Solve each system. (Hint: In Exercises 69–72, let 1/x = t and 1/y = u.)
2/x + 1/y = 3/2
3/x - 1/y = 1

Verified step by step guidance

1. Start by using the hint given: let \( t = \frac{1}{x} \) and \( u = \frac{1}{y} \). This substitution will transform the system into a simpler linear system in terms of \( t \) and \( u \).
2. Rewrite each equation using the substitutions: the first equation becomes \( 2t + u = \frac{3}{2} \), and the second equation becomes \( 3t - u = 1 \).
3. Add the two equations to eliminate \( u \): \( (2t + u) + (3t - u) = \frac{3}{2} + 1 \). Simplify this to find an equation with only \( t \).
4. Solve the resulting equation for \( t \), then substitute this value back into one of the original substituted equations (either \( 2t + u = \frac{3}{2} \) or \( 3t - u = 1 \)) to solve for \( u \).
5. Finally, recall that \( t = \frac{1}{x} \) and \( u = \frac{1}{y} \). Solve for \( x \) and \( y \) by taking the reciprocals of \( t \) and \( u \) respectively.

Key Concepts

Substitution Method in Systems of Equations

The substitution method involves replacing variables with new expressions to simplify a system. In this problem, letting 1/x = t and 1/y = u transforms the original system into a linear one in terms of t and u, making it easier to solve.

Reciprocal Functions and Variable Transformation

Reciprocal functions involve replacing variables with their inverses (1/x, 1/y). This transformation can simplify nonlinear equations into linear forms, allowing standard algebraic techniques to be applied effectively.

Solving Linear Systems of Equations

Once the system is rewritten in terms of t and u, it becomes a linear system. Solving linear systems typically involves methods like substitution, elimination, or matrix operations to find the values of variables that satisfy all equations.