SOLUTION: {(6 x -5 y=4),(6 x + 3 y=3)}addition method??

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Question 395443: {(6 x -5 y=4),(6 x + 3 y=3)}addition method??
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

6x+-5y=4
6x+%2B+3y=3..........addition method

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


Lets start with the given system of linear equations

6%2Ax-5%2Ay=4
6%2Ax%2B3%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 6 and 6 to some equal number, we could try to get them to the LCM.

Since the LCM of 6 and 6 is 6, 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%286%2Ax-5%2Ay%29=%284%29%2A1 Multiply the top equation (both sides) by 1
-1%2A%286%2Ax%2B3%2Ay%29=%283%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
6%2Ax-5%2Ay=4
-6%2Ax-3%2Ay=-3

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%29-5%2Ay-3%2Ay%29=4-3

%286-6%29%2Ax-5-3%29y=4-3

cross%286%2B-6%29%2Ax%2B%28-5-3%29%2Ay=4-3 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:

-8%2Ay=1

y=1%2F-8 Divide both sides by -8 to solve for y



y=-1%2F8 Reduce


Now plug this answer into the top equation 6%2Ax-5%2Ay=4 to solve for x

6%2Ax-5%28-1%2F8%29=4 Plug in y=-1%2F8


6%2Ax%2B5%2F8=4 Multiply



6%2Ax%2B5%2F8=4 Reduce



6%2Ax=4-5%2F8 Subtract 5%2F8 from both sides

6%2Ax=32%2F8-5%2F8 Make 4 into a fraction with a denominator of 8

6%2Ax=27%2F8 Combine the terms on the right side

cross%28%281%2F6%29%286%29%29%2Ax=%2827%2F8%29%281%2F6%29 Multiply both sides by 1%2F6. This will cancel out 6 on the left side.


x=9%2F16 Multiply the terms on the right side


So our answer is

x=9%2F16, y=-1%2F8

which also looks like

(9%2F16, -1%2F8)

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

6%2Ax-5%2Ay=4
6%2Ax%2B3%2Ay=3

we get



graph of 6%2Ax-5%2Ay=4 (red) 6%2Ax%2B3%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 (9%2F16,-1%2F8). This verifies our answer.