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Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 5 - Systems and MatricesProblem 69
Chapter 6, Problem 69

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 \( u \).
Rewrite each equation using the substitutions: the first equation becomes \( 2t + u = \frac{3}{2} \), and the second equation becomes \( 3t - u = 1 \).
Add the two equations to eliminate \( u \): \( (2t + u) + (3t - u) = \frac{3}{2} + 1 \). Simplify this to find an equation with only \( t \).
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 \( u \).
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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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Related Practice
Textbook Question

Given A=[4−231],B=[510−237]A = \(\left\)[ \(\begin{matrix}\) 4 & -2 \\ 3 & 1 \(\end{matrix}\) \(\right\)], \(\quad\) B = \(\left\)[ \(\begin{matrix}\) 5 & 1 \\ 0 & -2 \\ 3 & 7 \(\end{matrix}\) \(\right\)] and C=[−541036]C = \(\left\)[ \(\begin{matrix}\) -5 & 4 & 1 \\ 0 & 3 & 6 \(\end{matrix}\) \(\right\)], find each product, if possible. See Examples 5–7. BA

Textbook Question

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7. 1.5

x + 3y = 5

2x + 4y = 3

Textbook Question

Solve each problem. The supply and demand equations for a certain commodity are given. supply: p = 2000/(2000 - q) and demand: p = (7000 - 3q)/2q.

Find the equilibrium price (in dollars).

Textbook Question

Solve each problem. Find the radius and height (to the nearest thousandth) of an open-ended cylinder with volume 50 in.3 and lateral surface area 65 in.2.

Textbook Question

Find each product, if possible.

[−2−3−42−104−23][01412−132−2]\(\left\)[ \(\begin{matrix}\) -2 & -3 & -4 \\ 2 & -1 & 0 \\ 4 & -2 & 3 \(\end{matrix}\) \(\right\)] \(\left\)[ \(\begin{matrix}\) 0 & 1 & 4 \\ 1 & 2 & -1 \\ 3 & 2 & -2 \(\end{matrix}\) \(\right\)]

Textbook Question

Solve each problem. The supply and demand equations for a certain commodity are given. supply: p = 2000/(2000 - q) and demand: p = (7000 - 3q)/2q.

Find the equilibrium demand.