Problem 31 Explain when it might be more co... [FREE SOLUTION]

Chapter 1: Problem 31

Explain when it might be more convenient to use the elimination method than the substitution method to solve a linear system. Give an example to support your claim.

Short Answer

Expert verified

The elimination method is often more convenient when the coefficients of the variables can be easily manipulated to eliminate one variable by adding or subtracting equations, particularly when dealing with equations featuring fractions or irrational numbers. For example, in the system \(2x - 3y = 5\) and \(4x + 3y = 8\) it's clear that the elimination method by adding the equations is faster and more straightforward.

Step by step solution

Identify the conveniences of the elimination method

The elimination method can be more convenient when the coefficients of the variables in the equation can be easily matched to eliminate one of the variables when the equations are either added or subtracted. Unlike substitution, this method often requires fewer calculation steps.

Identify the inconveniences of the substitution method

The substitution method can be less convenient as it often requires more computation, especially when the coefficients of the equations are fractions or irrational numbers. It requires to rearrange an equation to solve for a variable, substitute this expression into the other equation, and then continue solving. The process might lead to more complicated calculations.

Provide Example

Consider the system of equations, \(2x - 3y = 5\) and \(4x + 3y = 8\). Here, the elimination method can be immediately applied by adding these two equations to remove variable \(y\). Adding the equations gives \(6x = 13\), which leads to \(x = \frac{13}{6}\) after division by 6. Substituting \(x\) in the first equation quickly gives \(y = -\frac{1}{3}\), without dealing with fractions in the coefficients. If the substitution method was to be used instead, it would require extra steps to isolate a variable, leading to more complicated calculations.

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.

Linear Systems

Linear systems consist of two or more linear equations that represent straight lines on a graph. These systems aim to find the point or points where these lines intersect, which refers to the solution of the equations.
Linear systems can be characterized by:

Having equations with variables raised only to the first power, meaning no squares or cubes.
The goal of solving these systems is to find the values of the variables that satisfy all equations simultaneously.
These systems can have a single solution, infinitely many solutions, or no solution at all.

To solve linear systems, we can use different methods such as the elimination method or substitution method. Each method can be more suitable than the other depending on the system's layout and coefficients.
Understanding which method to choose is crucial for efficient problem-solving.

Substitution Method

The substitution method is a common technique used to solve linear systems by replacing one variable with an expression derived from another equation. This involves several steps:

First, solve one of the equations for one of the variables. This usually means getting a variable by itself on one side of the equation.
Then substitute this expression into the other equation. This reduces the system to a single equation with only one variable.
Solve this single-variable equation to find the value of one variable.
Finally, plug the value back into the expression obtained for the first variable to find its corresponding value.

While the substitution method is straightforward when coefficients are simple, it becomes cumbersome when dealing with fractions or irrational numbers. The need to isolate a variable for substitution can lead to more complex calculations, particularly if the equations are not originally solved for one of the variables.

Solving Equations

Solving equations involves finding the unknown values of the variables that make the mathematical statement true.
This process often includes:

Identifying the type of equation (linear, quadratic, etc.) to choose the best method of solving.
Rearranging the equation where necessary to isolate the variable of interest, especially in the context of substitution.
Performing algebraic manipulations like adding, subtracting, multiplying, or dividing both sides of the equation to simplify or eliminate terms.
Checking the solution by substituting the found values back into the original equations to verify they satisfy all components of the system.

Solving equations efficiently requires a good understanding of algebraic principles and the ability to recognize when certain methods are more advantageous than others. The elimination and substitution methods are both valuable tools in this task, with their own sets of pros and cons depending on the form of the equations.

91影视

Explain when it might be more convenient to use the elimination method than the substitution method to solve a linear system. Give an example to support your claim.

Short Answer

Step by step solution

Identify the conveniences of the elimination method

Identify the inconveniences of the substitution method

Provide Example

Key Concepts

Linear Systems

Substitution Method

Solving Equations

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Decision Maths

Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.