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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 26
Chapter 6, Problem 26

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
x2+y2=02x^2+y^2=02
x2−3y2=0x^2-3y^2=0

Verified step by step guidance
1
Start by writing down the system of equations clearly: \[x^2 + y^2 = 0\] \[2x^2 - 3y^2 = 0\]
From the first equation, observe that the sum of two squares equals zero. Since squares of real numbers are nonnegative, this implies both \(x^2\) and \(y^2\) must be zero for real solutions. However, since complex solutions are allowed, consider the possibility of nonzero complex values.
Express one variable in terms of the other using the first equation. For example, from \[x^2 + y^2 = 0\], we get \[y^2 = -x^2\].
Substitute \[y^2 = -x^2\] into the second equation \[2x^2 - 3y^2 = 0\] to eliminate \[y^2\] and get an equation in terms of \[x^2\] only: \[2x^2 - 3(-x^2) = 0\].
Simplify the resulting equation to solve for \[x^2\], then use \[y^2 = -x^2\] to find \[y^2\]. Finally, take square roots (considering both positive and negative roots) to find all possible values of \[x\] and \[y\], including complex solutions.

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Key Concepts

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

Nonlinear Systems of Equations

A nonlinear system consists of two or more equations where at least one equation is nonlinear, such as involving variables raised to powers other than one. Solving these systems requires finding all variable values that satisfy every equation simultaneously, including real and complex solutions.
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Complex Numbers and Imaginary Solutions

When solving equations like x² + y² = 0, solutions may involve complex numbers, which include the imaginary unit i where i² = -1. Understanding how to work with complex numbers allows finding solutions beyond the real number line, especially when sums of squares equal zero.
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Substitution and Elimination Methods

These algebraic techniques help solve systems by expressing one variable in terms of another (substitution) or combining equations to eliminate a variable (elimination). Applying these methods simplifies nonlinear systems into solvable forms, enabling the determination of all possible solutions.
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Related Practice
Textbook Question

Find each sum or difference, if possible.

[6−92413]+[−8256−34]\(\left\)[ \(\begin{matrix}\) 6 & -9 & 2 \\ 4 & 1 & 3 \(\end{matrix}\) \(\right\)] + \(\left\)[ \(\begin{matrix}\) -8 & 2 & 5 \\ 6 & -3 & 4 \(\end{matrix}\) \(\right\)]

Textbook Question

Solve each system, using the method indicated.

5x + 2y = -10

3x - 5y = -6 (Gauss-Jordan)

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Textbook Question

Evaluate each determinant.

∣1−23000110−12∣\(\left\)| \(\begin{matrix}\) 1 & -2 & 3 \\ 0 & 0 & 0 \\ 1 & 10 & -12 \(\end{matrix}\) \(\right\)|

Textbook Question

Solve each system by elimination. In systems with fractions, first clear denominators.

x/2+ y/3 = 4

3x/2+3y/2 = 15

Textbook Question

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary.

6x - 3y - 4 = 0

3x + 6y - 7= 0

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Textbook Question

Graph each inequality. x2 + (y + 3)2 ≤ 16

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