Solving Fractional Systems of EquationsDate: 06/08/98 at 19:16:56 From: Lisa Mannon Subject: Algebra I need help solving the following system of equation. The fractions have confused me: 1/x + 3/y = 7 4/x - 2/y = 1 Thanks. Date: 06/08/98 at 20:20:44 From: Doctor Gary Subject: Re: Algebra You might want to "play" with the graph of this relation a bit. It will come in handy later, when we're checking our answer. When x is 1, 1/x is 1. Then 7 - 1 is 6, so 3/y must be 6. If 3/y is 6, then 3 is 6y, and y is 3/6 = 1/2. When x is 1/2, y is 3/5 When x is 1/4, y is 1 When x is 1/8, y is -3 Because division by zero is undefined, there can be no points on either the x or y axis. So, since there are points in more than one quadrant, the graph is discontinuous. When fractions are getting you down, try re-expressing the equation(s) by multiplying both sides by the denominator(s) - so long as we remember that neither x nor y can be zero. Then: 1/x + 3/y = 7 is equivalent to: y + 3x = 7xy 3x = 7xy - y 3x = y(7x-1) (3x)/(7x-1) = y We can use the same re-expression technique on the second equation: 4/x - 2/y = 1 Again, remembering thant neither x nor y can be equal to zero, we can re-express this as: 4y - 2x = xy -2x = xy - 4y -2x = y(x-4) (-2x)(x-4) = y (2x)/(4-x) = y Since two different expressions involving x are both equal to y, our solution(s) will be found when the expressions of x are equal in value: (3x)/(7x-1) = (2x)/(4-x) Since multiplying both sides of this equation by both denominators will create a cubic equation, let's see if there's a shortcut. Remember that we can always multiply any expression by 1 without changing its value. If we multiply (3x)/(7x-1) by 2/2 and multiply (2x)/(4-x) by 3/3, we can re-express our equation as: (6x)/(14x-2) = (6x)/(12-3x) Since the numerators are equal (and, remember, nonzero), the fractions are equal only if the denominators are equal. 14x - 2 = 12 - 3x 17x = 14 x = 14/17 Let's get those equations down here and see if x being equal to 14/17 "works": Using 1/x + 3/y = 7, when x is 14/17: 1/x is equal to 17/14 3/y must be equal to 81/14, so y must be 14/27 Using 4/x -2/y = 1, when x is 14/17: 4/x is equal to 68/14 2/y must be equal to 54/14, so y must by 14/27 -Doctor Gary, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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