Skip to main content
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 20
Chapter 6, Problem 20

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
y=6x+x2y=6x+x^2
4x−y=−34x-y=-3

Verified step by step guidance
1
Start with the given system of equations: \(y = 6x + x^2\) and \(4x - y = -3\).
Substitute the expression for \(y\) from the first equation into the second equation to eliminate \(y\). This gives: \(4x - (6x + x^2) = -3\).
Simplify the equation obtained after substitution: \(4x - 6x - x^2 = -3\) which simplifies to \(-2x - x^2 = -3\).
Rewrite the equation in standard quadratic form by moving all terms to one side: \(-x^2 - 2x + 3 = 0\) or equivalently \(x^2 + 2x - 3 = 0\) (multiply both sides by -1).
Solve the quadratic equation \(x^2 + 2x - 3 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a=1\), \(b=2\), and \(c=-3\). After finding the values of \(x\), substitute each back into the original equation \(y = 6x + x^2\) to find the corresponding \(y\) values.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m

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 beyond simple substitution or elimination used for linear systems, often involving substitution or factoring to find all possible solutions.
Recommended video:
3:21
Nonlinear Inequalities

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, which can then be solved using algebraic techniques, including solving quadratic equations.
Recommended video:
04:03
Choosing a Method to Solve Quadratics

Complex Solutions

When solving nonlinear equations, solutions may include nonreal complex numbers, especially if the quadratic equation has a negative discriminant. Understanding how to find and interpret complex solutions is essential for providing all possible answers to the system.
Recommended video:
05:33
Complex Conjugates
Related Practice
Textbook Question

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

1
views
Textbook Question

Find the inverse, if it exists, for each matrix. [−1−234]\(\left\)[ \(\begin{matrix}\) -1 & -2 \\ 3 & 4 \(\end{matrix}\) \(\right\)]

Textbook Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (x3 + 4)/(9x3 - 4x)

3
views
Textbook Question

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

4x + y = -23

x - 2y = -17

2
views
Textbook Question

Find the inverse, if it exists, for each matrix. [510−3−6]\(\left\)[ \(\begin{matrix}\) 5 & 10 \\ -3 & -6 \(\end{matrix}\) \(\right\)]

Textbook Question

Graph each inequality. y < 3x2 + 2

4
views