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Question 59409: What are some easy steps in solving 2 equations with 2 variables by canceling out one variable while adding both equations together? Apparently when you get your answer, it is suppose to be (x,y) form and nothing else. I am just so confused because the curriculum I have for homeschooling doesn't explain it very well. So, I am in need of some assistance if you can help! I hope I explained my delemma to where you understand it. Thank you...I hope I hear from you soon!
Thanks a bunch!
Stephanie
Found 2 solutions by uma, funmath:
Answer by uma(370)   (Show Source):
You can put this solution on YOUR website! Hi Stephanie,
There are many ways of solving two linear equations with two unknowns.
The simpler one is eliminating one variable and arriving at one equation with one variable.
Let me take an example and explain it to you.
Consider the two equations
2x+3y = 7------------(1)
x+7y = 9--------------(2)
For these two equations when we add or subtract them directly, we find that none of the variable gets cancelled.
So to eliminate one variable , we need to make the coefficients of one variable equal in both the equations.
Let us choose to eliminate the variable x.
Multiplying (2) by 2 [because 2 is the coefficient of x in the first equation]gives us 2x + 14y = 18 ----------------------(3)
2x + 3y = 7----------------------(1)
Once we subtract these two equations, we get..
14y - 3y = 18 - 9
==> 11y = 11 [An equation in 1 variable]
==> y = 1 [dividing by 11 both the sides]
Now we have the value of 1 variable y = 1.
Now plug in this value of y in any of the above three equations to arrive at the value of x.
Let us substitute y = 1 in eqn(1)
==> 2x + 3(1) = 7
==> 2x + 3 = 7
==> 2x = 4 [subtracting 3 from both the sides of the equation]
==> x = 2 [dividing both the sides of the equation by 2]
Thus we have solved the given equations arriving (x,y) as (2,1).
We could plug in these values in the given equations to check if we have worked it out right.
Steps to be followed:
1> Choose to eliminate one variable
2> Make the coefficients of that variable equal in both the equations.[by multiplying]
3> Add or subtract to cancel them.
4> solve the resulting equation for 1 variable.
5> plug in this in one of the equations to arrive at the value of the other variable.
Hope it is clear to you.
Good Luck!!!
Answer by funmath(2933) (Show Source):
You can put this solution on YOUR website! Here's an example of a problem that I solved for another student, I explained the steps. I hope that makes things clearer.
:
Solve the following systems by addition. If a unique solution does not exist, I need to state whether the system is inconsistent or dependent.
:
L1) 2x + 3y = 1
L2) 5x + 3y = 16
:
Multiply L2 by -1 and add it to L2 and the y's will be eliminated.
-1(5x+3y)=-1(16) --->> -5x-3y=-16
:
2x +3y=1
-5x-3y=-16
____________
-3x+0y=-15
-3x=-15
-3x/-3=-15/-3
x=5
Substitute x=5 into L1 and solve for y:
2(5)+3y=1
10+3y=1
-10+10+3y=1-10
3y=-9
3y/3=-9/3
y=-3
:
The solution is (x,y)=(5,-3)
:
FYI, if you go to eliminate one variable and both variables get eliminated and you get two numbers that aren't equal like 4=5, then the system has no solution and is inconsistent. If the same thing happens, but the numbers left do equal each other like 4=4, then the system has an infinite number of solutions and the system is dependent.
:
Happy Calculating!!!
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