SOLUTION: I need help with the following problems. All of these problems ar inside of the curly brackets.. example { } Solve by any convenient method : 4x + 12y = 24 2x + 6y = 12

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Question 88885: I need help with the following problems. All of these problems ar inside of the curly brackets.. example { }
Solve by any convenient method :
4x + 12y = 24
2x + 6y = 12



Solve by any convenient method :
8x + 4y =7
x = 2-2y

Solve by Elimination :
2x -3y =-1
3x + y +15

Solve by Substitution :
3x + 8y = 7
x- 4y =9
Found 3 solutions by stanbon, jim_thompson5910, malakumar_kos@yahoo.com:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Solve by any convenient method :
4x + 12y = 24
2x + 6y = 12
--------------
Multiply the 2nd equation by 2 and you will see the two equations are the same.
So the solution for the system of equations is 2x+6y=12
or x+3y=6 or y=-(1/3)x+2
---------------------------------
Solve by any convenient method :
8x + 4y =7
x = 2-2y
------
Substitute x=2-2y into the 1st equation to solve for y:
8(2-2y)+4y=7
16-16y+4y=7
-12y = -9
y = 3/4
-----------------
Solve by Elimination :
2x -3y =-1
3x + y =15
-------------
Multiply the 2nd equation by 3 to get:
9x+3y=45
Add that to the 1st equation so you can solve for x:
11x=44
x=4
-------
Substitute that into 3x+y=15 to solve for y:
3*4+y=15
y = 3
------------------------
Solve by Substitution :
3x + 8y = 7
x- 4y =9
--
Solve the 2nd equation for x: x=4y+9
Substitute into the 1st equation so you can solve for y:
3(4y+9)+8y = 7
12y+27+8y = 7
20y = -20
y = -1
-------
Substitute that into x=4y+9 to solve for x:
x=4*-1+9
x=5
============
cheers,
Stan H.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
"Solve by any convenient method :
4x + 12y = 24
2x + 6y = 12 "

Lets solve by substitution:

Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

4%2Ax%2B12%2Ay=24
2%2Ax%2B6%2Ay=12

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

12%2Ay=24-4%2AxSubtract 4%2Ax from both sides

y=%2824-4%2Ax%29%2F12 Divide both sides by 12.


Which breaks down and reduces to



y=2-%281%2F3%29%2Ax Now we've fully isolated y

Since y equals 2-%281%2F3%29%2Ax we can substitute the expression 2-%281%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


2%2Ax%2B6%2Ahighlight%28%282-%281%2F3%29%2Ax%29%29=12 Replace y with 2-%281%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

2%2Ax%2B6%2A%282%29%2B6%28-1%2F3%29x=12 Distribute 6 to 2-%281%2F3%29%2Ax

2%2Ax%2B12-%286%2F3%29%2Ax=12 Multiply



2%2Ax%2B12-2%2Ax=12 Reduce any fractions

2%2Ax-2%2Ax=12-12 Subtract 12 from both sides


2%2Ax-2%2Ax=0 Combine the terms on the right side



0%2Ax=0 Now combine the terms on the left side.
0=0 Since this expression is true for any x, we have an identity.


So there are an infinite number solutions. The simple reason is the 2 equations represent 2 lines that overlap each other. So they intersect each other at an infinite number of points.

If we graph 4%2Ax%2B12%2Ay=24 and 2%2Ax%2B6%2Ay=12 we get

+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%2824-4%2Ax%29%2F12%29+ graph of 4%2Ax%2B12%2Ay=24


+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%2812-2%2Ax%29%2F6+%29+ graph of 2%2Ax%2B6%2Ay=12 (hint: you may have to solve for y to graph these)

we can see that these two lines are the same. So this system is dependent



--------------------------------------------------------------------------------
"Solve by any convenient method :
8x + 4y =7
x = 2-2y "

Lets solve by substitution:

