SOLUTION: I dont understand how to solve by addition 3x+4y=25 2x+y=10

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: I dont understand how to solve by addition 3x+4y=25 2x+y=10      Log On


   

Question 88380: I dont understand how to solve by addition
3x+4y=25
2x+y=10
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

3%2Ax%2B4%2Ay=25
2%2Ax%2B1%2Ay=10

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

Since the LCM of 3 and 2 is 6, we need to multiply both sides of the top equation by 2 and multiply both sides of the bottom equation by -3 like this:

2%2A%283%2Ax%2B4%2Ay%29=%2825%29%2A2 Multiply the top equation (both sides) by 2
-3%2A%282%2Ax%2B1%2Ay%29=%2810%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
6%2Ax%2B8%2Ay=50
-6%2Ax-3%2Ay=-30

Notice how 6 and -6 add to zero (ie 6%2B-6=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%286%2Ax-6%2Ax%29%2B%288%2Ay-3%2Ay%29=50-30

%286-6%29%2Ax%2B%288-3%29y=50-30

cross%286%2B-6%29%2Ax%2B%288-3%29%2Ay=50-30 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:

5%2Ay=20

y=20%2F5 Divide both sides by 5 to solve for y



y=4 Reduce


Now plug this answer into the top equation 3%2Ax%2B4%2Ay=25 to solve for x

3%2Ax%2B4%284%29=25 Plug in y=4


3%2Ax%2B16=25 Multiply



3%2Ax=25-16 Subtract 16 from both sides

3%2Ax=9 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%289%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=3 Multiply the terms on the right side


So our answer is

x=3, y=4

which also looks like

(3, 4)

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

3%2Ax%2B4%2Ay=25
2%2Ax%2B1%2Ay=10

we get



graph of 3%2Ax%2B4%2Ay=25 (red) 2%2Ax%2B1%2Ay=10 (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 (3,4). This verifies our answer.