SOLUTION: Solve the system by the elimination method 5x+3y=-7 7x-2y=19

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Question 146322: Solve the system by the elimination method
5x+3y=-7
7x-2y=19
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Start with the given system of equations:

system%285x%2B3y=-7%2C7x-2y=19%29



Now in order to solve this system by using elimination/addition, we need to solve (or isolate) one variable. I'm going to solve for y.





In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).


So lets eliminate x. In order to do that, we need to have both x coefficients that are equal in magnitude but have opposite signs (for instance 2 and -2 are equal in magnitude but have opposite signs). This way they will add to zero. By adding to zero, they can be eliminated.



So to make the x coefficients equal in magnitude but opposite in sign, we need to multiply both x coefficients by some number to get them to an common number. So if we wanted to get 5 and 7 to some equal number, we could try to get them to the LCM.



Since the LCM of 5 and 7 is 35, we need to multiply both sides of the top equation by 7 and multiply both sides of the bottom equation by -5 like this:




7%285x%2B3y%29=7%28-7%29 Multiply the top equation (both sides) by 7
-5%287x-2y%29=-5%2819%29 Multiply the bottom equation (both sides) by -5




Distribute and multiply

35x%2B21y=-49
-35x%2B10y=-95


Now add the equations together. In order to add 2 equations, group like terms and combine them

%2835x-35x%29%2B%2821y%2B10y%29=-49-95

Combine like terms and simplify



cross%2835x-35x%29%2B31y=-144 Notice how the x terms cancel out




31y=-144 Simplify




y=-144%2F31 Divide both sides by 31 to isolate y



Now plug this answer into the top equation 5x%2B3y=-7 to solve for x

5x%2B3y=-7 Start with the first equation



5x%2B3%28-144%2F31%29=-7 Plug in y=-144%2F31




5x-432%2F31=-7 Multiply



%2831%29%285x-432%2F31%29=%2831%29%28-7%29 Multiply both sides by the LCM of 31. This will eliminate the fractions (note: if you need help with finding the LCM, check out this solver)



155x-432=-217 Distribute and multiply the LCM to each side



155x=-217%2B432Add 432 to both sides


155x=215 Combine like terms on the right side


x=%28215%29%2F%28155%29 Divide both sides by 155 to isolate x



x=43%2F31 Reduce




So our answer is
x=43%2F31 and y=-144%2F31



which forms the ordered pair