essential mathematical methods

[...]... a mathematical statement that contains an inequality symbol rather than an equals sign, for example 2x + 1 < 4 To solve linear inequations, proceed exactly as for equations except that: when multiplying or dividing both sides by a negative number, the ‘direction’ of the inequality symbol is reversed P1: FXS/ABE P2: FXS 0521609976c01.xml 14 CUAT006-EVANS August 3, 2005 13:19 Essential Mathematical Methods. .. August 3, 2005 13:19 Essential Mathematical Methods Units 1 & 2 Chapter summary A linear equation is one in which the variable has exponent 1 It is often helpful to look at how the equation has been constructed so that the steps necessary to ‘undo’ the equation can be identified It is most important that the steps taken to solve the equation are done in the correct order An inequation is a mathematical statement...P1: FXS/ABE P2: FXS 0521609976c01.xml 10 CUAT006-EVANS August 3, 2005 13:19 Essential Mathematical Methods Units 1 & 2 2(−3 − 2y) − y = 4 (reducing it to one equation −6 − 4y − y = 4 in one unknown) −5y = 10 y = −2 Substituting the value of y into (2) x + 2(−2) = −3 x =1 Equation (1) then... The techniques for transposing formulae include those used for solving linear equations detailed in section 1.1 P1: FXS/ABE P2: FXS 0521609976c01.xml 16 CUAT006-EVANS August 3, 2005 13:19 Essential Mathematical Methods Units 1 & 2 Example 17 Transpose the formula v = u + at, to make a the subject Solution v = u + at v − u = at v −u =a t Subtract u from both sides Divide both sides by t If we wish... left-hand side of the equation 1 b−a = ab c−2 Take the reciprocal of both sides ab =c−2 b−a Therefore c= ab +2 b−a 17 P1: FXS/ABE P2: FXS 0521609976c01.xml CUAT006-EVANS 18 August 3, 2005 13:19 Essential Mathematical Methods Units 1 & 2 Exercise 1F 1 For each of the following find the value of the letter in parentheses: a c = ab, a = 6, b = 3 c c = ab, a = 6, c = 18 √ (c) e c = a, a = 9 u g p = , u = 10,... Substitute in equation (1) 60 + y = 24 Hence y = −36 The two numbers are 60 and −36 Check in (2): 60 − (−36) = 96 P1: FXS/ABE P2: FXS 0521609976c01.xml CUAT006-EVANS 12 August 3, 2005 13:19 Essential Mathematical Methods Units 1 & 2 Example 12 3 kilograms of jam and 2 kilograms of butter cost $29 and 6 kilograms of jam and 3 kilograms of butter cost $54 Find the cost of 1 kilogram of jam and 1 kilogram... 4 and t = −3 2 Review 6 I bought x CDs for $a and y DVDs for $b The average price paid, in dollars, is 21 P1: FXS/ABE P2: FXS 0521609976c01.xml Review 22 CUAT006-EVANS August 3, 2005 13:19 Essential Mathematical Methods Units 1 & 2 5 A number d is equal to the square of a number e plus twice a number f a Find a formula for d in terms of e and f b Make f the subject of the formula c Find f when d =... line y Solution Let (x1 , y1 ) = (0, 3) and (x2 , y2 ) = (2, 0) y2 − y1 m= x2 − x1 0−3 = 2−0 3 =− 2 3 2 1 x 0 1 2 P1: FXS/ABE P2: FXS 0521609976c02.xml CUAT006-EVANS 26 August 3, 2005 13:22 Essential Mathematical Methods Units 1 & 2 It should be noted that the gradient of a line that slopes upwards from left to right is positive, as illustrated in Example 1, and the gradient of a line that slopes downwards... y, is called the gradient form Let x = 0, and thus then y = m(0) + c y=c i.e the y-axis intercept is equal to c P1: FXS/ABE P2: FXS 0521609976c02.xml 28 CUAT006-EVANS August 3, 2005 13:22 Essential Mathematical Methods Units 1 & 2 Example 4 Find the gradient and y-axis intercept of the graph of y = 3x − 4 Solution The value of m is 3 and the value of c is −4 Therefore the gradient of the above line... and y-axis intercept are known Example 7 Find the equation if m = −3 and c = 10 Solution Equation is y = −3x + 10 P1: FXS/ABE P2: FXS 0521609976c02.xml 30 CUAT006-EVANS August 3, 2005 13:22 Essential Mathematical Methods Units 1 & 2 Example 8 Find the equation of the straight line with gradient −3 which passes through the point with coordinates (−5, 10) Solution The general equation of lines with gradient . reversed. P1: FXS/ABE P2: FXS 05 216 09976c 01. xml CUAT006-EVANS August 3, 2005 13 :19 14 Essential Mathematical Methods Units1&2 Example 13 Solve the inequation 2x + 1 < 4. Solution 2x + 1 <. 3) 15 − 5(3 −4x) 15 > 2 Therefore 3(2x + 3) − 5(3 − 4x) > 30 and 6x + 9 − 15 + 20x > 30 Therefore 26x − 6 > 30 and x > 36 26 ∴ x > 18 13 P1: FXS/ABE P2: FXS 05 216 09976c 01. xml. equations detailed in section 1. 1. P1: FXS/ABE P2: FXS 05 216 09976c 01. xml CUAT006-EVANS August 3, 2005 13 :19 16 Essential Mathematical Methods Units1&2 Example 17 Transpose the formula v =