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Question 133632: This problem deals with the addition method, and I need help.
2a + 3b = -1
3a + 5b = -2
I need a lot of help, and thanks a lot!
Found 3 solutions by checkley71, nycsharkman, solver91311:
Answer by checkley71(8403)   (Show Source):
You can put this solution on YOUR website! 2a+3b=-1 multiply this equation by -3
3a+5b=-2 multiply this equation by 2
-6a-9b=3
6a+10b=-4
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b=-1 answer.
2a+3*-1=-1
2a-3=-1
2a=-1+3
2a=2
a=2/2
a=1 answer.
Answer by nycsharkman(136) (Show Source):
You can put this solution on YOUR website! This problem deals with the addition method, and I need help.
2a + 3b = -1
3a + 5b = -2
You have two equations in two variables. The variables are a and b.
What you are looking for is a point in the (a,b) form that indicates where the two equations meet on the coordinate plane. The point where the two equations meet is the solution of this system of equations. In order to know the a and b values, you were told to use the addition method. In this method, you must use a number (it could be fraction) to multiply either questions by for the sole purpose of doing away with either a or b as step one. The fun thing here is the fact that YOU choose which letter you want to delete first.
I want to remove letter a first.
To do this, I need to multiply EITHER equation by a number (or fraction) that will CANCEL OUT a in BOTH given equations.
Which number? Which fraction?
I will call 2a + 3b = -1 Equation A and
3a + 5b = -2 Equation B for you to follow my notes easier.
2a + 3b = -1...Equation A
3a + 5b = -2...Equation B
I will multiply Equation B by -2/3. Using -2/3 will create a -2a in Equation B, which will then allow me to do away with the a letter in both equations.
Are you with me so far?
-2/3 times 3a = -2a
-2/3 times 5b = -10b/3
-2/3 times -2 = 4/3
We now have this:
2a + 3b = -1...Equation A
-2a -10b/3 = 4/3...This is our new Equation B.
Do you see that NOW I can delete or do away with 2a and -2a. These two terms cancel out because one is positive and the other is negative.
We now add the rest of the terms in Equations A and B.
+3b - 10b/3 = -b/3
-1 + 4/3 = 1/3
We now have this equation:
-b/3 = 1/3
To remove the fractions, simply multiply BOTH sides of the equation by the denominator 3.
Doing so, we get this:
-b = 1
Now, divide BOTH sides by -1 to find the value of b.
b = 1/-1
b = -1
We just found the value of b.
Here comes the easy part.
Plug the value of b I just found into EITHER Equation A or B to find the value of a.
I will use Equation A.
Let b = -1
2a + 3(-1) = -1
2a - 3 = -1
2a = -1 + 3
2a = 2
a = 2/2
a = 1
The solution for your equations is: a = 1 and b = -1.
Did you follow?
NOTE: What does a = 1 and b = -1 really mean? It means that BOTH equations given to you will meet or touch each other when graphed on the coordinate plane at the point (1, -1). Is this clear?
Answer by solver91311(24713)  (Show Source):
You can put this solution on YOUR website! In order to perform the addition method, the coefficient on one of the variables in one equation must be the additive inverse of the coefficient on the same variable in the other equation.
The lowest common multiple of 2 and 3 (the coefficients on a) is 6, so we need to multiply the first equation by 3 so that the coefficient on a is 6 and multiply the second equation by -2 so that the coefficient on a is the additive inverse of 6, or -6. Or vice-versa (1st equation by -3, second by 2).
Now you can add the two left sides together and set the result equal to the sum of the right sides. That is because if  and  , then 
Now you can do either of two things. You could take this newly derived value for b and substitute it into either of the original equations and then solve the result for a, or you could recognize that the LCM for 5 and 3 is 15, then multiply eq 1 by 5 and eq 2 by -3, add the equations and determine a that way. Since b came out to be a nice tidy integer, let's do the substitution thing.
Therefore the solution set is the ordered pair (a,b)=(1,-1)
Check your answer:
The derived solution makes both equations true, answer checks.
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