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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 30
Chapter 6, Problem 30

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
3x2+5y2=173x^2 + 5y^2 = 17
2x2−3y2=52x^2 - 3y^2 = 5

Verified step by step guidance
1
Start by writing down the system of equations clearly: \[3x^2 + 5y^2 = 17\] \[2x^2 - 3y^2 = 5\]
Introduce new variables to simplify the system: let \[a = x^2\] and \[b = y^2\]. This transforms the system into: \[3a + 5b = 17\] \[2a - 3b = 5\]
Solve the system of linear equations for \[a\] and \[b\] using either substitution or elimination. For example, multiply the second equation by 5 and the first by 3 to align coefficients of \[b\], then add or subtract to eliminate \[b\].
Once you find the values of \[a\] and \[b\], recall that \[a = x^2\] and \[b = y^2\]. Solve for \[x\] and \[y\] by taking the square roots of \[a\] and \[b\] respectively, remembering to consider both positive and negative roots as well as complex roots if \[a\] or \[b\] are negative.
Write down all possible solutions for \[x\] and \[y\], including real and nonreal complex solutions, based on the values obtained for \[a\] and \[b\].

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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 involving variables raised to powers other than one or multiplied together. Solving such systems requires methods beyond simple substitution or elimination used for linear systems, often involving algebraic manipulation or substitution to reduce the system to solvable equations.
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Substitution and Elimination Methods

These are algebraic techniques used to solve 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. Both methods help reduce the system to a single equation in one variable.
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Complex Number Solutions

When solving nonlinear systems, solutions may include complex numbers if the equations yield negative values under square roots or other operations. Complex solutions have a real part and an imaginary part, and including them ensures all possible solutions to the system are found.
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Related Practice
Textbook Question

Find each sum or difference, if possible.

[246]+[−2−4−6]\(\left\)[ \(\begin{matrix}\) 2 & 4 & 6 \(\end{matrix}\) \(\right\)] + \(\left\)[ \(\begin{matrix}\) -2 \\ -4 \\ -6 \(\end{matrix}\) \(\right\)]

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.

x + y - 5z = -18

3x - 3y + z = 6

x + 3y - 2z = -13

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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.

(3/8)x - (1/2)y = 7/8

-6x + 8y = -14

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Textbook Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (2x5 + 3x4 - 3x3 - 2x2 + x)/(2x2 + 5x + 2)

Textbook Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (3x - 1)/(x(2x2 + 1)2)

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Textbook Question

Solve each system of equations. State whether it is an inconsistent system or has infinitely many solutions. If a system has infinitely many solutions, write the solution set with x arbitrary.

9x - 5y = 1

-18x + 10y = 1