SOLUTION: solve the system by addition, x+y=2 x-y=4 5x-3y=13 4x-3y=11

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Question 82595: solve the system by addition,
x+y=2
x-y=4
5x-3y=13
4x-3y=11
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Lets start with
x%2By=2
x-y=4

Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

1%2Ax%2B1%2Ay=2
1%2Ax-1%2Ay=4

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 1 to some equal number, we could try to get them to the LCM.

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

1%2A%281%2Ax%2B1%2Ay%29=%282%29%2A1 Multiply the top equation (both sides) by 1
-1%2A%281%2Ax-1%2Ay%29=%284%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
1%2Ax%2B1%2Ay=2
-1%2Ax%2B1%2Ay=-4

Notice how 1 and -1 add to zero (ie 1%2B-1=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%281%2Ax-1%2Ax%29%2B%281%2Ay%2B1%2Ay%29=2-4

%281-1%29%2Ax%2B%281%2B1%29y=2-4

cross%281%2B-1%29%2Ax%2B%281%2B1%29%2Ay=2-4 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:

2%2Ay=-2

y=-2%2F2 Divide both sides by 2 to solve for y



y=-1 Reduce


Now plug this answer into the top equation 1%2Ax%2B1%2Ay=2 to solve for x

1%2Ax%2B1%28-1%29=2 Plug in y=-1


1%2Ax-1=2 Multiply



1%2Ax=2%2B1 Subtract -1 from both sides

1%2Ax=3 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ax=%283%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.


x=3 Multiply the terms on the right side


So our answer is

x=3, y=-1

which also looks like

(3, -1)

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

1%2Ax%2B1%2Ay=2
1%2Ax-1%2Ay=4

we get



graph of 1%2Ax%2B1%2Ay=2 (red) 1%2Ax-1%2Ay=4 (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 (3,-1). This verifies our answer.




Now lets solve
5x-3y=13
4x-3y=11

Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

5%2Ax%2B3%2Ay=13
4%2Ax-3%2Ay=11

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

Since the LCM of 5 and 4 is 20, we need to multiply both sides of the top equation by 4 and multiply both sides of the bottom equation by -5 like this:

4%2A%285%2Ax%2B3%2Ay%29=%2813%29%2A4 Multiply the top equation (both sides) by 4
-5%2A%284%2Ax-3%2Ay%29=%2811%29%2A-5 Multiply the bottom equation (both sides) by -5


So after multiplying we get this:
20%2Ax%2B12%2Ay=52
-20%2Ax%2B15%2Ay=-55

Notice how 20 and -20 add to zero (ie 20%2B-20=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2820%2Ax-20%2Ax%29%2B%2812%2Ay%2B15%2Ay%29=52-55

%2820-20%29%2Ax%2B%2812%2B15%29y=52-55

cross%2820%2B-20%29%2Ax%2B%2812%2B15%29%2Ay=52-55 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:

27%2Ay=-3

y=-3%2F27 Divide both sides by 27 to solve for y



y=-1%2F9 Reduce


Now plug this answer into the top equation 5%2Ax%2B3%2Ay=13 to solve for x

5%2Ax%2B3%28-1%2F9%29=13 Plug in y=-1%2F9


5%2Ax-3%2F9=13 Multiply



5%2Ax-1%2F3=13 Reduce



5%2Ax=13%2B1%2F3 Subtract -1%2F3 from both sides

5%2Ax=39%2F3%2B1%2F3 Make 13 into a fraction with a denominator of 3

5%2Ax=40%2F3 Combine the terms on the right side

cross%28%281%2F5%29%285%29%29%2Ax=%2840%2F3%29%281%2F5%29 Multiply both sides by 1%2F5. This will cancel out 5 on the left side.


x=8%2F3 Multiply the terms on the right side


So our answer is

x=8%2F3, y=-1%2F9

which also looks like

(8%2F3, -1%2F9)

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

5%2Ax%2B3%2Ay=13
4%2Ax-3%2Ay=11

we get



graph of 5%2Ax%2B3%2Ay=13 (red) 4%2Ax-3%2Ay=11 (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 (8%2F3,-1%2F9). This verifies our answer.