Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 4: Problem 99
(a) For which values \(b, c\) does the following system of equations have a unique solution? $$ x+y+z=3, \quad x y+y z+z x=b, \quad x^{2}+y^{2}+z^{2}=c $$ (b) For which values \(a, b, c\) does the following system of equations have a unique solution? $$ x+y+z=a, \quad x y+y z+z x=b, \quad x^{2}+y^{2}+z^{2}=c $$
Short Answer
Expert verified
(a) Unique solutions exist when \(c = 9 - 2b\). (b) Unique solutions exist when \(c = a^2 - 2b\).
Step by step solution
01
Understanding the Uniqueness of Solutions (a)
To determine when the system:\[x + y + z = 3, \quad x y + y z + z x = b, \quad x^2 + y^2 + z^2 = c\]has a unique solution in terms of variables \(b\) and \(c\), we need to understand what having a unique solution entails. Typically, for a system of equations, a unique solution means that there is precisely one set of values for \(x\), \(y\), and \(z\) that satisfies all equations. This requires that the determinant of the system's equivalent matrix representation does not lead to any indeterminate forms or explicitly accounts for one unique form of solution.
02
Expressing the System with Known Conditions (a)
Assume \(x, y, z\) are real numbers. Use the identity that for three variables, the system:\[S_1 = x+y+z = 3, \quad S_2 = xy + yz + zx = b, \quad S_3 = x^2 + y^2+ z^2 = c\]can be transformed with these known equations:\[S_3 = S_1^2 - 2S_2\].Substitute \(S_1 = 3\) into the equation:\[c = 3^2 - 2b\].Rearrange this to yield:\[c = 9 - 2b\].
03
Unique Solution Condition (a)
To ensure a unique solution exists, any relationship between \(b\) and \(c\) derived from the transformations above must be such that no further variables interact to cause indeterminacy. The equation:\[c = 9 - 2b\]should be kept intact and uncoupled from constraints or further dependencies externally imposed.
04
Result for (a)
This condition implies that all values \(b\) and \(c\) that satisfy:\[c = 9 - 2b\]lead to systems with potentially unique solutions, assuming no further constraints are introduced.
05
Understanding the Uniqueness of Solutions (b)
Extend the logic for system (b) with variables \(a, b, c\):\[x + y + z = a, \quad xy + yz + zx = b, \quad x^2 + y^2+ z^2 = c\]Apply the known transformation relationships. For real numbers \(x, y, z\), we use:\[c = a^2 - 2b\].This simplifies to:\[c = a^2 - 2b\], which must hold for a unique determination of \(x, y, z\) given self-contained parameters.
06
Unique Solution Condition (b)
The system equation:\[c = a^2 - 2b\]must hold to guarantee uniqueness. Any discrepancies or solutions for \(a, b, c\) that lead to multiple realizations of the variable sets \(x, y, z\) are avoided by ensuring compatibility and correctness per this equation.
07
Result for (b)
Thus, for any \(a, b, c\) that satisfy:\[c = a^2 - 2b\]with this formeter completely adherent to the system identity, the parameter values lead to a determinable, unique solution such as is compatible.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Systems of Equations
A system of equations involves finding values for a set of variables that satisfy multiple equations simultaneously.
This means if you have three equations in three variables, you need to find those unique values that work for all the equations. Systems of equations are everywhere in mathematics, often used to model real-world situations. The goal is to find solutions that make all equations true. Here are some components of systems of equations:
This means if you have three equations in three variables, you need to find those unique values that work for all the equations. Systems of equations are everywhere in mathematics, often used to model real-world situations. The goal is to find solutions that make all equations true. Here are some components of systems of equations:
- Equations: These are the mathematical statements that express relationships among variables.
- Variables: These are the unknowns we want to solve for.
- Solution: A set of values for the variables that satisfies all the equations in the system.
Unique Solution
A system has a unique solution when exactly one set of values satisfies all the equations.
This is different from systems that might have no solution or infinitely many solutions.For a unique solution to exist, certain conditions must be met:
This ensures controlled parameters and prevents ambiguity.
This is different from systems that might have no solution or infinitely many solutions.For a unique solution to exist, certain conditions must be met:
- No contradictory or redundant equations.
- Usually a determinant or discriminator that鈥檚 not zero in a linear algebra context.
This ensures controlled parameters and prevents ambiguity.
Real Numbers
Real numbers are a fundamental concept in mathematics encompassing all 鈭� rational and irrational numbers 鈭� located on the number line.
These numbers include integers, fractions, and decimals, which you encounter in everyday life.In linear algebra and systems of equations, variables like \(x, y, \text{and } z\) are assumed to be real numbers.This ensures the solutions fall within a predictable and comprehensive set of numbers:
Numbers can approach infinity but always satisfy the properties of being real and measurable.
These numbers include integers, fractions, and decimals, which you encounter in everyday life.In linear algebra and systems of equations, variables like \(x, y, \text{and } z\) are assumed to be real numbers.This ensures the solutions fall within a predictable and comprehensive set of numbers:
- Rational Numbers: Numbers that can be expressed as a fraction, like \(\frac{1}{2}\).
- Irrational Numbers: Numbers that cannot be expressed precisely as a fraction, like \(\pi\) or \(\sqrt{2}\).
Numbers can approach infinity but always satisfy the properties of being real and measurable.
Matrix Representation
Matrix representation is one of the most powerful tools in linear algebra. It transforms systems of linear equations into a compact form, making complex calculations manageable.
By organizing coefficients and constants into matrices, one can employ various techniques for solutions.Here's a quick breakdown:
In essence, matrix representation simplifies systems, offering a structured approach to understand and solve equations.
By organizing coefficients and constants into matrices, one can employ various techniques for solutions.Here's a quick breakdown:
- Matrices: A rectangular array of numbers arranged in rows and columns.
- Determinants: A special number calculated from a square matrix that indicates certain properties about the matrix and the system it represents.
- Row Operations: Moves allowed on matrices that simplify solving or finding the uniqueness of solutions.
In essence, matrix representation simplifies systems, offering a structured approach to understand and solve equations.
