/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 79 The system of linear equations \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 26: Problem 79

The system of linear equations \(x+y+z=2\) \(2 x+3 y+2 z=5\) \(2 x+3 y+\left(a^{2}-1\right) z=a+1 \quad\) [Jan 09 2019I] (a) is inconsistent when \(\mathrm{a}=4\) (b) has a unique solution for \(|\mathrm{a}|=\sqrt{3}\) (c) has infinitely many solutions for \(\mathrm{a}=4\) (d) is inconsistent when \(|\mathrm{a}|=\sqrt{3}\)

Short Answer

Expert verified
(c) has infinitely many solutions for \\ a=4.

Step by step solution

01

Understanding the Problem

We have a system of three linear equations with parameters that can change the behavior of the solutions. The goal is to determine conditions for inconsistency or types of solutions based on different values of 'a'.
02

Write the System of Equations

Write the given system of linear equations as: 1. \( x + y + z = 2 \) 2. \( 2x + 3y + 2z = 5 \) 3. \( 2x + 3y + (a^2 - 1)z = a + 1 \)
03

Check Consistency for a = 4

Substitute \(a = 4\) into the third equation: \[(a^2 - 1)z = (4^2 - 1)z = 15z\] and \[(a + 1) = (4 + 1) = 5\]So the third equation becomes \(2x + 3y + 15z = 5\). Compare with equations 1 and 2 to check for contradictions.
04

Solve for a = 4

Substitute \(a = 4\) and solve the system:1. \( x + y + z = 2 \)2. \( 2x + 3y + 2z = 5 \)3. \( 2x + 3y + 15z = 5 \)Subtract equation 2 from equation 3: \[ (2x + 3y + 15z) - (2x + 3y + 2z) = 5 - 5 \]\[13z = 0 \Rightarrow z = 0 \]Substitute \(z = 0\) in equations 1 and 2 to find \(x\) and \(y\).
05

Solving for a Solution when a = 4

Substitute \(z = 0\) into equations 1 and 2:Equation 1 becomes \(x + y = 2\).Equation 2 becomes \(2x + 3y = 5\).Solve this system of 2 equations in 2 variables with substitution or elimination to find a consistent solution, indicating infinite solutions exist due to a leading zero row appearing only in a third equation trial test.
06

Check Consistency for |a| = sqrt(3)

Substitute \(|a| = \sqrt{3}\) into the third equation:\((a^2 - 1)z = (3 - 1)z = 2z\)and\(a + 1 = \sqrt{3} + 1\)The third equation becomes \(2x + 3y + 2z = \sqrt{3} + 1\). Compare and analyze for solution consistency with equations 1 and 2.
07

Determine Solutions for |a| = sqrt(3)

The system 1. \( x + y + z = 2 \)2. \( 2x + 3y + 2z = 5 \)3. \( 2x + 3y + 2z = \sqrt{3} + 1 \) has a conflict when substituting equations meaning there is consistency only when \(|a| = \sqrt{3} = 1.732\), leaving a reduced possibility for unique or no solutions.

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.

System of Equations
A system of equations is like a set of mathematical puzzles all tied together. Each equation represents a rule that relates different variables. In a linear system, these rules are linear, meaning each variable gets multiplied by a constant and then summed up. Linear relationships look like straight lines when plotted on a graph.

Our given system consists of three equations:
  • \(x + y + z = 2\)
  • \(2x + 3y + 2z = 5\)
  • \(2x + 3y + (a^2 - 1)z = a + 1\)
Here, the equations incorporate a parameter \(a\) which affects the overall behavior of the system. Depending on the value of \(a\), the solutions to the system can change significantly. Understanding this setup is crucial as it lays the foundation for analyzing the range of potential solutions.
Consistency in Systems of Equations
Consistency in a system of equations refers to whether there is at least one solution that satisfies all equations simultaneously. When all equations in a system can share at least one common solution, the system is consistent. Otherwise, it's inconsistent.

For example, when \(a = 4\), the system becomes:
  • \(x + y + z = 2\)
  • \(2x + 3y + 2z = 5\)
  • \(2x + 3y + 15z = 5\)
By substituting \(a = 4\), we notice that the third equation simplifies to match the second, except with different coefficients for \(z\). This results in conditions where the system appears consistent, offering solutions for \(x\), \(y\), and \(z\). Meanwhile, substituting other values might introduce contradictions and inconsistencies, such as \(|a| = \sqrt{3}\), where the equations conflict and offer no solution.
Solutions to Linear Systems
Solutions to a linear system depend on how the equations in that system intersect:
  • Unique solution: Occurs when the lines or planes intersect at a single point. This happens when the number of independent equations matches the number of unknowns, and they aren't parallel or coincide.
  • Infinitely many solutions: This is seen when the lines or planes overlap exactly; in 3D, this means an entire line or plane is common to all and the system cannot pinpoint one particular solution.
  • No solution: Describes a system where the lines or planes never meet, such as parallel planes in three dimensions.
When \(a = 4\), solving the system leaves us with consistent equations, suggesting infinitely many solutions rather than one specific solution. Conversely, when \(|a| = \sqrt{3}\), the system becomes inconsistent, meaning there isn't any solution that satisfies all equations. Understanding these different types of solutions is important for interpreting results in practical situations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The image of the line \(\frac{x-1}{3}=\frac{y-3}{1}=\frac{z-4}{-5}\) in the plane \(2 x-y+z+3=0\) is the line: (a) \(\frac{x-3}{3}=\frac{y+5}{1}=\frac{z-2}{-5}\) (b) \(\frac{x-3}{-3}=\frac{y+5}{-1}=\frac{z-2}{5}\) (c) \(\frac{x+3}{3}=\frac{y-5}{1}=\frac{z-2}{-5}\) (d) \(\frac{x+3}{-3}=\frac{y-5}{-1}=\frac{z+2}{5}\)

The equation of a plane through the line of intersection of the planes \(x+2 y=3, y-2 z+1=0\), and perpendicular to the first plane is : [Online April 25, 2013] (a) \(2 x-y-10 z=9\) (b) \(2 x-y+7 z=11\) (c) \(2 x-y+10 z=11\) (d) \(2 x-y-9 z=10\)

If the line, \(\frac{x-3}{2}=\frac{y+2}{-1}=\frac{z+4}{3}\) lies in theplane, \(l x+m y-z=\) 9, then \(\mathrm{l}^{2}+\mathrm{m}^{2}\) is equal to : (a) 5 (b) 2 (c) 26 (d) 18

The plane which bisects the line joining the points \((4,-2,3)\) and \((2,4,-1)\) at right angles also passes through the point: \([\) Sep. \(03,2020(\mathrm{II})]\) (a) \((4,0,1)\) (b) \((0,-1,1)\) (c) \((4,0,-1)\) (d) \((0,1,-1)\)

If an angle between the line, \(\frac{x+1}{2}=\frac{y-2}{1}=\frac{z-3}{-2}\) and the plane, \(x-2 y-k x=3\) is \(\cos ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)\), then a value of \(k\) is [Jan. 12, 2019 (II)] (a) \(\sqrt{\frac{5}{3}}\) (b) \(\sqrt{\frac{3}{5}}\)(c) \(-\frac{3}{5}\) (d) \(-\frac{5}{3}\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.