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 answer

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Question 93420: 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. Can you show me how to work this problem so I can do my homwwork?
4x + y =4
2x + 8y =0
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

4%2Ax%2B1%2Ay=4
2%2Ax%2B8%2Ay=0

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

Since the LCM of 4 and 2 is 4, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -2 like this:

1%2A%284%2Ax%2B1%2Ay%29=%284%29%2A1 Multiply the top equation (both sides) by 1
-2%2A%282%2Ax%2B8%2Ay%29=%280%29%2A-2 Multiply the bottom equation (both sides) by -2


So after multiplying we get this:
4%2Ax%2B1%2Ay=4
-4%2Ax-16%2Ay=0

Notice how 4 and -4 add to zero (ie 4%2B-4=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%284%2Ax-4%2Ax%29%2B%281%2Ay-16%2Ay%29=4%2B0

%284-4%29%2Ax%2B%281-16%29y=4%2B0

cross%284%2B-4%29%2Ax%2B%281-16%29%2Ay=4%2B0 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:

-15%2Ay=4

y=4%2F-15 Divide both sides by -15 to solve for y



y=-4%2F15 Reduce


Now plug this answer into the top equation 4%2Ax%2B1%2Ay=4 to solve for x

4%2Ax%2B1%28-4%2F15%29=4 Plug in y=-4%2F15


4%2Ax-4%2F15=4 Multiply



4%2Ax-4%2F15=4 Reduce



4%2Ax=4%2B4%2F15 Subtract -4%2F15 from both sides

4%2Ax=60%2F15%2B4%2F15 Make 4 into a fraction with a denominator of 15

4%2Ax=64%2F15 Combine the terms on the right side

cross%28%281%2F4%29%284%29%29%2Ax=%2864%2F15%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4 on the left side.


x=16%2F15 Multiply the terms on the right side


So our answer is

x=16%2F15, y=-4%2F15

which also looks like

(16%2F15, -4%2F15)

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

4%2Ax%2B1%2Ay=4
2%2Ax%2B8%2Ay=0

we get



graph of 4%2Ax%2B1%2Ay=4 (red) 2%2Ax%2B8%2Ay=0 (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 (16%2F15,-4%2F15). This verifies our answer.