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 41

Chapter 6, Problem 41

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

Verified step by step guidance

Rewrite both equations in a standard form to make them easier to work with. For the first equation,

2x = 3y + 4

, subtract

3y

and 4 from both sides to get

2x - 3y = 4

. For the second equation,

4x = 3 - 5y

, add

5y

to both sides and subtract 3 to get

4x + 5y = 3

Choose a method to solve the system: substitution or elimination. Here, elimination might be efficient. Multiply the first equation by 2 to align the coefficients of

x

2(2x - 3y) = 2(4)

which simplifies to

4x - 6y = 8

Now subtract the second equation

4x + 5y = 3

from the new equation

4x - 6y = 8

to eliminate

x

. This gives

(4x - 6y) - (4x + 5y) = 8 - 3

, simplifying to

-11y = 5

Solve for

y

by dividing both sides by

-11

y = \(\frac{5}{-11}\) = -\(\frac{5}{11}\)

Substitute the value of

y

back into one of the original equations, for example

2x - 3y = 4

, to solve for

x

. Replace

y

with

-\(\frac{5}{11}\)

and solve the resulting equation for

x

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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 all variable values 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 involves analyzing the equations' relationships, such as parallel lines for no solution or coincident lines for infinite solutions.

Set Notation for Solution Sets

Set notation expresses the solution set clearly and concisely. For example, a single solution is written as {(x, y)}, no solution as the empty set ∅, and infinitely many solutions as a set describing all points satisfying the system, often using a parameter.

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Related Practice

Textbook Question

Write the partial fraction decomposition of each rational expression. (x³-4x²+9x-5)/(x² -2x+3)²

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Textbook Question

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

Textbook Question

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

$\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.

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

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Textbook Question

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

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