8%282-2y%29%2B4y=7 Plug in x=2-2y

16-16y%2B4y=7 Distribute

16-12y=7 Combine like terms

-12y=-9 Subtract 16 from both sides

-12y=-9%2F-12 Subtract

y=3%2F4 Reduce


x=2-2%283%2F4%29 Now plug in y=3%2F4

x=2-3%2F2 Multiply

x=1%2F2 Combine like terms

So we have x=1%2F2 and y=3%2F4
--------------------------------------------------------------------------------

"Solve by Elimination :
2x -3y =-1
3x + y +15 "

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


Lets start with the given system of linear equations

2%2Ax-3%2Ay=-1
3%2Ax%2B1%2Ay=15

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 2 and 3 to some equal number, we could try to get them to the LCM.

Since the LCM of 2 and 3 is 6, we need to multiply both sides of the top equation by 3 and multiply both sides of the bottom equation by -2 like this:

3%2A%282%2Ax-3%2Ay%29=%28-1%29%2A3 Multiply the top equation (both sides) by 3
-2%2A%283%2Ax%2B1%2Ay%29=%2815%29%2A-2 Multiply the bottom equation (both sides) by -2


So after multiplying we get this:
6%2Ax-9%2Ay=-3
-6%2Ax-2%2Ay=-30

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%29-9%2Ay-2%2Ay%29=-3-30

%286-6%29%2Ax-9-2%29y=-3-30

cross%286%2B-6%29%2Ax%2B%28-9-2%29%2Ay=-3-30 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=-33

y=-33%2F-11 Divide both sides by -11 to solve for y



y=3 Reduce


Now plug this answer into the top equation 2%2Ax-3%2Ay=-1 to solve for x

2%2Ax-3%283%29=-1 Plug in y=3


2%2Ax-9=-1 Multiply



2%2Ax=-1%2B9 Subtract -9 from both sides

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

cross%28%281%2F2%29%282%29%29%2Ax=%288%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2 on the left side.


x=4 Multiply the terms on the right side


So our answer is

x=4, y=3

which also looks like

(4, 3)

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

2%2Ax-3%2Ay=-1
3%2Ax%2B1%2Ay=15

we get



graph of 2%2Ax-3%2Ay=-1 (red) 3%2Ax%2B1%2Ay=15 (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 (4,3). This verifies our answer.



--------------------------------------------------------------------------------
"Solve by Substitution :
3x + 8y = 7
x- 4y =9"

Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

3%2Ax%2B8%2Ay=7
1%2Ax-4%2Ay=9

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

8%2Ay=7-3%2AxSubtract 3%2Ax from both sides

y=%287-3%2Ax%29%2F8 Divide both sides by 8.


Which breaks down and reduces to



y=7%2F8-%283%2F8%29%2Ax Now we've fully isolated y

Since y equals 7%2F8-%283%2F8%29%2Ax we can substitute the expression 7%2F8-%283%2F8%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B-4%2Ahighlight%28%287%2F8-%283%2F8%29%2Ax%29%29=9 Replace y with 7%2F8-%283%2F8%29%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax-4%2A%287%2F8%29-4%28-3%2F8%29x=9 Distribute -4 to 7%2F8-%283%2F8%29%2Ax

1%2Ax-28%2F8%2B%2812%2F8%29%2Ax=9 Multiply



1%2Ax-7%2F2%2B%283%2F2%29%2Ax=9 Reduce any fractions

1%2Ax%2B%283%2F2%29%2Ax=9%2B7%2F2Add 7%2F2 to both sides


1%2Ax%2B%283%2F2%29%2Ax=18%2F2%2B7%2F2 Make 9 into a fraction with a denominator of 2


1%2Ax%2B%283%2F2%29%2Ax=25%2F2 Combine the terms on the right side



%282%2F2%29%2Ax%2B%283%2F2%29x=25%2F2 Make 1 into a fraction with a denominator of 2

%285%2F2%29%2Ax=25%2F2 Now combine the terms on the left side.


cross%28%282%2F5%29%285%2F2%29%29x=%2825%2F2%29%282%2F5%29 Multiply both sides by 2%2F5. This will cancel out 5%2F2 and isolate x

