[PDF] [PDF] HELM 3: Equations Inequalities and Partial Fractions

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ContentsContents

& Partial FractionsEquations, Inequalities3.1 Solving Linear Equations 2

3.2 Solving Quadratic Equations 13

3.3 Solving Polynomial Equations 31

3.4 Solving Simultaneous Linear Equations 42

3.5 Solving Inequalities 50

3.6 Partial Fractions 60

Learning

In this Workbook you will learn about solving single equations, mainly linear and quadratic, but also cubic and higher degree, and also simultaneous linear equations. Such equations often arise as part of a more complicated problem. In order to gain confidence in mathematics you will need to be thoroughly familiar with these basis topics. You will also study how to manipulate inequalities. You will also be introduced to partial fractions which will enable you to re-express an algebraic fraction in terms of simpler fractions. This will prove to be extremely useful in later studies on integration. outcomes

Solving Linear

Equations

3.1

Introduction

Many problems in engineering reduce to the solution of an equation or a set of equations. An equation

is a type of mathematical expression which contains one or more unknown quantities which you will be required to nd. In this Section we consider a particular type of equation which contains a single unknown quantity, and is known as a linear equation. Later Sections will describe techniques for solving other types of equations.

Prerequisites

Before starting this Section you should:::be able to add, subtract, multiply and divide fractions be able to transpose formulae

Learning Outcomes

On completion you should be able to:::recognise and solve a linear equation

2HELM (2015):

Workbook 3: Equations, Inequalities & Partial Fractions

1. Linear equations

Key Point 1

Alinear equationis an equation of the form

ax+b= 0a6= 0

whereaandbare known numbers andxrepresents an unknown quantity to be found.In the equationax+b= 0, the numberais called thecoecient ofx, and the numberbis called

theconstant term.

The following are examples of linear equations

3x+ 4 = 0;2x+ 3 = 0;12

x3 = 0 Note that the unknown,x, appears only to the rst power, that is asx, and not asx2,px,x1=2etc. Linear equations often appear in a non-standard form, and also dierent letters are sometimes used for the unknown quantity. For example

2x=x+ 1 3t7 = 17;13 = 3z+ 1;112

y= 3 21:5 = 0 are all examples of linear equations. Where necessary the equations can be rearranged and written in the formax+b= 0. We will explain how to do this later in this Section. TaskWhich of the following are linear equations and which are not linear? (a)3x+ 7 = 0, (b)3t+ 17 = 0, (c)3x2+ 7 = 0, (d)5p= 0 The equations which can be written in the formax+b= 0are linear.Your solution (a) (b) (c) (d)Answer

(a) linear inx(b) linear int(c) non-linear - quadratic inx(d) linear inp, constant is zeroTo solve a linear equation means to nd the value ofxthat can be substituted into the equation so

that the left-hand side equals the right-hand side. Any such value obtained is known as asolution orrootof the equation and the value ofxis said tosatisfythe equation.

HELM (2015):

Section 3.1: Solving Linear Equations3

Example 1

Consider the linear equation3x2 = 10.

(a)

Check that x= 4is a solution.

(b)

C heckthat x= 2isnota solution.Solution

(a) T oche ckthat x= 4is a solution we substitute the value forxand see if both sides of the equation are equal. Evaluating the left-hand side we nd3(4)2which equals 10, the same as the right-hand side. So,x= 4is a solution. We say thatx= 4satises the equation. (b) Substituting x= 2into the left-hand side we nd3(2)2which equals 4. Clearly the left-hand side is not equal to 10 and sox= 2is not a solution. The numberx= 2does not satisfy the equation. TaskTest which of the given values are solutions of the equation

184x= 26

(a)x= 2;(b)x=2;(c)x= 8 (a) Substitutingx= 2, the left-hand side equalsYour solution

Answer

1842 = 10:But106= 26sox= 2is not a solution.(b) Substitutingx=2, the left-hand side equals:Your solution

Answer

184(2) = 26. This is the same as the right-hand side, sox=2is a solution.(c) Substitutingx= 8, the left-hand side equals:Your solution

Answer

184(8) =14. But146= 26and sox= 8is not a solution.4HELM (2015):

Workbook 3: Equations, Inequalities & Partial Fractions

Exercises

(a)

W ritedo wnthe general fo rmof a linea requation.

(b) Explain what i smeant b ythe ro oto rs olutionof a linea requation. In questions 2-8 verify that the given value is a solution of the given equation.

2.3z7 =28,z=7

3.8x3 =11,x=1

4.2s+ 3 = 4,s=12

x+43 = 2,x= 2

6.7t+ 7 = 7,t= 0

7.11x1 = 10,x= 1

8.0:01t1 = 0,t= 100.Answers

(a) The general fo rmis ax+b= 0whereaandbare known numbers andxrepresents the unknown quantity.quotesdbs_dbs7.pdfusesText_5

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