SOLUTION: Solve the following systems by addition. If a unique solution does not exist, state whether the system is inconsistent or dependent. x + 5y = 10 2x - 10y=20

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Question 91965: Solve the following systems by addition. If a unique solution does not exist, state whether the system is inconsistent or dependent.
x + 5y = 10
2x - 10y=20
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

1%2Ax%2B5%2Ay=10
2%2Ax-10%2Ay=20

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 but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 1 and 2 to some equal number, we could try to get them to the LCM.

Since the LCM of 1 and 2 is 2, we need to multiply both sides of the top equation by 2 and multiply both sides of the bottom equation by -1 like this:

2%2A%281%2Ax%2B5%2Ay%29=%2810%29%2A2 Multiply the top equation (both sides) by 2
-1%2A%282%2Ax-10%2Ay%29=%2820%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
2%2Ax%2B10%2Ay=20
-2%2Ax%2B10%2Ay=-20

Notice how 2 and -2 add to zero (ie 2%2B-2=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%282%2Ax-2%2Ax%29%2B%2810%2Ay%2B10%2Ay%29=20-20

%282-2%29%2Ax%2B%2810%2B10%29y=20-20

cross%282%2B-2%29%2Ax%2B%2810%2B10%29%2Ay=20-20 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

20%2Ay=0

y=0%2F20 Divide both sides by 20 to solve for y



y=0 Reduce


Now plug this answer into the top equation 1%2Ax%2B5%2Ay=10 to solve for x

1%2Ax%2B5%280%29=10 Plug in y=0


1%2Ax%2B0=10 Multiply



1%2Ax=10-0 Subtract 0 from both sides

1%2Ax=10 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ax=%2810%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.


x=10 Multiply the terms on the right side


So our answer is

x=10, y=0

which also looks like

(10, 0)

Notice if we graph the equations (if you need help with graphing, check out this solver)

1%2Ax%2B5%2Ay=10
2%2Ax-10%2Ay=20

we get



graph of 1%2Ax%2B5%2Ay=10 (red) 2%2Ax-10%2Ay=20 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (10,0). This verifies our answer.