SOLUTION: use elimination to find all points of intersection of the graphs of 3x+2y=10 and 2x+5y=3.

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Question 778382: use elimination to find all points of intersection of the graphs of 3x+2y=10 and 2x+5y=3.
Answer by MathLover1(20849) 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%2B2%2Ay=10
2%2Ax%2B5%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 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%2B2%2Ay%29=%2810%29%2A2 Multiply the top equation (both sides) by 2
-3%2A%282%2Ax%2B5%2Ay%29=%283%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
6%2Ax%2B4%2Ay=20
-6%2Ax-15%2Ay=-9

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%284%2Ay-15%2Ay%29=20-9

%286-6%29%2Ax%2B%284-15%29y=20-9

cross%286%2B-6%29%2Ax%2B%284-15%29%2Ay=20-9 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=11

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



y=-1 Reduce


Now plug this answer into the top equation 3%2Ax%2B2%2Ay=10 to solve for x

3%2Ax%2B2%28-1%29=10 Plug in y=-1


3%2Ax-2=10 Multiply



3%2Ax=10%2B2 Subtract -2 from both sides

3%2Ax=12 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%2812%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=4 Multiply the terms on the right side


So our answer is

x=4, y=-1

which also looks like

(4, -1)

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

3%2Ax%2B2%2Ay=10
2%2Ax%2B5%2Ay=3

we get



graph of 3%2Ax%2B2%2Ay=10 (red) 2%2Ax%2B5%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 (4,-1). This verifies our answer.