/*! 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 32 Solve each system. $$\begin{al... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 32

Solve each system. $$\begin{aligned}x-y+z &=2 \\\y-z &=3 \\\x &=4\end{aligned}$$

Short Answer

Expert verified
The system has no solution because it is inconsistent.

Step by step solution

01

Express x

The third equation already gives the value of x. It states that \(x = 4\).
02

Substitute x into the first equation

Replace x with 4 in the first equation: \[4 - y + z = 2\] Simplify to obtain: \[-y + z = -2\].
03

Simplify the first equation

Rewrite the modified first equation: \[-y + z = -2\] as \[y - z = 2\].
04

Compare the modified first equation with the second equation

Now, the system is: \[y - z = 2\] \[y - z = 3\] Clearly, the system is inconsistent because \[y - z\] cannot equal both 2 and 3 simultaneously.

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.

Inconsistent Systems
Inconsistent systems are systems of equations that have no solutions. This means there are no values for the variables that can satisfy all the equations at the same time.
An example of an inconsistent system is:\[y - z = 2\] and \[y - z = 3\]
Mathematically, it's impossible for \(y - z\) to equal both 2 and 3 simultaneously. Hence, this system of equations cannot be solved.
When you encounter such systems, it indicates that there is a contradiction in the equations, leading to no possible solution.
Substitution Method
The substitution method is a technique to solve a system of equations by replacing one variable with an expression involving the other variable.
Here are the steps:
  • Step 1: Solve one of the equations for one variable in terms of the other variables.
  • Step 2: Substitute this expression into the other equations.
  • Step 3: Simplify and solve the resulting equation.
  • Step 4: Substitute back to find the remaining variables.
In our example, we start with: \[x = 4\]
Substituting \(x = 4\) into the first equation: \[4 - y + z = 2\] simplifies to \(-y + z = -2\). This substitution aids in reducing the system to simpler forms, making it easier to identify inconsistencies or solve the equations.
System of Equations
A system of equations consists of multiple equations that share the same set of variables. Solving the system means finding values for the variables that satisfy all given equations.
For example, in our exercise, we had:
  • \(x - y + z = 2\)
  • \(y - z = 3\)
  • \(x = 4\)
When solving a system of equations, it's essential to use methods like substitution or elimination to combine and simplify the equations.
You determine whether the system:
  • Has a unique solution (consistent and independent)
  • Has infinitely many solutions (consistent and dependent)
  • Has no solution (inconsistent)
In our exercise, the system is inconsistent, as clarified by the final comparison step where the modified equations demand contradicting equalities.

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

Use Cramer's rule to solve each system. \(x+y+z=6\) \(x-y+z=2\) \(2 x+y+z=7\)

Solve each problem by using a system of three equations in three unknowns. In three days, Katy traveled 146 miles down the Mississippi River in her kayak with 30 hours of paddling. The first day she averaged 6{mph}, the second day 5{mph}, and the third day 4 {mph}. If her distance on the third day was equal to her distance on the first day, then for how many hours did she paddle each day?

Solve each system using the Gauss-Jordan elimination method. $$ \begin{array}{r} x+2 y=1 \\ 3 x+6 y=3 \end{array} $$

Solve each problem by using a system of three equations in three unknowns. Working overtime. To make ends meet, Ms. Farns by works three jobs. Her total income last year was 48,000 dollars. Her income from teaching was just 6000 dollars more than her income from house painting. Royalties from her textbook sales were one-seventh of the total money she received from teaching and house painting. How much did she make from each source last year?

A truck carrying 3600 cubic feet of cargo consisting of washing machines and refrigerators was hijacked. The washing machines are worth \(\$ 300\) each and are shipped in 36 -cubic-foot cartons. The refrigerators are worth \(\$ 900\) each and are shipped in 45 -cubic-foot cartons. If the total value of the cargo was \(\$ 51,000,\) then how many of each were there on the truck?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

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.