im a little bit confused with this number problem?

Question

Twice the sum of two numbers exceeds three times their difference by 8; while half the sum is one more than the difference. What are the numbers?


can you show a clear solution for these?!

please show how you get the answer!

Answer 1

The two numbers are x and y.

"Twice the sum of the two numbers" = 2(x + y)
"Three times their difference" = 3(x - y)

So..
"Twice the sum of two numbers exceeds three times their difference by 8" =
2(x + y) = 3(x - y) + 8

And..
"half the sum" = (1/2)(x + y)

So...
"half the sum is one more than the difference" means
(1/2)(x + y) = 1 + (x - y)

So, now you have two equations and two unknowns:
#1: 2(x + y) = 3(x - y) + 8
#2: (1/2)(x + y) = 1 + (x - y)

Simplify them both:
#1: 2(x + y) = 3(x - y) + 8
2x + 2y = 3x - 3y + 8
0 = x - 5y + 8
x - 5y = -8

#2: (1/2)(x + y) = 1 + (x - y)
(1/2)x + (1/2)y = 1 + x - y
0 = 1 + (1/2)x - (3/2)y
0 = 2 + x - 3y
x - 3y = -2

So now we have:
#1: x - 5y = -8
#2: x - 3y = -2

Subtract:
#1 - #2
(x - 5y) - (x - 3y) = (-8) - (-2)
x - 5y - x + 3y = -8 + 2
-2y = -6
y = 3

Plug that back in to find x:
x - 5y = -8
x - 5(3) = -8
x - 15 = -8
x = 7

So... x = 7 and y = 3.

Answer 2

Let's call the two numbers 'x' and 'y,' and we'll just set up the two equations that we are given:

2(x + y) = 3(x - y) + 8
(1/2)(x + y) = (x - y) + 1

Let's just go ahead and solve for 'y' in terms of 'x,' so we'll start with the first equation:

2(x + y) = 3(x - y) + 8
==> distribute constants
2x + 2y = 3x - 3y + 8
==> subtract 2y from both sides
2x = 3x - 5y + 8
==> subtract 3x from both sides
-x = -5y + 8
==> divide by -1
x = 5y - 8

Plug this into the second equation:

(1/2)((5y - 8) + y) = ((5y - 8) - y) + 1
==> combine like terms
(1/2)(6y - 8) = (4y - 7)
==> multiply both sides by 2
6y - 8 = 8y - 14
==> subtract 6y from both sides
-8 = 2y -14
==> add 14 from both sides
6 = 2y
==> divide by 2 on both sides
3 = y

Now, plug this into the equation we solved for 'x':

x = 5y - 8 = 5(3) - 8 = 15 - 8 = 7

So, our answers are that x = 7, y = 3.

Go ahead and check:

2(x + y) = 3(x - y) + 8
2(7 + 3) = 3(7 - 3) + 8
2(10)= 3(4) + 8
20 = 20 ....... GOOD

(1/2)(x + y) = (x - y) + 1
(1/2)(7 + 3) = (7 - 3) + 1
(1/2)(4) = 4 + 1
5 = 5 ....... GOOD

Answer 3

Let the numbers be x and y
2(x + y) - 8 = 3(x - y)
2x + 2y - 8 = 3x - 3y
- x + 5y = 8

(1/2)(x + y) - 1 = x - y
x + y - 2 = 2x - 2y
x - 3y = - 2

- x + 5y = 8
x - 3y = - 2-----ADD

2y = 6
y = 3

x - 9 = - 2
x = 7

x = 7 , y = 3

Answer 4

Let the nos. be x and y
Hence their sum and difference are x+y and x-y respectively.
By the conditions of the problem
2(x+y)=3(x-y)+8
2x+2y=3x-3y+8
or 2x-3x+2y+3y=8
or-x+5y=8.............(equation 1)
By the second condtion of the problem,
(1/2)(x+y)=(x-y)+1
x+y=2(x-y)+2 [multiplying both sides by 2]
x+y=2x-2y+2
x+y-2x+2y=2
-x+3y=2..........(eqquation 2
Subtracting equation 2 from eqn 1
2y=6
y=3
Plugging the value of y in eqn 2
-x+9=2
-x=2-9= -7
x=7
Hence the numbers are 7 and 3

Answer 5

Translate the given info

2(x+y) = 3(x-y) + 8

1/2(x+y) = (x-y) + 1

2x + 2y = 3x - 3y + 8

-x + 5y = 8

1/2(x+y) = (x-y) + 1

x + y = 2x - 2y + 2

-x + 3y = 2

-x + 5y = 8
-x + 3y = 2

2y = 6

y = 3

-x + 15 = 8

-x = -7

x = -7

Answer 6

let's say the numbers are x and y

2(x+y) = 3(x-y)+8
1/2(x+y) = (x-y)+1
2x+2y=3x-3y+8
x-5y+8=0..........1

x+y=2x-2y+2 (multiply both sides by 2)

x-3y+2=0.....2
x = 3y-2
3y-2-5y+8=0 (pluging the value of x back into equation 1)
-2y=-6
y=3
x=3*3-2
x = 7

test it out
2(x+y) = 3(x-y)+8
2(7+3)=3(7-3)+8
20 = 20

Answer 7

7 and 3

since 2(x+y) = 3(x-y)+8
so 2x+2y = 3x-3y+8
so 5y-x=8

and (x+y)/2 = x-y+1 [2]
so x+y=2x-2y+2
so 3y-x = 2

so 2y=6, so y=3, so x = 7
.

Answer 8

2(x+y)=3(x-y)+8 (i)
(x+y)/2=(x-y) + 1 (ii)

solving (i)
2x + 2y = 3x - 3y + 8
5y - x = 8
5y = x+8
x= 5y - 8
from (ii)
x + y = 2x - 2y + 2
5y - 8 + y = 10y - 16 -2y + 2
-2y = -6
y = 3
x = 7

Answer 9

2(A + B ) = 3 (A - B) + 8 . . . equation 1
2A + 2B = 3A - 3B + 8
A = 5B - 8 . . . . . . . . . . also equation 1
1/2(A + B) = (A - B) + 1 . . . equation 2
(A + B) = 2 (A - B) + 2 . . . equation 2 multiply by 2
A + B = 2A - 2B + 2
A = 3B -2 . . . . substitute to equation 1
3B -2 = 5B - 8
2B = 6
B = 3
A = 3(3) - 2 = 7