So when we multiply 25%2F2 and 2%2F5 (and simplify) we get



x=5 <---------------------------------One answer

Now that we know that x=5, lets substitute that in for x to solve for y

1%285%29-4%2Ay=9 Plug in x=5 into the 2nd equation

5-4%2Ay=9 Multiply

-4%2Ay=9-5Subtract 5 from both sides

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

cross%28%281%2F-4%29%28-4%29%29%2Ay=%284%2F1%29%281%2F-4%29 Multiply both sides by 1%2F-4. This will cancel out -4 on the left side.

y=4%2F-4 Multiply the terms on the right side


y=-1 Reduce


So this is the other answer


y=-1<---------------------------------Other answer


So our solution is

x=5 and y=-1

which can also look like

(5,-1)

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

3%2Ax%2B8%2Ay=7
1%2Ax-4%2Ay=9

we get


graph of 3%2Ax%2B8%2Ay=7 (red) and 1%2Ax-4%2Ay=9 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (5,-1). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (5,-1) into the system of equations


Let x=5 and y=-1. Now plug those values into the equation 3%2Ax%2B8%2Ay=7

3%2A%285%29%2B8%2A%28-1%29=7 Plug in x=5 and y=-1


15-8=7 Multiply


7=7 Add


7=7 Reduce. Since this equation is true the solution works.


So the solution (5,-1) satisfies 3%2Ax%2B8%2Ay=7



Let x=5 and y=-1. Now plug those values into the equation 1%2Ax-4%2Ay=9

1%2A%285%29-4%2A%28-1%29=9 Plug in x=5 and y=-1


5%2B4=9 Multiply


9=9 Add


9=9 Reduce. Since this equation is true the solution works.


So the solution (5,-1) satisfies 1%2Ax-4%2Ay=9


Since the solution (5,-1) satisfies the system of equations


3%2Ax%2B8%2Ay=7
1%2Ax-4%2Ay=9


this verifies our answer.

Answer by malakumar_kos@yahoo.com(315) About Me  (Show Source):
You can put this solution on YOUR website!
Solve by any convenient method :
4x + 12y = 24 1)4x+12y = 24 ..........eq'n(1)
2x + 6y = 12 2x+6y = 12...........eq'n(2)
divide eq'n (1) by 4 andeq'n (2) by 2
we get x+3y = 6 and x+3y =6
there is no solution for this set of eq'ns as the
given eq'ns are inconsistent. (on solving the values of x & y are 0 which is meaningless)





Solve by any convenient method : 8x+4y = 7..........(1)
8x + 4y =7
x = 2-2y x+2y = 2............(2)
Solve by Elimination : multiply eq'n (2) by 2
2x -3y =-1
3x + y +15 we get 2x+4y = 4.........eqe'n(3)
Solve by Substitution : subtract eq'n(3) from eq'n(1) we get
3x + 8y = 7
x- 4y =9 6x = 3 or x = 3/6 = 1/2
substitute for x in eq'n(2)
2y = 2-x = 2-1/2 = 4-1/2 = 3/2
solution is x= 1/2 and y = 3/2



3)2x-3y = -1......eq'n(1)
3x+y = 15......eq'n(2)
multiply eq'n(2) by 3, we get 9x+3y = 45....(3)
add eq''n(1) &eq'n(3) we get 11x = 44 or x = 4
substitute for x in eq'n (2) we get 3(4)+y= 15
12+y = 15 or y = 15-12 = 3
solution is x = 4 and y = 3




4)3x+8y = 7.........eq'n(1)
x-4y = 9.........eq'n(2)
from eq'n(2) x = 9+4y.....eq'n(3)
3(9+4y)+8y = 7 (by substituting for x)
27+12y+8y = 7
20y = 7-27 = -20 therefore y = -1
substituting for y in eq'n(2) we get x = 9+4(-1)
x = 9-4 = 5
therefore the solution is x = 5 and y = -1