The variables x and y satisfy the equations
2/x−24/y=−1
and
6/x+33/y=2.
Find the product xy.
6/x +33/y = 2
6/x -72/y = -3 (after tripling your first equation)
105/y = 5
y = 21
2/x -24/21 = -1
2/x = 3/21 = 1/7
x = 14
xy = 294
To find the value of the product xy, we will use the given equations and solve them simultaneously. Let's start by rearranging the equations to isolate one variable in terms of the other.
Equation 1: 2/x − 24/y = -1
Multiply both sides by xy to get rid of the fractions:
2y - 24x = -xy
Equation 2: 6/x + 33/y = 2
Multiply both sides by xy:
6y + 33x = 2xy
Now, we have a system of linear equations to solve:
2y - 24x = -xy
6y + 33x = 2xy
We can solve this system of equations by elimination or substitution. Let's use the elimination method.
Multiply the first equation by 3 and the second equation by 4 to create opposite coefficients for x:
6y - 72x = -3xy
24y + 132x = 8xy
Add the two equations together to eliminate x:
(6y - 72x) + (24y + 132x) = (-3xy) + (8xy)
30y + 60x = 5xy
Now, simplify the equation:
30y + 60x - 5xy = 0
Factor out the common factor:
5(6y + 12x - xy) = 0
Since the product xy cannot be equal to zero (as it gives us the trivial solution x = 0, y = 0), we can conclude that:
6y + 12x - xy = 0
Rearrange the equation to express xy:
xy = 6y + 12x
We have derived the value of xy, which is equal to 6y + 12x.
At this point, we don't have enough information to find the exact value of xy since the values of x and y are not given. However, we now have an expression for xy in terms of x and y.
Hence, the product xy is given by xy = 6y + 12x.