SOLUTION: Determine the solution sets of the following systems of linear equation by elimination. 3x-8y=-18 x+y=1

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Question 911966: Determine the solution sets of the following systems of linear equation by elimination.

3x-8y=-18
x+y=1
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-8%2Ay=-18
1%2Ax%2B1%2Ay=1

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

Since the LCM of 3 and 1 is 3, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -3 like this:

1%2A%283%2Ax-8%2Ay%29=%28-18%29%2A1 Multiply the top equation (both sides) by 1
-3%2A%281%2Ax%2B1%2Ay%29=%281%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
3%2Ax-8%2Ay=-18
-3%2Ax-3%2Ay=-3

Notice how 3 and -3 add to zero (ie 3%2B-3=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%283%2Ax-3%2Ax%29-8%2Ay-3%2Ay%29=-18-3

%283-3%29%2Ax-8-3%29y=-18-3

cross%283%2B-3%29%2Ax%2B%28-8-3%29%2Ay=-18-3 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:

-11%2Ay=-21

y=-21%2F-11 Divide both sides by -11 to solve for y



y=21%2F11 Reduce


Now plug this answer into the top equation 3%2Ax-8%2Ay=-18 to solve for x

3%2Ax-8%2821%2F11%29=-18 Plug in y=21%2F11


3%2Ax-168%2F11=-18 Multiply



3%2Ax-168%2F11=-18 Reduce



3%2Ax=-18%2B168%2F11 Subtract -168%2F11 from both sides

3%2Ax=-198%2F11%2B168%2F11 Make -18 into a fraction with a denominator of 11

3%2Ax=-30%2F11 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%28-30%2F11%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=-10%2F11 Multiply the terms on the right side


So our answer is

x=-10%2F11, y=21%2F11

which also looks like

(-10%2F11, 21%2F11)

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

3%2Ax-8%2Ay=-18
1%2Ax%2B1%2Ay=1

we get



graph of 3%2Ax-8%2Ay=-18 (red) 1%2Ax%2B1%2Ay=1 (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 (-10%2F11,21%2F11). This verifies our answer.