SOLUTION: Use the Gauss-Jordan method to solve the following system of equations x+y=9 4x+3y=30

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Question 902277: Use the Gauss-Jordan method to solve the following system of equations
x+y=9 4x+3y=30
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
1,1,9
4,3,30

add down (-4) *row 1 to row 2
1,1,9
0,-1,-6
divide row 2 by -1
1,1,9
0,1,6
We now have the value for the last variable.
We will work our way up and get the other solutions.
add up (-1) *row 2 to row 1
1,0,3
0,1,6
final
1,0,3
0,1,6
1,0,3
0,1,6
"3","6"
(3,6)
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=9
4%2Ax%2B3%2Ay=30

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

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

4%2A%281%2Ax%2B1%2Ay%29=%289%29%2A4 Multiply the top equation (both sides) by 4
-1%2A%284%2Ax%2B3%2Ay%29=%2830%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
4%2Ax%2B4%2Ay=36
-4%2Ax-3%2Ay=-30

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%284%2Ay-3%2Ay%29=36-30

%284-4%29%2Ax%2B%284-3%29y=36-30

cross%284%2B-4%29%2Ax%2B%284-3%29%2Ay=36-30 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:

1%2Ay=6

y=6 Divide both sides by 1 to solve for y



y=6 Reduce


Now plug this answer into the top equation 1%2Ax%2B1%2Ay=9 to solve for x

1%2Ax%2B1%286%29=9 Plug in y=6


1%2Ax%2B6=9 Multiply



1%2Ax=9-6 Subtract 6 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=6

which also looks like

(3, 6)

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

1%2Ax%2B1%2Ay=9
4%2Ax%2B3%2Ay=30

we get



graph of 1%2Ax%2B1%2Ay=9 (red) 4%2Ax%2B3%2Ay=30 (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 (3,6). This verifies our answer.