Problem 43 Match the lettered items in colu... [FREE SOLUTION]

91影视

Understanding Elementary Algebra with Geometry

Lewis Hirsch, Arthur Goodman

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 6: Problem 43

Match the lettered items in column II with the numbered items in column I. Column I (1) Simultaneous equations in two variables with exactly one solution (2) An equation in one variable with no solutions (3) An equation in one variable with exactly one solution (4) Simultancous equations in two variables with infinitely many solutions (5) Simultaneous equations in two variables with no solutions (6) An equation in one variable with infinitely many solutions. Column II (a) Conditional (b) Consistent (c) Dependent (d) Inconsistent (e) Contradiction (f) Identity

Short Answer

Expert verified

(1) (b), (2) (e), (3) (a), (4) (c), (5) (d), (6) (f)

Step by step solution

- Identify term definitions

Examine each labeled item in Column II and understand its meaning. (a) Conditional: True under certain conditions. (b) Consistent: Has at least one solution. (c) Dependent: Infinitely many solutions, overlaps completely. (d) Inconsistent: No solutions. (e) Contradiction: No solution possible. (f) Identity: Always true for all variable values.

- Match Column I (1)

Simultaneous equations in two variables with exactly one solution: These are consistent because they meet at exactly one point. So, (1) matches with (b).

- Match Column I (2)

An equation in one variable with no solutions: This indicates a contradiction because it is never true. So, (2) matches with (e).

- Match Column I (3)

An equation in one variable with exactly one solution: This is a conditional equation. So, (3) matches with (a).

- Match Column I (4)

Simultaneous equations in two variables with infinitely many solutions: These equations are dependent, as they overlap entirely. So, (4) matches with (c).

- Match Column I (5)

Simultaneous equations in two variables with no solutions: These equations are inconsistent as they never intersect. So, (5) matches with (d).

- Match Column I (6)

An equation in one variable with infinitely many solutions: This is an identity, true for any value. So, (6) matches with (f).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simultaneous Equations

Simultaneous equations are a set of equations with multiple variables which are solved together. In other words, we are looking for values of the variables that satisfy all equations simultaneously. For example, solving the system of equations $ 2x + y = 5 $ and $ x - y = 1 $ means finding values of $ x $ and $ y $ that make both equations true at the same time. Solving simultaneous equations is important as it helps to understand relationships between different variables in a system.

Solutions of Equations

Finding the solution to an equation means determining the values of the variable that make the equation true. For example, in the equation $ x + 3 = 5 $, the solution is $ x = 2 $. When it comes to simultaneous equations, we look for a set of values that satisfy all equations in the system. For example, the solution to $ 2x + 3y = 6 $ and $ x - y = 1 $ is $ x = 3 $ and $ y = 2 $. Solutions are crucial as they allow us to solve real-world problems modeled by equations.

Equation Classification

Equations can be classified based on the number of solutions they have:

Conditional Equations: These are true only under specific conditions or specific values of the variables. For instance, $ x + 2 = 4 $ has the solution $ x = 2 $.
Identities: These are equations that are always true, no matter what the values of the variables are. An example is $ x = x $.
Contradictions: These are equations that have no solutions. An example is $ x + 2 = x $, which simplifies to $ 2 = 0 $, an impossibility.

Understanding the classification of equations helps in effectively solving them.

Consistent and Inconsistent Systems

Systems of equations can be either consistent or inconsistent.

Consistent System: This has at least one solution. For example, the system $ x + y = 2 $ and $ x - y = 0 $ is consistent because the solution $ x = 1, \ y = 1 $ satisfies both equations.
Inconsistent System: This has no solutions because the equations contradict each other. An example is $ x + y = 2 $ and $ x + y = 3 $, which can never be satisfied simultaneously.

Recognizing if a system is consistent or inconsistent is important as it determines whether a solution can be found.

Dependent Equations

Dependent equations are a specific type of system where the equations overlap completely, resulting in infinitely many solutions. For example, the system $ x + y = 2 $ and $ 2x + 2y = 4 $ is dependent, as the second equation is simply a multiple of the first. This means every solution of $ x + y = 2 $ is also a solution of $ 2x + 2y = 4 $. Identifying dependent equations helps to understand situations where there are infinite solutions.

Conditional Equations

Conditional equations are true only for certain values of the variables. For example, the equation $ x - 5 = 10 $ is conditional because it is true only when $ x = 15 $. Such equations are important in many mathematical and real-world problems, where solutions are valid only under certain conditions. Solving conditional equations involves finding the specific variable values that meet the equation's requirements.

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