Question

Solve the following pair of equations by converting these into linear pair of equations: \[ \frac{1}{2x} - \frac{1}{3y} = 2, \quad \frac{1}{3x} + \frac{1}{2y} = \frac{13}{16}. \]

Hint:

When dealing with systems of equations involving fractions, substituting new variables can simplify the process.

Answer 1

We are given the system of equations: \[ \frac{1}{2x} - \frac{1}{3y} = 2 \quad \text{(Equation 1)} \] and \[ \frac{1}{3x} + \frac{1}{2y} = \frac{13}{16} \quad \text{(Equation 2)}. \] Step 1: Convert the equations into linear form. Let \( u = \frac{1}{x} \) and \( v = \frac{1}{y} \), so we have: \[ \frac{1}{2x} = \frac{u}{2}, \quad \frac{1}{3y} = \frac{v}{3}. \] Substitute these into the equations: \[ \frac{u}{2} - \frac{v}{3} = 2 \quad \text{(Equation 1)} \] and \[ \frac{u}{3} + \frac{v}{2} = \frac{13}{16} \quad \text{(Equation 2)}. \] Step 2: Eliminate fractions by multiplying through. Multiply Equation 1 by 6 and Equation 2 by 6 to clear the denominators: \[ 3u - 2v = 12 \quad \text{(Equation 3)} \] and \[ 2u + 3v = \frac{13}{16} \times 6 = \frac{78}{16} = 4.875 \quad \text{(Equation 4)}. \] Step 3: Solve the system of linear equations. We now solve the system of linear equations (Equation 3 and Equation 4): 1. \( 3u - 2v = 12 \) 2. \( 2u + 3v = 4.875 \) Multiply the first equation by 3 and the second equation by 2 to make the coefficients of \( v \) equal: \[ 9u - 6v = 36 \] \[ 4u + 6v = 9.75 \] Now, add the two equations: \[ (9u - 6v) + (4u + 6v) = 36 + 9.75 \] \[ 13u = 45.75 \] \[ u = \frac{45.75}{13} = 3.515 \quad \text{(value of \( u \))}. \] Step 4: Solve for \( v \). Substitute \( u = 3.515 \) into one of the original equations, say \( 3u - 2v = 12 \): \[ 3(3.515) - 2v = 12, \] \[ 10.545 - 2v = 12, \] \[ -2v = 12 - 10.545 = 1.455, \] \[ v = -\frac{1.455}{2} = -0.7275. \] Step 5: Solve for \( x \) and \( y \). Since \( u = \frac{1}{x} \) and \( v = \frac{1}{y} \), we have: \[ x = \frac{1}{u} = \frac{1}{3.515} \approx 0.284, \] \[ y = \frac{1}{v} = \frac{1}{-0.7275} \approx -1.376. \]
Conclusion:
The solutions for \( x \) and \( y \) are approximately \( x \approx 0.284 \) and \( y \approx -1.376 \).