SOLUTION: i don't understand what to do with these equations 4x+3y=-2 and 3x+2y=-3 . The directions are Solve by linear combinations

Algebra ->  Linear-equations -> SOLUTION: i don't understand what to do with these equations 4x+3y=-2 and 3x+2y=-3 . The directions are Solve by linear combinations      Log On


   

Question 83185: i don't understand what to do with these equations 4x+3y=-2 and 3x+2y=-3 . The directions are Solve by linear combinations
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
If you want to solve by addition then...

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


Lets start with the given system of linear equations

4%2Ax%2B3%2Ay=-2
3%2Ax%2B2%2Ay=-3

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

Since the LCM of 4 and 3 is 12, we need to multiply both sides of the top equation by 3 and multiply both sides of the bottom equation by -4 like this:

3%2A%284%2Ax%2B3%2Ay%29=%28-2%29%2A3 Multiply the top equation (both sides) by 3
-4%2A%283%2Ax%2B2%2Ay%29=%28-3%29%2A-4 Multiply the bottom equation (both sides) by -4


So after multiplying we get this:
12%2Ax%2B9%2Ay=-6
-12%2Ax-8%2Ay=12

Notice how 12 and -12 add to zero (ie 12%2B-12=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2812%2Ax-12%2Ax%29%2B%289%2Ay-8%2Ay%29=-6%2B12

%2812-12%29%2Ax%2B%289-8%29y=-6%2B12

cross%2812%2B-12%29%2Ax%2B%289-8%29%2Ay=-6%2B12 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 4%2Ax%2B3%2Ay=-2 to solve for x

4%2Ax%2B3%286%29=-2 Plug in y=6


4%2Ax%2B18=-2 Multiply



4%2Ax=-2-18 Subtract 18 from both sides

4%2Ax=-20 Combine the terms on the right side

cross%28%281%2F4%29%284%29%29%2Ax=%28-20%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4 on the left side.


x=-5 Multiply the terms on the right side


So our answer is

x=-5, y=6

which also looks like

(-5, 6)

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

4%2Ax%2B3%2Ay=-2
3%2Ax%2B2%2Ay=-3

we get



graph of 4%2Ax%2B3%2Ay=-2 (red) 3%2Ax%2B2%2Ay=-3 (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 (-5,6). This verifies our answer.