Divide 56 into two parts such that 3/8 of the greater exceeds 1/4 of the smaller by 6.
G+S=56
3G/8 -6=S/4
Put 56-G in for S in the second equation, and solve for G
can you explain this in more depth for me, please?
To solve this problem, we need to break it down step by step. Let's assume the two parts we need to find are x and y. According to the information given, we can form the following equation:
(3/8) * x = (1/4) * y + 6
Now, we can simplify this equation by multiplying both sides by the least common multiple (LCM) of 8 and 4, which is 8:
8 * (3/8) * x = 8 * (1/4) * y + 8 * 6
Simplifying further, we have:
3x = 2y + 48
Next, we can rearrange the equation to isolate one unknown variable:
3x - 2y = 48
Now, we also know that the sum of the two parts is equal to 56:
x + y = 56
To solve this system of equations, we can use the method of elimination. Multiply the first equation by 2 and the second equation by 3 to cancel out the coefficients of x and y:
(2) * (3x - 2y) = (2) * (48)
(3) * (x + y) = (3) * (56)
This simplifies to:
6x - 4y = 96
3x + 3y = 168
Now, subtract the second equation from the first:
6x - 4y - (3x + 3y) = 96 - 168
6x - 4y - 3x - 3y = -72
Combine like terms:
3x - 7y = -72
Now, we have a new equation to work with. Let's rewrite the equation x + y = 56 by isolating x:
x = 56 - y
Next, substitute this value of x into the equation 3x - 7y = -72:
3(56 - y) - 7y = -72
Distribute 3:
168 - 3y - 7y = -72
Combine like terms:
-10y = -240
Divide the equation by -10 to solve for y:
y = -240 / -10
y = 24
Now that we have y, we can substitute it back into x = 56 - y to find x:
x = 56 - 24
x = 32
Therefore, the two parts that divide 56 as specified in the problem are 32 and 24.