Solve each system. (Hint: In Exercises 69–72, let 1/x = t and 1/y = u.)2/x + 1/y = 3/23/x - 1/y = 1

Verified step by step guidance

1. Substitute \( t = \frac{1}{x} \) and \( u = \frac{1}{y} \) into the equations, transforming them into: \( 2t + u = \frac{3}{2} \) and \( 3t - u = 1 \).
2. Add the two equations: \( (2t + u) + (3t - u) = \frac{3}{2} + 1 \).
3. Simplify the resulting equation to find \( t \).
4. Substitute the value of \( t \) back into one of the original transformed equations to solve for \( u \).
5. Once \( t \) and \( u \) are found, substitute back to find \( x = \frac{1}{t} \) and \( y = \frac{1}{u} \).

Key Concepts

Systems of Equations

A system of equations consists of two or more equations that share the same variables. The goal is to find the values of these variables that satisfy all equations simultaneously. Solutions can be found using various methods, including substitution, elimination, or graphical representation.

Substitution Method

The substitution method involves solving one equation for a variable and then substituting that expression into the other equation(s). This technique simplifies the system, making it easier to solve for the remaining variables. In this problem, the hint suggests substituting 1/x with t and 1/y with u to facilitate the process.

Variable Transformation

Variable transformation is a technique used to simplify complex equations by substituting new variables for existing ones. In this case, letting 1/x = t and 1/y = u transforms the original equations into a more manageable form. This approach can help clarify relationships between variables and make solving the system more straightforward.