Skip to main content
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 41
Chapter 6, Problem 41

In Exercises 29–42, solve each system by the method of your choice. {x2+y2+3y=222x+y=−1\(\begin{cases}\)x^2 + y^2 + 3y = 22 \\2x + y = -1\(\end{cases}\)

Verified step by step guidance
1
Start by examining the given system of equations: \(x^2 + y^2 + 3y = 22\) and \(2x + y = -1\).
From the linear equation \(2x + y = -1\), solve for \(y\) in terms of \(x\): \(y = -1 - 2x\).
Substitute the expression for \(y\) into the first equation to eliminate \(y\): \(x^2 + (-1 - 2x)^2 + 3(-1 - 2x) = 22\).
Expand and simplify the resulting equation to form a quadratic equation in terms of \(x\) only.
Solve the quadratic equation for \(x\), then substitute each solution back into \(y = -1 - 2x\) 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
Was this helpful?

Key Concepts

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

Solving Systems of Equations

A system of equations consists of two or more equations with the same variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. Methods include substitution, elimination, and graphing, each useful depending on the system's form.
Recommended video:
Guided course
5:48
Solving Systems of Equations - Substitution

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, making it easier to solve. It is especially effective when one equation is linear.
Recommended video:
04:03
Choosing a Method to Solve Quadratics

Quadratic Equations and Circles

The equation x² + y² + 3y = 22 represents a circle after completing the square for y. Understanding how to manipulate and solve quadratic equations is essential to find the points of intersection with the linear equation. This helps determine the system's solutions.
Recommended video:
5:18
Circles in Standard Form
Related Practice
Textbook Question

In Exercises 31–42, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. 2x = 3y + 4 4x = 3 - 5y

2
views
Textbook Question

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

{x+y>4x+y<−1\(\begin{cases}\)x + y > 4 \(\x\) + y < -1\(\end{cases}\)

Textbook Question

In Exercises 39–45, graph each inequality. y ≤ (-1/2)x + 2

Textbook Question

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

{x−y≤1x≥2\(\begin{cases}\)x - y \(\leq\) 1 \(\x\) \(\geq\) 2\(\end{cases}\)

1
views
Textbook Question

In Exercises 43–46, let x represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The sum of two numbers is 10 and their product is 24. Find the numbers.

2
views
Textbook Question

Write the partial fraction decomposition of each rational expression. (4x2+3x+14)/(x3 - 8)

1
views