Addition Method...?

Question

My mind is going blank this morning as I try and finish my overdue homework!! Please walk me thru a couple problems using the Addition Method..

1) x+y=1
2x-y=5

2) x-2y=7
3x-2y=9

3) 4x-8y=5
8x+2y=1

4)3x+6y=7
2x+4y=5

5) 3x-4y=0
4x-7y=0

THANKS A MILLION FOR ANY HELP!

Answer 1

Hi,


1) Add these as is and the y terms drop out. Solve for x. Then plug that answer back in an equation and solve for y.
x+y=1
2x-y=5
----------
3x = 6
x = 2
If x = 2 then x+y=1 becomes 2+y = 1. This solves to y = -1. So the answer is (2,-1).


2) On this problem, multiply the first equation all by -1 so you will have +2y in one equation and -2y in the other. Now add the equations so y drops out. Solve for x. Plug it back in to get y.

x-2y=7
3x-2y=9

-1( x-2y=7)
3x-2y=9

-x+2y= -7
3x-2y=9
-------------
2x = 2
x = 1
If x = 1, then x - 2y = 7 becomes 1 - 2y = 7. This becomes -2y = 6, so y = -3.

The answer is (1,-3).

3) You can really eliminate either letter, so to be different we will eliminate the "x" this time. To do that, multiply the first equation by -2 so that the equations would have a -8x and 8x. Add them together and the x terms drop out. Solve for y and plug it back in to find x.

4x-8y=5
8x+2y=1

-2(4x-8y=5)
8x+2y=1

-8x+16y=-10
8x+2y=1
-------------
18y = -9
y = -½
If y = -½, then 4x - 8y = 5 becomes 4x -8(-½) = 5. this simplifies to 4x + 4 = 5. This becomes 4x = 1 so x = ¼.
The answer is (¼,-½).


4) This time we have to multiply both equations to get our terms to drop out. Multiply the entire top equation by 2 to get a +12y and multiply the entire bottom equation by -3 to get a -12y. Then the y terms will drop out when the equations are added together. Solve for x and use it to solve for y.

3x+6y=7
2x+4y=5

2(3x+6y=7)
-3(2x+4y=5)

6x+12y=14
-6x-12y=-15
------------------
0 = -1 This time when we expected the y terms to drop out the x terms dropped out too, giving us the equation 0 = -1.We know this is never true. That means this system of equations NEVER has a solution either. So the answer is this system of equations has "No Solution".

If both the x and y terms had dropped out and given an equation like 0 = 0 that is always true, then the answer would have been "All Real Numbers" would work in that system of equations.

5) This time we have to multiply both equations to get our terms to drop out. Multiply the entire top equation by 7 to get a -28y and multiply the entire bottom equation by -4 to get a 28y. Then the y terms will drop out when the equations are added together. Solve for x and use it to solve for y.

3x-4y=0
4x-7y=0

7(3x-4y=0)
-4(4x-7y=0)

21x-28y=0
-16x+28y=0
------------------
5x = 0
x = 0
If x = 0, then 3x-4y=0 becomes 3(0) -4y=0. This simplifies to -4y=0 and solves to y = 0.
The answer is (0,0).

I hope those help!! :-)

Answer 2

1)

