SOLUTION: Solve the following system by addition. If a unique solution does not exist, state whether the system is inconsistent or dependent: 2x+3y=1 5x+3y=16

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Question 120650: Solve the following system by addition. If a unique solution does not exist, state whether the system is inconsistent or dependent:
2x+3y=1
5x+3y=16
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

2%2Ax%2B3%2Ay=1
5%2Ax%2B3%2Ay=16

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 2 and 5 to some equal number, we could try to get them to the LCM.

Since the LCM of 2 and 5 is 10, we need to multiply both sides of the top equation by 5 and multiply both sides of the bottom equation by -2 like this:

5%2A%282%2Ax%2B3%2Ay%29=%281%29%2A5 Multiply the top equation (both sides) by 5
-2%2A%285%2Ax%2B3%2Ay%29=%2816%29%2A-2 Multiply the bottom equation (both sides) by -2


So after multiplying we get this:
10%2Ax%2B15%2Ay=5
-10%2Ax-6%2Ay=-32

Notice how 10 and -10 add to zero (ie 10%2B-10=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2810%2Ax-10%2Ax%29%2B%2815%2Ay-6%2Ay%29=5-32

%2810-10%29%2Ax%2B%2815-6%29y=5-32

cross%2810%2B-10%29%2Ax%2B%2815-6%29%2Ay=5-32 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:

9%2Ay=-27

y=-27%2F9 Divide both sides by 9 to solve for y



y=-3 Reduce


Now plug this answer into the top equation 2%2Ax%2B3%2Ay=1 to solve for x

2%2Ax%2B3%28-3%29=1 Plug in y=-3


2%2Ax-9=1 Multiply



2%2Ax=1%2B9 Subtract -9 from both sides

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

cross%28%281%2F2%29%282%29%29%2Ax=%2810%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2 on the left side.


x=5 Multiply the terms on the right side


So our answer is

x=5, y=-3

which also looks like

(5, -3)

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

2%2Ax%2B3%2Ay=1
5%2Ax%2B3%2Ay=16

we get



graph of 2%2Ax%2B3%2Ay=1 (red) 5%2Ax%2B3%2Ay=16 (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 (5,-3). This verifies our answer.