Chapter 15
Solving Linear Equations

Previous section: Making Systems of Equations, Triangular

Here are some more examples in which we solve equations. Our aim is to become familiar or at ease with handling and manipulating equations. So we look at the algebraic solution of equations containing one or more unknown numbers.

5  Examples with Three Unknowns

Example: Solve the following equations (a1), (a2) and (a3).

A Solution:   First subtract equation (a1) from (a2). This implies yields equation (b2) below.

Second, add equation (b1) to equation (b3). This gives

The above steps have eliminated z from the last two equations.

Third, from equation (c2) subtract two times equation (c3). This implies

Finally, we may change the order of equations. This yields the more suggestive system of equations:

This last step was optional. Now we can do the following.

 

• Solve equation (d2) for x. This gives

  x = 8/2 = 4

• Solve equation (d3) for y. This yields

  y = 1 3 (11-2x) = 1 3 ·(11-8) = 1

• Solve equation (d1) for z. This implies

  z = 1 2 (27-5x-2y) = 1 2 (27-20-2) = 1 2 (5) = 5 2

Exercise:   Solve

Note that you can and should check your answer (values for x, y and z) satisfy each equation. If one is not satisfied then there is an error somewhere in your work, the solution or the check.

Exercise:   Solve

See the difference a "small" change in the problem makes.

More Chapter Sections: [ Up ] [ 15 Algebra Solutions ] [ 15 Triangular Systems ] [ 15 Making Triangular ] [ 15 With 3 Unknowns ] [ 15 Rules and Advice ]

Chapters and Appendices

Home
Postscript: The 4-th Skill For Algebra
Foreword
1. Introduction
2. Implication Rules
3. Chains of Reason
4. Romeo and Juliet
4. Induction Mathematical
5 Knowledge Islands
6  Old Language
7  Arith Skill Check
7. The Next Chapters
8 The Three Skills
8 VNR-Concise-Encyclopedia
PS. What is a Variable
9. Algebra Talk
10 Two More Skills
11 Why Shorthand
12 Shorthand Usage
13 What's Next
14 Compound Interest
15 Linear Equations
PS I.  Distributive Law
PS II. Polynomials
16 Painless Proofs
17 Pythagoras
18 Rules of Algebra
19  Functions & Sets
20 Degrees & Radians
21 What's Next
22. Arith & Geometric Sums
23 Summation Notation
24 Your Money
25 Induction & Recursion
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
A. Advice For Learning

Words Before Symbols: 
What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent or Independent
Variable, a Matter of Choice

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