Solve each system. (Hint: In Exercises 69–72, let $1/x=t$ and $1/y=u$.)
$\frac{2}{x}+\frac{3}{y}=18$
$\frac{4}{x}-\frac{5}{y}=-8$

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 linear system in terms of \( t \) and \( u \).
2. Rewrite each equation by substituting \( \frac{1}{x} = t \) and \( \frac{1}{y} = u \). The system becomes: \[ 2t + 3u = 18 \] \[ 4t - 5u = -8 \]
3. Solve the new system of linear equations for \( t \) and \( u \) using either the substitution method or the elimination method. For example, you can multiply the first equation by 4 and the second by 2 to align coefficients for elimination.
4. Once you find the values of \( t \) and \( u \), recall that \( t = \frac{1}{x} \) and \( u = \frac{1}{y} \). Use these relationships to solve for \( x \) and \( y \) by taking the reciprocal of \( t \) and \( u \) respectively.
5. Check your solutions by substituting \( x \) and \( y \) back into the original equations to ensure they satisfy both equations.

Key Concepts

Substitution Method

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

Solving Systems of Linear Equations

Once the substitution is made, the system becomes linear in t and u. Solving systems of linear equations involves finding values for variables that satisfy all equations simultaneously, typically using methods like substitution, elimination, or matrix operations.

Back-Substitution

After finding the values of t and u, back-substitution is used to find the original variables x and y by reversing the substitution: x = 1/t and y = 1/u. This step is crucial to interpret the solution in terms of the original variables.