SOLUTION: Solve the system of equations using the addition (elimination) method. If the answer is a unique solution, present it as an ordered pair: (x, y). If not, specify whether the answe

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Question 179448: Solve the system of equations using the addition (elimination) method.
If the answer is a unique solution, present it as an ordered pair: (x, y). If not, specify whether the answer is "no solution" or "infinitely many solutions" and state how you arrived at that conclusion.
4x + 10y = 2
3x + 5y = 5
Answer by solver91311(24713) About Me  (Show Source):
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The elimination method requires that you multiply one or both of the equations by some constant so that the coefficients on one of the variables in each of the equations are additive inverses, that is, their sum is zero.





For this problem, it is convenient to multiply Equation 2 by -2 to obtain Equation 3:



Now add Equation 3 to Equation 1 term-by-term:



Now we have a zero coefficient on the y term, eliminating that term (hence the name of the method) and we have an equation in a single variable that can be solved by ordinary methods.





Now that we have determined the value of x, we can proceed one of two ways. We can either go back to the original equations and find a multiplier or multipliers that would allow us to eliminate the x variable and solve for y (in this case you could multiply Equation 1 by 3 and Equation 2 by -4 so the coefficients on x would become 12 and -12) or you can simply take the value of x we determined when we eliminated the y variable and substitute it into either of the original equations.

Let's substitute into Equation 1:











Therefore the solution set consists of the ordered pair

Check your answer by substituting both coordinate values into the other original equation and verify that you still have a true statement:



, True statement, answer checks.

Now let's go back and eliminate the x variable to assure ourselves that the method works this way as well:





Multiply (1) by 3 and (2) by -4:





Add (3) to (4):





Which is the same result we obtained by substitution of the value of x earlier, thus verifying the method.


John