How do you solve the following system using substitution?: 2/x - 2/y = 1/2, 1/x + 5/y = 3/4

1 Answer
Noah G
Feb 8, 2016

You will have to solve for one of the variables. I prefer to do y, so that's how I'll show it to you.

Explanation:

1/x + 5/y = 3/4

Put on equal denominators:

(1(4y))/(x(4y)) + (5(4x))/(y(4x)) = (3(xy))/(4(xy))

We can now eliminate the denominators because we're on equal denominators.

4y + 20x = 3xy

Solving for y:

4y - 3xy = -20x

Factor out y:

y(4 - 3x) = -20x

y = (-20x) / (4 - 3x)

Substitute (-20x) / (4 - 3x) for y in the other equation

2/x - 2/((-20x) / (4 - 3x)) = 1/2

Once placed on a common denominator of (-40x^2)/(4 - 3x):

(-80x)/(4 - 3x) - 4x = (-20x^2)/(4 - 3x)

Now, we must place the -4x on the new common denominator of 4 - 3x.

(-80x)/(4 - 3x) - (4x(4 - 3x))/(4 - 3x) = (-20x^2)/(4 - 3x)

Eliminating the denominators, we get the quadratic equation -80x - 16x + 12x^2 = -20x^2

32x^2 - 96x = 0

32x(x - 3) = 0

x = 0 and x = 3

x = 0 cannot be a solution since division by 0 is non defined.

Solving the equation to find y:

1/3 + 5/y = 3/4

(4y)/(12y) + 60/(12y) = (9y)/(12y)

4y - 9y = 60

-5y = 60

y = -12

So, the solution set is (3, -12)

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