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Question 223050: please help me solve by elimination method
5x+6y=3
10x+12y=6
*the solution is an ordered pair use integers or fractions*
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! The key to the Elimination Method is "opposites". We need to get both equations to have opposites for one pair of the variable terms. We want either opposite x terms or opposite y terms. If the system does not start with opposites then we have to figure out how to create them.
To create opposites we multiply both sides of one (or both) equations by some number(s).
Let's look at your system:
5x+6y=3
10x+12y=6
The x terms (5x and 10x) are not opposites. The y terms (6y and 12y) are no opposites. So we will have to create opposites. We will have to figure out how to multiply one (or both) equations so that we end up with opposites. With some thought I hope it is obvious that one way to get opposites is to multiply both sides of the first equation by -2. (If this is difficult to see, see below for a method which requires no cleverness.) So we'll multiply both sides of the first equation by -2:
-10x + -12y = -6
10x + 12y = 6
Now we have opposites. Once we have opposites add the two equations together. In this problem everything adds up to 0!
0 + 0 = 0
0 = 0
There are no variables left! This does not normally happen. Normally only one pair of variable terms are opposites (which then add up to 0 and "disappear").
So what does it mean when both variables disappear like this? Let's remember what the solution of a system represents. The solution of a system represents the point or points where the graphs of the two equations intersect. When both variables disappear like this it means one of two things:- There is no solution because the graphs to not intersect. In fancy terms this is called "inconsistent".
- There are an infinite number of solutions because the two equations are actually different forms of the same equation. In fancy terms this is called consistent and dependent.
So how do tell which one of these we have? It depends on the truth or correctness of the equation with no variables.- If the equation is false likethen the system is inconsistent (i.e. there are no solutions)
- If the equation true likethen the system is consistent and dependent (i.e. there are an infinite number of solutions)
Since our variable-less equation, 0=0, is true, then our system is consistent and dependent (infinite solutions).
Procedure for creating opposites (which requires no imagination/creativity):- Transform both equations so they are in the form: Ax + By = C.
- Multiply both sides of the first equation by the second equation's coefficient of x. and multiply both sides of the second equation by the negative of the first equation's coefficient of x.
Let's try this out on this system:
-2x = 3y + -4
3x + 4y = 6
Step 1: Transform
Add -3y to both sides of the first equation:
-2x + (-3y ) = -4
3x + 4y = 3
Step 2: Multiply each equation.... (Note that the 1st eq.'s x coefficient is -2 and the negative of -2 is 2.):
3(-2x + (-3y)) = 3(-4)
(2)(3x + 4y) = (2)(6)
Multiplying these out we get:
-6x + (-9y) = -12
6x + 8y = 12
And we have opposite x terms!
If we want to finish solving this, add the two equations:
0 + (-y) = 0
-y = 0
y = 0
(Unlike your problem, both variables did not disappear. This is how most systems will work out.)
Substitute back into one of the original equations:
-2x + 3(0) = -4
Solve
-2x = -4
x = 2
Solution: (-2, 0)
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