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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 17
Chapter 6, Problem 17

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components.
y = x2 - 2x + 1
x - 3y = -1

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1
Start with the given system of equations: \(y = x^{2} - 2x + 1\) and \(x - 3y = -1\).
Substitute the expression for \(y\) from the first equation into the second equation to eliminate \(y\). This gives: \(x - 3(x^{2} - 2x + 1) = -1\).
Expand and simplify the equation: \(x - 3x^{2} + 6x - 3 = -1\) which simplifies to \(-3x^{2} + 7x - 3 = -1\).
Bring all terms to one side to form a quadratic equation: \(-3x^{2} + 7x - 3 + 1 = 0\) which simplifies to \(-3x^{2} + 7x - 2 = 0\).
Solve the quadratic equation for \(x\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) where \(a = -3\), \(b = 7\), and \(c = -2\). After finding the values of \(x\), substitute each back into the original equation for \(y\) to find the corresponding \(y\) values.

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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 involves at least one equation that is not linear, such as quadratic or higher-degree polynomials. Solving these systems requires methods that handle curves and more complex relationships, unlike linear systems which involve straight lines.
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Substitution Method

The substitution method involves solving one equation for one variable and substituting that expression into the other equation. This reduces the system to a single equation with one variable, making it easier to solve, especially when one equation is already solved for a variable.
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Complex Solutions

When solving polynomial equations, solutions may include nonreal complex numbers involving the imaginary unit i. Recognizing and including complex solutions ensures all possible roots are found, which is essential for a complete solution to nonlinear systems.
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Related Practice
Textbook Question

Write the system of equations associated with each augmented matrix . Do not solve.

[10020103001−2]\(\left\)[ \(\begin{matrix}\) 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & -2 \(\end{matrix}\) \(\right\)]

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Textbook Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.

y=x2+6x+9y=x^2+6x+9

x+2y=−2x+2y=-2

Textbook Question

Find the cofactor of each element in the second row of each matrix.

[−201120421]\(\left\)[ \(\begin{matrix}\) -2 & 0 & 1 \\ 1 & 2 & 0 \\ 4 & 2 & 1 \(\end{matrix}\) \(\right\)]

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Textbook Question

Graph each inequality. 2x > 3 - 4y

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Textbook Question

Find the inverse, if it exists, for each matrix. [−12−2−1]\(\left\)[ \(\begin{matrix}\) -1 & 2 \\-2 & -1 \(\end{matrix}\) \(\right\)]

Textbook Question

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

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