SOLUTION: hi, can you please help me solve this problem: using the elimination method: 1. 3x-2y=5 2x+3y=12 2. 4x+3y+3=0 8x=9y-1 this ones either substitution,

Algebra ->  Linear-equations -> SOLUTION: hi, can you please help me solve this problem: using the elimination method: 1. 3x-2y=5 2x+3y=12 2. 4x+3y+3=0 8x=9y-1 this ones either substitution,      Log On


   

Question 425022: hi, can you please help me solve this problem:
using the elimination method:
1. 3x-2y=5
2x+3y=12
2. 4x+3y+3=0
8x=9y-1
this ones either substitution, elimination or comparison methods:
3. 0.3x-0.5y=1.2
0.7x-0.2y=-0.1
4. x/3+y/4=2
2x/3-y/2=0

thank you so much for your help..ana
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

the elimination method:
1. 3x-2y=5
2x%2B3y=12

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=5
2%2Ax%2B3%2Ay=12

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=%285%29%2A2 Multiply the top equation (both sides) by 2
-3%2A%282%2Ax%2B3%2Ay%29=%2812%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
6%2Ax-4%2Ay=10
-6%2Ax-9%2Ay=-36

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=10-36

%286-6%29%2Ax-4-9%29y=10-36

cross%286%2B-6%29%2Ax%2B%28-4-9%29%2Ay=10-36 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 3%2Ax-2%2Ay=5 to solve for x

3%2Ax-2%282%29=5 Plug in y=2


3%2Ax-4=5 Multiply



3%2Ax=5%2B4 Subtract -4 from both sides

3%2Ax=9 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%289%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=3 Multiply the terms on the right side


So our answer is

x=3, y=2

which also looks like

(3, 2)

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

3%2Ax-2%2Ay=5
2%2Ax%2B3%2Ay=12

we get



graph of 3%2Ax-2%2Ay=5 (red) 2%2Ax%2B3%2Ay=12 (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 (3,2). This verifies our answer.



2. 4x%2B3y%2B3=0
8x=9y-1
or 4x%2B3y=-3
8x-9y=-1

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


Lets start with the given system of linear equations

4%2Ax%2B3%2Ay=-3
8%2Ax-9%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 4 and 8 to some equal number, we could try to get them to the LCM.

Since the LCM of 4 and 8 is 8, we need to multiply both sides of the top equation by 2 and multiply both sides of the bottom equation by -1 like this:

2%2A%284%2Ax%2B3%2Ay%29=%28-3%29%2A2 Multiply the top equation (both sides) by 2
-1%2A%288%2Ax-9%2Ay%29=%28-1%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
8%2Ax%2B6%2Ay=-6
-8%2Ax%2B9%2Ay=1

Notice how 8 and -8 add to zero (ie 8%2B-8=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%288%2Ax-8%2Ax%29%2B%286%2Ay%2B9%2Ay%29=-6%2B1

%288-8%29%2Ax%2B%286%2B9%29y=-6%2B1

cross%288%2B-8%29%2Ax%2B%286%2B9%29%2Ay=-6%2B1 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:

15%2Ay=-5

y=-5%2F15 Divide both sides by 15 to solve for y



y=-1%2F3 Reduce


Now plug this answer into the top equation 4%2Ax%2B3%2Ay=-3 to solve for x

4%2Ax%2B3%28-1%2F3%29=-3 Plug in y=-1%2F3


4%2Ax-3%2F3=-3 Multiply



4%2Ax-1=-3 Reduce



4%2Ax=-3%2B1 Subtract -1 from both sides

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

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


x=-1%2F2 Multiply the terms on the right side


So our answer is

x=-1%2F2, y=-1%2F3

which also looks like

(-1%2F2, -1%2F3)

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

4%2Ax%2B3%2Ay=-3
8%2Ax-9%2Ay=-1

we get



graph of 4%2Ax%2B3%2Ay=-3 (red) 8%2Ax-9%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 (-1%2F2,-1%2F3). This verifies our answer.



this ones either substitution, elimination or comparison methods:
3. 0.3x-0.5y=1.2
0.7x-0.2y=-0.1
Solved by pluggable solver: SOLVE linear system by SUBSTITUTION
Solve:
We'll use substitution. After moving -0.5*y to the right, we get:
0.3%2Ax+=+1.2+-+-0.5%2Ay, or x+=+1.2%2F0.3+-+-0.5%2Ay%2F0.3. Substitute that
into another equation:
0.7%2A%281.2%2F0.3+-+-0.5%2Ay%2F0.3%29+%2B+-0.2%5Cy+=+-0.1 and simplify: So, we know that y=-3. Since x+=+1.2%2F0.3+-+-0.5%2Ay%2F0.3, x=-1.

Answer: system%28+x=-1%2C+y=-3+%29.



4. x%2F3%2By%2F4=2
2x%2F3-y%2F2=0
or
%281%2F3%29x%2B%281%2F4%29y=2
%282%2F3%29x-%281%2F2%29y=0

Solved by pluggable solver: SOLVE linear system by SUBSTITUTION
Solve:
We'll use substitution. After moving 0.25*y to the right, we get:
0.333333333333333%2Ax+=+2+-+0.25%2Ay, or x+=+2%2F0.333333333333333+-+0.25%2Ay%2F0.333333333333333. Substitute that
into another equation:
and simplify: So, we know that y=0. Since x+=+2%2F0.333333333333333+-+0.25%2Ay%2F0.333333333333333, x=6.00000000000001.

Answer: system%28+x=6.00000000000001%2C+y=0+%29.