SOLUTION: How to use an elimination strategy to solve the linear system: 5/3x +1/4y = 10 1/3x + 1/2y = 5 ans: (5/3x +1/4y = 10 ) * 12 --> 30x+3y=120 (1/3x + 1/2y = 5) * 6 -->

Algebra ->  Coordinate Systems and Linear Equations  -> Lessons -> SOLUTION: How to use an elimination strategy to solve the linear system: 5/3x +1/4y = 10 1/3x + 1/2y = 5 ans: (5/3x +1/4y = 10 ) * 12 --> 30x+3y=120 (1/3x + 1/2y = 5) * 6 -->      Log On


   

Question 962387: How to use an elimination strategy to solve the linear system:
5/3x +1/4y = 10
1/3x + 1/2y = 5


ans:
(5/3x +1/4y = 10 ) * 12 --> 30x+3y=120
(1/3x + 1/2y = 5) * 6 --> 2x+3y=30
I never seem to get the answer right everything I check the solution. Plz help me correct it ??
Answer by MathLover1(20850) 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
TEST

%285%2F3%29%2Ax%2B%281%2F4%29%2Ay=10 Start with the first equation


12%28%285%2F3%29%2Ax%2B%281%2F4%29%2Ay%29=%2812%29%2A%2810%29 Multiply both sides by the LCD 12



20%2Ax%2B3%2Ay=120Distribute and simplify


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%281%2F3%29%2Ax%2B%281%2F2%29%2Ay=5 Start with the second equation


6%28%281%2F3%29%2Ax%2B%281%2F2%29%2Ay%29=%286%29%2A%285%29 Multiply both sides by the LCD 6



2%2Ax%2B3%2Ay=30 Distribute and simplify



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Lets start with the given system of linear equations

20%2Ax%2B3%2Ay=120
2%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 20 and 2 to some equal number, we could try to get them to the LCM.

Since the LCM of 20 and 2 is 20, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -10 like this:

1%2A%2820%2Ax%2B3%2Ay%29=%28120%29%2A1 Multiply the top equation (both sides) by 1
-10%2A%282%2Ax%2B3%2Ay%29=%2830%29%2A-10 Multiply the bottom equation (both sides) by -10


So after multiplying we get this:
20%2Ax%2B3%2Ay=120
-20%2Ax-30%2Ay=-300

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%283%2Ay-30%2Ay%29=120-300

%2820-20%29%2Ax%2B%283-30%29y=120-300

cross%2820%2B-20%29%2Ax%2B%283-30%29%2Ay=120-300 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=-180

y=-180%2F-27 Divide both sides by -27 to solve for y



y=20%2F3 Reduce


Now plug this answer into the top equation 20%2Ax%2B3%2Ay=120 to solve for x

20%2Ax%2B3%2820%2F3%29=120 Plug in y=20%2F3


20%2Ax%2B60%2F3=120 Multiply



20%2Ax%2B20=120 Reduce



20%2Ax=120-20 Subtract 20 from both sides

20%2Ax=100 Combine the terms on the right side

cross%28%281%2F20%29%2820%29%29%2Ax=%28100%29%281%2F20%29 Multiply both sides by 1%2F20. This will cancel out 20 on the left side.


x=5 Multiply the terms on the right side


So our answer is

x=5, y=20%2F3

which also looks like

(5, 20%2F3)

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

20%2Ax%2B3%2Ay=120
2%2Ax%2B3%2Ay=30

we get



graph of 20%2Ax%2B3%2Ay=120 (red) 2%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 (5,20%2F3). This verifies our answer.