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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 47
Chapter 6, Problem 47

In Exercises 47–52, solve each system by the method of your choice. {2x2+xy=6x2+2xy=0\(\begin{cases}\)2x^2 + xy = 6 \(\x\)^2 + 2xy = 0\(\end{cases}\)

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Start by writing down the system of equations clearly: \(2x^{2} + xy = 6\) and \(x^{2} + 2xy = 0\).
From the second equation \(x^{2} + 2xy = 0\), factor out the common term \(x\): \(x(x + 2y) = 0\).
Set each factor equal to zero to find possible cases: either \(x = 0\) or \(x + 2y = 0\).
For the case \(x = 0\), substitute into the first equation \(2(0)^{2} + 0 imes y = 6\) and check if it holds true.
For the case \(x + 2y = 0\), solve for \(y\) as \(y = -\frac{x}{2}\), then substitute this expression for \(y\) into the first equation \(2x^{2} + x\left(-\frac{x}{2}\right) = 6\) to find values of \(x\).

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Key Concepts

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

Systems of Nonlinear Equations

A system of nonlinear equations involves two or more equations with variables raised to powers other than one or multiplied together. Solving such systems requires finding values for the variables that satisfy all equations simultaneously, often involving substitution or elimination methods adapted for nonlinear terms.
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Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can be solved using algebraic techniques, especially useful when one equation can be easily rearranged.
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Factoring and Quadratic Equations

Factoring is a technique to rewrite expressions as products of simpler expressions, often used to solve quadratic equations. In systems involving quadratic terms, factoring helps find roots or solutions by setting each factor equal to zero, simplifying the process of solving nonlinear systems.
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Related Practice
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 47–48, solve each system by the method of your choice. (x + 2)/2 - (y + 4)/3 = 3 (x + y)/5 = (x - y)/2 - 5/2

Textbook Question

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

{x2+y2≤16x+y>2\(\begin{cases}\)x^2 + y^2 \(\leq\) 16 \(\x\) + y > 2\(\end{cases}\)

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Textbook Question

In Exercises 47–48, solve each system by the method of your choice. (x - y)/3 = (x + y)/2 - 1/2 (x + 2)/2 - 4 = (y + 4)/3

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Textbook Question

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

{x2+y2>1x2+y2<16\(\begin{cases}\)x^2 + y^2 > 1 \(\x\)^2 + y^2 < 16\(\end{cases}\)

Textbook Question

In Exercises 46–55, graph the solution set of each system of inequalities or indicate that the system has no solution.


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is a piecewise function. Refer to the textbook.