Twice the sum of two numbers exceed three times their difference by 8, while half the sum is one more than the difference. What are the numbers?
2(x+y) = 3(x-y)+8
(x+y)/2 = (x-y)+1
The numbers are 7 and 3
Let's assume that the two numbers are represented by x and y.
According to the given information:
1) "Twice the sum of two numbers exceed three times their difference by 8":
2(x + y) = 3(x - y) + 8
2) "Half the sum is one more than the difference":
0.5(x + y) = (x - y) + 1
Now, let's solve these equations step-by-step to find the values of x and y:
Expanding equation (1):
2x + 2y = 3x - 3y + 8
Rearranging terms:
2y + 3y = 2x - 3x + 8
Combining like terms:
5y = -x + 8
Similarly, expanding equation (2):
0.5x + 0.5y = x - y + 1
Rearranging terms:
0.5y + y = x - 0.5x + 1
Combining like terms:
1.5y = 0.5x + 1
Now, we have a system of equations:
5y = -x + 8
1.5y = 0.5x + 1
Let's solve these equations using the method of substitution:
From equation (1), let's solve for x:
5y = -x + 8
x = -5y + 8
Substituting x in equation (2):
1.5y = 0.5(-5y + 8) + 1
Expanding the expression:
1.5y = -2.5y + 4 + 1
Combining like terms:
1.5y = -2.5y + 5
Adding 2.5y to both sides:
1.5y + 2.5y = 5
Combining like terms:
4y = 5
Dividing both sides by 4:
y = 5/4
Substituting y back into equation (1) to find x:
5(5/4) = -x + 8
25/4 = -x + 8
Multiplying both sides by 4:
25 = -4x + 32
Subtracting 32 from both sides:
-7 = -4x
Dividing both sides by -4 (keeping in mind that dividing by negative changes the sign):
7/4 = x
Therefore, the numbers are x = 7/4 and y = 5/4.
To solve this problem, we can start by assigning variables to the two numbers. Let's call the first number "x" and the second number "y".
According to the given information, "Twice the sum of two numbers exceed three times their difference by 8." We can translate this into the following equation:
2(x + y) = 3(x - y) + 8
The second piece of information states that "half the sum is one more than the difference." This can be represented by the equation:
0.5(x + y) = (x - y) + 1
Now we have a system of two equations with two variables. We can solve this system to find the values of x and y.
Let's begin by simplifying the first equation:
2x + 2y = 3x - 3y + 8
Next, let's simplify the second equation:
0.5x + 0.5y = x - y + 1
To eliminate decimals, we can multiply both sides of the second equation by 2:
x + y = 2x - 2y + 2
Now we have a system of equations:
2x + 2y = 3x - 3y + 8
x + y = 2x - 2y + 2
We can solve this system by isolating one variable and substituting it into the other equation. Let's isolate x in the second equation:
x = 2y - y + 2
x = y + 2
Now we substitute this expression for x in the first equation:
2(y + 2) + 2y = 3(y + 2) - 3y + 8
2y + 4 + 2y = 3y + 6 - 3y + 8
4y + 4 = 6 + 8
4y + 4 = 14
4y = 14 - 4
4y = 10
y = 10/4
y = 2.5
Now that we have the value of y, we can substitute it back into the second equation to find x:
x + 2.5 = 2(x - 2.5) + 2
x + 2.5 = 2x - 5 + 2
x + 2.5 = 2x - 3
2.5 + 3 = 2x - x
5.5 = x
Therefore, the two numbers are x = 5.5 and y = 2.5.