Ch. 5 - Systems of Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

Problem 35

Chapter 6, Problem 35

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. 3x - 2y = − 5 4x + y = 8

Verified step by step guidance

Start by writing down the system of equations clearly:

3x - 2 y = - 5

and

4 x + y = 8

Choose a method to solve the system. Here, substitution or elimination are both good options. For elimination, aim to eliminate one variable by making the coefficients of either

x

y

opposites.

Multiply the second equation by 2 to align the coefficients of

y

2(4x + y) = 2(8)

which gives

8x + 2y = 16

Add the first equation

3x - 2y = -5

and the new equation

8x + 2y = 16

to eliminate

y

(3x + 8x) + (-2y + 2y) = -5 + 16

Solve the resulting equation for

x

, then substitute this value back into one of the original equations to find

y

. Finally, check if the system has one solution, no solution, or infinitely many solutions by analyzing the consistency of the equations.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Solving Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. Common methods include substitution, elimination, and graphing.

Types of Solutions for Systems

Systems of linear equations can have one solution (consistent and independent), no solution (inconsistent), or infinitely many solutions (dependent). Identifying the type depends on the relationships between the equations, such as parallel lines or coincident lines.

Expressing Solution Sets Using Set Notation

Set notation is a concise way to represent the solution(s) of a system. For a unique solution, it lists the ordered pair; for infinitely many solutions, it expresses the solution in terms of a parameter; and for no solution, the solution set is the empty set, denoted by ∅.

Recommended video:

07:07

Introduction to Exponents

Related Practice

Textbook Question

Write the partial fraction decomposition of each rational expression. x+4/x² (x²+4)

views

Textbook Question

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

$\begin{cases}\)x $\leq$ 2 $\y$ $\geq$ -1\(\end{cases}$

views

Textbook Question

In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}\)x^3 + y = 0 $\x$^2 - y = 0\(\end{cases}$

Textbook Question

The perimeter of a rectangle is 26 meters and its area is 40 square meters. Find its dimensions.

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. x + 3y = 2 3x + 9y = 6

Textbook Question

Write the partial fraction decomposition of each rational expression. 6x²-x+1/(x³ + x²+ x +1)