let s and w represent positive integers where x,y satisfy x/s+y/w=1 and s/x+w/y=4, find x+y in terms of s and w
It's really hard even for the teachers it took them a while. So I I was wondering how to solve it
x/s + y/w = 1
s/x + w/y = 4
You can use substitution.
From the first equation,
wx + sy = sw
x = (sw - sy)/w
Substitute to the second equation:
s/x + w/y = 4
sy + wx = 4xy
sy + (w)((sw - sy)/w) = 4((sw - sy)/w)y
sy + sw - sy = 4y(sw - sy)/w
sw^2 = 4swy - 4sy^2
4sy^2 - 4swy + sw^2 = 0
s(4y^2 - 4wy + w^2) = 0
Factoring,
s(2y - w)(2y - w) = 0
y = w/2
Substituting this to x:
y = w/2:
x = (sw - sy)/w
x = (sw - s(w/2))/w
x = (sw - sw/2)/w
x = (1/2)s
x = s/2
Thus, x + y = w/2 + s/2
Hope this helps :)
To solve this problem, we can use a system of equations and apply algebraic manipulation.
Given the equations:
1) x/s + y/w = 1
2) s/x + w/y = 4
Let's solve for x and y in terms of s and w separately:
Equation 1:
x/s + y/w = 1
Multiply both sides of Equation 1 by s and w to eliminate the denominators:
xw + ys = sw
Now, let's solve for y:
ys = sw - xw
y = (sw - xw)/s
Equation 2:
s/x + w/y = 4
Multiply both sides of Equation 2 by x and y to eliminate the denominators:
sy + wx = 4xy
Now, let's solve for x:
wx = 4xy - sy
xw = 4xy - sy
x = (4xy - sy)/w
To find x + y, substitute the value of x and y we obtained above into the expression:
x + y = [(4xy - sy)/w] + [(sw - xw)/s]
Now, let's simplify this expression:
x + y = (4xy - sy)/w + (sw - xw)/s
To simplify further, let's find a common denominator, which is "ws":
x + y = [(4xy - sy)s + (sw - xw)w] / (ws)
= (4xys - sys + sw^2 - xww) / (ws)
Therefore, x + y in terms of s and w is:
x + y = (4xys - sys + sw^2 - xww) / (ws)
Now you can use this expression to find the value of x + y given any specific values for s and w.
To find the value of x+y in terms of s and w, we'll solve the given system of equations step-by-step:
Step 1: Rearrange the first equation
x/s + y/w = 1
Multiply through by sw to eliminate the fractions:
wx + sy = sw
Step 2: Rearrange the second equation
s/x + w/y = 4
Multiply through by xy to eliminate the fractions:
ys + wx = 4xy
Step 3: Solve for x and y in terms of s and w
From equations (1) and (2), we have the system:
wx + sy = sw (equation 1)
ys + wx = 4xy (equation 2)
Let's solve equation 1 for x:
wx = sw - sy
x = (sw - sy) / w
Now, let's solve equation 2 for y:
ys = 4xy - wx
ys = 4xy - (sw - sy)
ys = 4xy - sw + sy
ys - sy = 4xy - sw
y(s - 1) = x(4y - s)
y = x(4y - s) / (s - 1)
Step 4: Substitute the value of y in terms of x into equation 1
wx + sy = sw
wx + s(x(4y - s) / (s - 1)) = sw
wx + (4x(sy) - s^2) / (s - 1)) = sw
wx(s - 1) + 4x(sy) - s^2 = sw(s - 1)
wx(s - 1) + 4x(sy) = sw(s - 1) + s^2
Step 5: Simplify the equation further
wx(s - 1) + 4x(sy) = sw(s - 1) + s^2
wxs - wx + 4xsy = sws - sw + s^2
Step 6: Rearrange the equation
wxs - 4xsy = wx - s^2 + sw - sws
x(ws - 4sy) = -s^2
Step 7: Solve for x in terms of s and w
x = -s^2 / (ws - 4sy)
Step 8: Substitute the value of x into the equation for y in terms of x
y = x(4y - s) / (s - 1)
y = (-s^2 / (ws - 4sy))(4y - s) / (s - 1)
y = (-4s^2y + s^3) / (ws - 4sy)(s - 1)
y(ws - 4sy)(s - 1) = -4s^2y + s^3
Step 9: Simplify the equation further
y(ws^2 - 4s^2y - ws + 4sy) = -4s^2y + s^3
y(ws^2 - 4s^2y - ws + 4sy + 4s^2y - s^3) = 0
y(ws^2 + 4sy - s^3 - ws) = 0
Finally, we have found the values of x and y in terms of s and w. x = -s^2 / (ws - 4sy) and y(ws^2 + 4sy - s^3 - ws) = 0.
Now, to find x + y in terms of s and w, we substitute these values back into the equation:
x + y = -s^2 / (ws - 4sy) + 0
x + y = -s^2 / (ws - 4sy)
Therefore, the value of x + y in terms of s and w is -s^2 / (ws - 4sy).