SOLUTION: SOLVE EACH SYSTEM OF EQUATIONS BY USING ELIMINATION 1. 3x-2y=4 2x+3y=7 2. 4x-y=10 5x+2y=6

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Question 471182: SOLVE EACH SYSTEM OF EQUATIONS BY USING ELIMINATION
1. 3x-2y=4 2x+3y=7

2. 4x-y=10 5x+2y=6

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

1.
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

3%2Ax-2%2Ay=4
2%2Ax%2B3%2Ay=7

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-2%2Ay%29=%284%29%2A2 Multiply the top equation (both sides) by 2
-3%2A%282%2Ax%2B3%2Ay%29=%287%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
6%2Ax-4%2Ay=8
-6%2Ax-9%2Ay=-21

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%29-4%2Ay-9%2Ay%29=8-21

%286-6%29%2Ax-4-9%29y=8-21

cross%286%2B-6%29%2Ax%2B%28-4-9%29%2Ay=8-21 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:

-13%2Ay=-13

y=-13%2F-13 Divide both sides by -13 to solve for y



y=1 Reduce


Now plug this answer into the top equation 3%2Ax-2%2Ay=4 to solve for x

3%2Ax-2%281%29=4 Plug in y=1


3%2Ax-2=4 Multiply



3%2Ax=4%2B2 Subtract -2 from both sides

3%2Ax=6 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%286%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=2 Multiply the terms on the right side


So our answer is

x=2, y=1

which also looks like

(2, 1)

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

3%2Ax-2%2Ay=4
2%2Ax%2B3%2Ay=7

we get



graph of 3%2Ax-2%2Ay=4 (red) 2%2Ax%2B3%2Ay=7 (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 (2,1). This verifies our answer.


2.
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

4%2Ax-1%2Ay=10
5%2Ax%2B2%2Ay=6

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

Since the LCM of 4 and 5 is 20, we need to multiply both sides of the top equation by 5 and multiply both sides of the bottom equation by -4 like this:

5%2A%284%2Ax-1%2Ay%29=%2810%29%2A5 Multiply the top equation (both sides) by 5
-4%2A%285%2Ax%2B2%2Ay%29=%286%29%2A-4 Multiply the bottom equation (both sides) by -4


So after multiplying we get this:
20%2Ax-5%2Ay=50
-20%2Ax-8%2Ay=-24

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%29-5%2Ay-8%2Ay%29=50-24

%2820-20%29%2Ax-5-8%29y=50-24

cross%2820%2B-20%29%2Ax%2B%28-5-8%29%2Ay=50-24 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:

-13%2Ay=26

y=26%2F-13 Divide both sides by -13 to solve for y



y=-2 Reduce


Now plug this answer into the top equation 4%2Ax-1%2Ay=10 to solve for x

4%2Ax-1%28-2%29=10 Plug in y=-2


4%2Ax%2B2=10 Multiply



4%2Ax=10-2 Subtract 2 from both sides

4%2Ax=8 Combine the terms on the right side

cross%28%281%2F4%29%284%29%29%2Ax=%288%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4 on the left side.


x=2 Multiply the terms on the right side


So our answer is

x=2, y=-2

which also looks like

(2, -2)

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

4%2Ax-1%2Ay=10
5%2Ax%2B2%2Ay=6

we get



graph of 4%2Ax-1%2Ay=10 (red) 5%2Ax%2B2%2Ay=6 (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 (2,-2). This verifies our answer.