5/x+7/y=31,7x+5y=29xy by comparision and subsitition method
To solve the system of equations 5/x + 7/y = 31 and 7x + 5y = 29xy using the comparison and substitution method, follow these steps:
Step 1: Rearrange the first equation to isolate one variable in terms of the other:
5/x + 7/y = 31
Multiply both sides by xy to eliminate the denominators:
5y + 7x = 31xy (Equation A)
Step 2: Solve the second equation for one variable in terms of the other:
7x + 5y = 29xy
Rewrite the equation as:
7x = 29xy - 5y
Divide both sides by y:
7x/y = 29x - 5
Rearrange to isolate x:
7x/y - 29x = -5
Factor out x:
x(7/y - 29) = -5
Divide both sides by (7/y - 29):
x = -5 / (7/y - 29) (Equation B)
Step 3: Substitute the expression for x from Equation B into Equation A:
5y + 7x = 31xy
5y + 7(-5 / (7/y - 29)) = 31y(-5 / (7/y - 29))
Simplify the expression by multiplying both sides by (7/y - 29):
5y(7/y - 29) + 7(-5) = 31y(-5)
Expand the left side:
35 - 145y + -35 = -155y
Combine like terms:
-145y = -155y - 35 + 35
Simplify the expression:
-145y = -155y
Rearrange the equation:
10y = 0
Divide both sides by 10:
y = 0
Step 4: Substitute the value of y back into Equation B to find x:
x = -5 / (7/y - 29)
Substitute y = 0:
x = -5 / (7/0 - 29)
Since dividing by zero is undefined, there is no solution for x.
Therefore, the system of equations 5/x + 7/y = 31 and 7x + 5y = 29xy has no solution.
To solve the system of equations 5/x + 7/y = 31 and 7x + 5y = 29xy using the comparison and substitution method, follow these steps:
Step 1: Set up the equations
Rewrite the given equations in a suitable format for comparison and substitution.
Equation 1: 5/x + 7/y = 31
Equation 2: 7x + 5y = 29xy
Step 2: Solve for one variable in terms of the other
Choose one equation to solve for one variable in terms of the other. Let's solve Equation 2 for x in terms of y.
From Equation 2:
7x + 5y = 29xy
7x = 29xy - 5y
x = (29xy - 5y)/7
Step 3: Substitute x in terms of y into the other equation
Substitute the expression for x from Step 2 into Equation 1. This will give you an equation in one variable (y) that you can solve.
Substituting x in terms of y into Equation 1:
5/((29xy - 5y)/7) + 7/y = 31
Now, simplify the equation.
Step 4: Solve the resulting equation
Multiply both sides of the equation by 7 to eliminate the denominator. This will help you solve for y.
7 * (5/((29xy - 5y)/7)) + 7 * (7/y) = 7 * 31
Simplifying:
35/((29xy - 5y)/7) + 49/y = 217
To remove the fraction, multiply both sides by the denominator ((29xy - 5y)/7) and divide by the common factors.
35 * 7 + 49 * ((29xy - 5y)/7y) = 217 * ((29xy - 5y)/7y)
Rearrange and simplify:
245 + 49(29xy - 5y)/7y = 217(29xy - 5y)/7y
Multiply through by 7y to eliminate the denominator:
1715y + 343(29xy - 5y) = 217(29xy - 5y)
Now, distribute and simplify further:
1715y + 9997xy - 1715y = 6303xy - 1085y
Combine like terms:
9997xy = 6303xy - 1085y
Move all terms involving y to one side of the equation:
9997xy - 6303xy + 1085y = 0
Combine like terms:
3694xy + 1085y = 0
Now, factor out y:
y(3694x + 1085) = 0
So, either y = 0 or 3694x + 1085 = 0.
If y = 0, substitute back into the original Equation 2 to solve for x:
7x + 5(0) = 29x(0)
7x = 0
x = 0
If 3694x + 1085 = 0, solve for x:
3694x = -1085
x = -1085/3694
Hence, the solution to the system of equations is x = 0 and y can be any value or x = -1085/3694 and y = 0.
5 y + 7 x = 31 x y
5 y + 7 x = 29 x y
well, (0,0) would work but then we would be dividing by zero in the original.
using substitution,
7/y = 31-5/x so
y = 7/(31-5/x) = 7x/(31x-5)
7x+5y=29xy
y = -7x/(5-29x)
7x + 35x/(31x-5) = 29x * 7x/(31x-5)
7x(31x-5) + 35x = 203x^2
14x^2 = 0
As Damon noted, x=0 is not allowed, so there are no solutions.
If you plot the difference between the two functions, you can see that it is zero only at x=0.
http://www.wolframalpha.com/input/?i=7x%2F%2831x-5%29%2B7x%2F%285-29x%29+for+x%3D+-0.1+to+0.1