (multiply everything on both sides by 4 so we don't change the balance of the
equation).
.
When multiplied by 4, the bottom equation becomes:
.
16x + 20y = 88
.
So now our two equations are:
.
40x - 20y = 80 and
16x + 20y = 88
.
Notice what we have done now. The two y terms are now the same size but opposite
in sign. We made that happen by multiplying the original top equation by 5 and the
original bottom equation by 4. Clever choice of multipliers on our part. Now we just
add these two equations together vertically and notice that the y terms cancel each other
out. We are left with 56x = 168 (from 40x + 16x and 80 + 88). If you divide both sides
by 56 you get that x = 3. This can then be put back into any of the original equations
to solve for y. But if you do, you should get y = +2 just as we found above. And when
you previously substituted +2 for y in one of the original equations in solving that equation
for x you should have gotten that x = +3.
.
Hope that this clarifies where the 2 came from in the work of the other tutor. He or she
wisely chose it as a means of making the 4x the same size as the 8x in the first equation
so that the variable in x would disappear when the two equations were combined.
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