well i'm stumped. i can't figure out how to solve this.
solve for (x,y):
3/x + 2/y = 0
5/x - 1/y = 13
i can't graph it, and i can't use a calculator. i keep getting stuck on the second equation b/c i get 5y -13xy -x=0 and since 13 is a prime number i cant figure out what to do next.
first:
3/x + 2/y = 0
times xy
3y + 2x = 0
y = -2x/3
in 2nd:
times xy
5y - x = 13xy
sub in y = -2x/3
5(-2x/3) - x = 13x(-2x/3)
-10x/3 - x = -26x^2/3
times 3
-10x - 3x = -26x^
26x^2 - 13x = 0
13x(2x - 1) = 0
x = 0 or x = 1/2
but x cannot be zero or else we divided by zero in each equation, that would be undefined
x = 1/2
then y = (-2/3)(1/2) = -1/3
To solve this system of equations without using a calculator or graphing, you can try solving it algebraically using elimination or substitution method. Let's use the elimination method in this case.
Given:
3/x + 2/y = 0 ...(Equation 1)
5/x - 1/y = 13 ...(Equation 2)
Step 1: Eliminate the fractions
Multiply both sides of Equation 1 by xy, and Equation 2 by -2xy to eliminate the fractions:
3y + 2x = 0 ...(Equation 3)
-10y + 2xy = -26 ...(Equation 4)
Step 2: Eliminate the variable 'y'
Multiply Equation 3 by 5 and Equation 4 by 3 to make the coefficients of the x term equal:
15y + 10x = 0 ...(Equation 5)
-30y + 6xy = -78 ...(Equation 6)
Add Equation 5 and Equation 6 together to eliminate 'y':
15y + 10x + (-30y + 6xy) = 0 + (-78)
15y - 30y + 10x + 6xy = -78
-15y + 10x + 6xy = -78 ...(Equation 7)
Step 3: Isolate the 'x' term
Rearrange Equation 7 to isolate the 'x' term:
10x +6xy = -78 + 15y
10x + 6xy = 15y - 78
Factor out 'x':
x(10 + 6y) = 15y - 78
x = (15y - 78) / (10 + 6y) ...(Equation 8)
Now you have an expression for 'x' in terms of 'y'.
Step 4: Substitute back into one of the original equations
Let's choose Equation 3:
3y + 2x = 0
Substitute the expression for 'x' from Equation 8 into this equation:
3y + 2[(15y - 78)/(10 + 6y)] = 0
Simplify and solve for 'y'.