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Transpositions

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CONTENTS

ITEM TYPE NUMBER
Rearranging equations Workout 59 slides
Transpositions Library 20 questions
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SAMPLE FROM THE WORKOUT

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SLIDE 1 - QUESTION 1

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SLIDE 2 - SOLUTION

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SAMPLE FROM THE LIBRARY

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QUESTION [difficulty 0.2]

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SOLUTION

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DEPENDENCIES

286: Algebraic factors
288: Simultaneous equations
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296: Transpositions
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300: Algebraic manipulations

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CONCEPTS

ITEM
 LEV.
Transposition, rearrangement, changing the subject 696.5
Fundamental rule of equations when transposing 696.8
Elementary transpositions - basic operations 697.0
Rearranging the equation of the straight line 697.2
Finding the gradient and intercept of a straight line 697.2
Transposing an algebraic fraction 697.7
Blunder in the manipulation of algebraic fractions 698.2
Cancelling algebraic fractions avoiding blunders 698.4
Tricky aspects of algebraic fractions 698.6
Transposition: factor and solve 698.8
Fractions appearing in formulas 699.1
Transposition: move as one 699.4
Transposing square roots and roots in general 699.7
Transposing the form y = (ax + b) / (cx + d) 700.6
Clearing a square root 701.1
Ratio problems leading to transposition problems 701.8

RAW CONTENT OF THE WORKOUT

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Transpositions SLIDE 1 Consolidation ? Find I when ? Find P when ? Find v when SLIDE 2 ? Find I when Solution ? Find P when Solution ? Find v when Solution SLIDE 1B Consolidation ? Find I when ? Find P when The answer is in horsepower (hp) ? Find v when SLIDE 2B ? Find I when Solution ? Find P when Solution ? Find v when Solution SLIDE 3 In the subject of the equation is F. Consolidation In each case, state the subject of the formula ? ? ? SLIDE 4 ? The subject of the equation is P ? The subject of the equation of K ? The subject of the equation is v SLIDE 5 Transposition To transpose is to make another symbol the subject of the equation. It is the same as rearranging the equation to bring another symbol to the left. Example Make F the subject of the formula It is not necessary to include all these steps. It is customary to go from the first to the last line in one step, though when you are learning the process, it can be helpful to include as many lines as you need. SLIDE 6 What is the fundamental rule for equations and how is this rule used in this process of transposing the equation for F? SLIDE 7 The fundamental rule for equations is what you do to (all of) one side of the equation, you also do to (all of) the other side of the equation. At line (2) we divide both sides by d. This is an example of the fundamental rule. We divide one side by d in order to “cancel it out”, so we divide the other side by d, at the same time. SLIDE 8 The fundamental rule for equations is what you do to one side of the equation, you also do to the other side of the equation. Annotate each of the following to show how the fundamental rule for equations has been used when changing the subject of the formula. ? ? ? SLIDE 9 The fundamental rule for equations is what you do to one side of the equation, you also do to the other side of the equation. ? ? ? SLIDE 10 Transpose each of the following formulae ? ? ? ? ? ? SLIDE 11 ? ? ? ? ? ? SLIDE 12 Rearranging the equation of the straight line Make y the subject of the equation Solution Putting the equation in the form We have SLIDE 13 Put each of the following equations in the form . Find in each case the gradient and intercept on the y-axis. ? ? ? SLIDE 14 ? ? ? SLIDE 15 Sketch the graph of SLIDE 16 Sketch the graph of Solution We first put in the form . SLIDE 17 Example Make x the subject of Solution SLIDE 18 ? Make t the subject of ? Rearrange to make y the subject SLIDE 19 ? Make t the subject of Solution ? Rearrange to make y the subject Solution SLIDE 20 Transpose for C SLIDE 21 Transpose for C Solution SLIDE 22 Error in manipulation of algebraic fractions What is the error in the following? SLIDE 23 This is a blunder. It arises because the student has not mastered the addition of fractions at an earlier stage. The correct manipulation is SLIDE 24 ? Complete the following algebraic description of the rule for the addition of fractions. ? Cancel the bottom factor into the top, being careful to avoid a blunder. SLIDE 25 ? ? SLIDE 26 If p is a positive number and q is a negative number, which of the following is true? A B C D SLIDE 27 If p is a positive number and q is a negative number, which of the following is true? A B C D Solution Solving this problem depends on the correct manipulation of fractions. Since p is positive and q is negative, is a negative number, possibly a fraction. Then This gives option C, . To show that B, , is incorrect, observe that to add any number to 1 (positive or negative) is to make it not equal to 1. SLIDE 28 Factor and solve Rearrange for k Solution Step 1 Bring everything with k to the left, and everything else to the right. Step 2 Factor for k Step 3 Solve for k SLIDE 29 ? Make k the subject of ? Make x the subject of SLIDE 30 ? Make k the subject of Solution ? Make x the subject of Solution SLIDE 31 Fractions appearing A small fraction appearing in front of a formula can cause difficulties. Example Make m the subject of . Solution SLIDE 32 ? Make x the subject of ? Make P the subject of SLIDE 33 ? Make x the subject of Solution ? Make P the subject of Solution SLIDE 34 Move “as one” Make m the subject of Here the may move as one expression. Solution SLIDE 35 ? Make S the subject of ? Make x the subject of SLIDE 36 ? Make S the subject of Solution ? Make x the subject of Solution SLIDE 37 Square roots, roots in general Make v the subject of Here we must take the square root to obtain v Solution SLIDE 38 ? Make x the subject of ? Make T the subject of SLIDE 39 ? Make x the subject of Solution ? Make P the subject of Solution SLIDE 40 Transpose ? ? SLIDE 41 ? Solution ? Solution SLIDE 42 Transpose ? ? SLIDE 43 ? Solution ? Solution SLIDE 44 Transpose ? ? SLIDE 45 ? Solution ? Solution SLIDE 46 Example Make x the subject of Solution SLIDE 47 Make x the subject of Annotate the following solution to this problem, explaining each step of the process. SLIDE 48 SLIDE 49 Rearrange ? ? SLIDE 50 ? ? SLIDE 51 Clearing a square root Make k the subject of Solution SLIDE 52 ? Make T the subject of ? Make x the subject of SLIDE 53 ? ? SLIDE 54 ? Make r the subject of ? Make x the subject of SLIDE 55 ? ? SLIDE 56 ? Transpose ? Make v the subject of SLIDE 57 ? ? SLIDE 58 The ratios and are the same. Find a in terms of b and k. SLIDE 59 The ratios and are the same. Find a in terms of b and k. Solution