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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblem 51
Chapter 6, Problem 51

In Exercises 47–52, solve each system by the method of your choice. {3x2+1y2=75x2−2y2=−3\(\begin{cases}\]\frac{3}{x^2}\) + \(\frac{1}{y^2}\) = 7 \(\frac{5}{x^2}\) - \(\frac{2}{y^2}\) = -3\(\end{cases}\)

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Identify the system of equations: \[ \frac{3}{x^2} + \frac{1}{y^2} = 7 \] \[ \frac{5}{x^2} - \frac{2}{y^2} = -3 \]
To simplify the system, introduce new variables: let \[ a = \frac{1}{x^2} \quad \text{and} \quad b = \frac{1}{y^2} \] This transforms the system into linear equations in terms of \(a\) and \(b\).
Rewrite the system using the new variables: \[ 3a + b = 7 \] \[ 5a - 2b = -3 \]
Solve the linear system for \(a\) and \(b\) using either substitution or elimination method. For example, multiply the first equation to align coefficients and eliminate one variable.
Once you find values for \(a\) and \(b\), substitute back to find \(x\) and \(y\) by solving: \[ a = \frac{1}{x^2} \implies x^2 = \frac{1}{a} \] \[ b = \frac{1}{y^2} \implies y^2 = \frac{1}{b} \] Then take square roots to find \(x\) and \(y\), remembering to consider both positive and negative roots.

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Key Concepts

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

Systems of Equations

A system of equations consists of two or more equations with multiple variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to manipulate and solve these systems is fundamental in algebra.
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Substitution and Elimination Methods

These are common techniques for solving systems of equations. Substitution involves solving one equation for a variable and substituting into the other, while elimination involves adding or subtracting equations to eliminate a variable. Choosing the appropriate method simplifies solving complex systems.
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Handling Rational Expressions and Variables in Denominators

When variables appear in denominators, it is important to rewrite the equations to avoid division by zero and simplify the system. This often involves substituting new variables for expressions like 1/x² or 1/y², turning the system into a more manageable form.
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Related Practice
Textbook Question

Solve each system for x and y, expressing either value in terms of a or b, if necessary. Assume that a ≠ 0 and b ≠ 0. For the linear function f(x) = mx + b, f(−2) = 11 and ƒ(3) = -9. Find m and b.

Textbook Question

Graph the solution set of each system of inequalities or indicate that the system has no solution.

{(x−1)2+(y+1)2<25(x−1)2+(y+1)2≥16\(\begin{cases}\)(x - 1)^2 + (y + 1)^2 < 25 \\(x - 1)^2 + (y + 1)^2 \(\geq\) 16\(\end{cases}\)

Textbook Question

In Exercises 49–50, solve each system for x and y, expressing either value in terms of a or b, if necessary. Assume that a ≠ 0, b ≠ 0 5ax + 4y = 17 ax + 7y = 22

Textbook Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x2+y2≤1, y−x2>0

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Textbook Question

Find the partial fraction decomposition for 1/x(x+1) and use the result to find the following sum:

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Textbook Question

In Exercises 47–52, solve each system by the method of your choice. {−4x+y=12y=x3+3x2\(\begin{cases}\)-4x + y = 12 \(\y\) = x^3 + 3x^2\(\end{cases}\)