Problem 44 Describe in general terms how to... [FREE SOLUTION]

Chapter 7: Problem 44

Describe in general terms how to solve a system in three variables.

Short Answer

Expert verified

The solution to a system of equations in three variables is found by forming the correct equation system, applying the elimination method to reduce the system to two variables, and solving the resulting system. The value of the third variable is then found by substitution.

Step by step solution

Form the Equation System

Identify the three equations in the system from the problem statement. The system should typically be in the form of \(ax + by + cz = d\), \(ex + fy + gz = h\), and \(ix + jy + kz = l\), where a, b, c, d, e, f, g, h, i, j, k, and l are constants and x, y, z are the variables we are trying to solve for.

Apply the Elimination Method

Choose two of the three equations and eliminate one variable. This can be done by either adding or subtracting the two equations or multiplying one or both equations by a suitable number then adding or subtracting. This will result in a new equation with only two variables.

Form New Two-variable System and Solve

Repeat Step 2 with a different pair of equations to eliminate the same variable, resulting in a second equation with the two remaining variables. You now have a system of two equations with two variables. Solve this system using substitution or elimination method to get the values of the two remaining variables.

Solve for the Third Variable

Substitute the values of the two solved variables back into one of the original three-variable equations to solve for the third variable.

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.

Elimination Method

The elimination method is a strategic approach used in solving systems of equations, particularly useful when dealing with multiple variables. In essence, this method involves removing one variable to simplify the system of equations. Here's how it works:

First, choose two equations from your system. The goal is to combine them in a manner that eliminates one of the variables.
You can do this by adding or subtracting the equations. Sometimes, you might need to multiply one or both equations by certain numbers to align the coefficients, ensuring a variable is eliminated when the equations are combined.
After eliminating one variable, you'll be left with a simpler equation system with only two variables.

This method is particularly powerful in a three-variable system because it reduces the complexity of the system, making it more manageable. For students, mastering the elimination method involves practice, particularly with identifying which equations to pair and how to manipulate their coefficients efficiently.

Substitution Method

The substitution method is another strategy for solving systems of equations, especially effective when one of the equations in the system is already solved for one variable, or can be easily solved for one variable. Here's a breakdown of the steps:

Start by selecting an equation and solve it for one of the variables. This involves expressing one variable in terms of the other two.
Substitute this expression into the other equations in the system. This reduces the number of variables, leaving a simpler equation system.
Continue substituting this expression into the remaining equations to solve the system step-by-step until all variables are found.

Substitution is a useful method to employ when equations lend themselves easily to being manipulated without complex calculations. It effectively reduces the system's equations until you find solutions for each variable. This method, when used in tandem with elimination, can simplify complex equations considerably.

Three Variables

Solving systems with three variables requires an understanding of how to manipulate multiple equations simultaneously. The main idea is to reduce the system from three to two variables, and then to one, or solve them directly depending on the structure of the equations. Here鈥檚 a simplified method for tackling these systems:

Begin with the three-variable system, usually structured as three separate equations.
Use either the elimination or substitution methods to reduce the system to two variables. Typically, this is done by eliminating the same variable from two separate pairs of equations.
This yields two equations with two variables. These can be solved using methods for two-variable systems, such as elimination or substitution.
Once two of the variables are known, substitute them back into one of the original equations to solve for the third variable.

Working through these steps requires practice to choose the best combinations of equations and methods. Understanding the interactions between the variables and equations is key to solving these systems efficiently.

91影视

Describe in general terms how to solve a system in three variables.

Short Answer

Step by step solution

Form the Equation System

Apply the Elimination Method

Form New Two-variable System and Solve

Solve for the Third Variable

Key Concepts

Elimination Method

Substitution Method

Three Variables

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Pure Maths

Discrete Mathematics

Statistics

Applied Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.