elimination by addition method

Elimination of y

x + y = 1- - - - - -Equation 1
2x - y = 5- - - - - -Equation 2
- - - - - - - -

3x = 6

Divide both sides of the equation by 3

3x / 3 = 6 / 3

x = 6/3

x = 2

Insert the x value into equation 1
- - - - - - - - -

x + y = 1

2 + y = 1

Transpose 2

2 + y - 2 = 1 - 2

y = 1 - 2

y = - 1

Insert the y value into equation 1
- - - - - - - - -

Check for equation 1

x + y = 1

2 + (- 1) = 1

Remove parenthkesis

2 - 1 = 1

1 = 1

- - - - - - -

Check for equation 2

2x - y = 5

2(2) - (- 1) = 5

4 + 1 = 5

5 = 5
- - - - - - -

Both equations balance

The solution set is { 2, - 1}

- - - - - - - s-

Answer 3

x+y = 1 ----Equation1
2x-y = 5 ----Equation2

Multiply both sides of equation 1 by 2,
2x+2y=2 ---- Equation 3

Equation3 - Equation 2
2y-y=2-5
y = -3

x-3 = 1
x = 4

2) x-2y=7 --Equation1
3x-2y=9 --Equation2

Equation 2-Equation1
3x-x = 9-7
2x = 2
x = 1

1-2y = 7
-2y = 6
y = -3

Answer 4

x+y =1 -------- 1
2x-y=5-------------2
multiply equ 1 with no 2 and add
2x+2y=2
2x-y=5
subtracting these equations
2x+2y- 2x-y= 2-5
2y-y=-3
y= -3
substuite y=-3
x-3=1
x=4
y=-3

2) x-2y=7 -------- 1
3x-2y=9 ----------2
subtract equation1 -equation2
x-2y-3x+2y=7-9
-2x= -2
x=1
substutie the value of x=1 in equation 1
1-2y=7
-2y=6
y= -3

3) 4x-8y=5
8x+2y=1
multiply equation 2 with 4 and add
4x-8y = 5
32x+8y=4
add both the equations
36x=9
x=1/4
substuite the value of x in euation 1
4(1/4) -8y=5
1-8y =5
-8y =4
y= -1/2
x=1/4

4)3x+6y=7
2x+4y=5
multiply equa 1 with 2 and equation 2 with 3
6x+12y=14
6x-12y=15
add
12x=29
x=29/12
substuite in 1
3(29/12) + 6y =7
29/4 +6y=7
6y=7-29/4
6y= 28-29//4
6y= -1/4
y=-1/24

5) 3x-4y=0
4x-7y=0

mult eq 1 with 4 and equ 2with 3
12x-16y= 0
12x-21y=0
sub both the equations
5y=0
y=0
x=0

Answer 5

1) x+y=1
2x-y=5
>add (2x and x, y and -y, 1 and 5)
3x+0y=6
3x=6
x=2
if u need to solve for y just substitute the value of x to any equation...
x+y=1
2+y=1
y=-1

2)x-2y=7
3x-2y=9

multiply -1 to any equation so that 2y will be eliminated if you add
-1(x-2y=7)
3x-2y=9

-x+2y=-7
3x-2y=9
add(-x and 3x, 2y and -2y, -7 and 9)
2x+0y=2
2x=2
x=1
substitute the value of x to any original equation if you want to solve y
x-2y=7
1-2y=7
-2y=6
y=-3

3) 4x-8y=5
8x+2y=1
decide if what variable you want to eliminate... for example x
so think of a number when you multiply it to an equation it can eliminate the variable x...
in the first equation u can multiply all by -2 so that 4x will be -8x and when you add x will be eliminated...
-2(4x-8y=5)
-8x+16y-5

-8x+16y=-10
8x+2y=1
add...
0x+18y=-9
y=-1/2
if you want to solve x substitute the value of y to any orig equation...
8x+2y=1
8x+2(-1/2)=1
8x-1=1
8x=2
x=1/4

4) 3x+6y=7
2x+4y=5
multiply 2 in the first equation and -3 in the second equation
2(3x+6y=7)
-3(2x+4y=5)

6x+12y=14
-6x-12y=-15
0x+0y=-1

this will result to 0=-1 and we know that 0 and -1 will never be equal therefore there is 'NO ANSWER or NO SOLUTION' in this question

5) 3x-4y=0
4x-7y=0
if u use your common sense x and y is 0 because all are equated to zero and the only value that when you multiply will result to zero is zero...

let's go back to addition method
multiply -4 in the first equation and 3 in the second equation to eliminate x

-4(3x-4y=0)
3(4x-7y=0)

-12x+16y=0
12x-21y=0
0x-5y=0
-5y=0
y=0
substitute y to any original equation...
3x-4y=0
3x-4(0)=0
3x-0=0
3x=0
x=0

Answer 6

1)x=2 2+(-1)=1 or 2-1=1
y=-1 2x2-(-1)=5 or 4+1=5

2)x=1
y=-3

3)x=1/4
y=-1/2

4)Sorry!!

5)x=8.78
y=-5

Answer 7

Yes I think I got all the answers..........You show me yours and I'll show you mine.