Pre-Pharmacy and Pre-Medicine 1 Mathematics PDF

DR IBRAHIM DUMBUYA HEAD OF DEPARTMENT INDUSTRIAL TECHNOLOGY NJALA UNIVERSITY MR HASSAN E J TURAY LECTURER COMAHS,USL PRE-PHARMACY AND PRE-MEDICINE 1 MATHEMATICS  PRE – CALCULUS 1. NUMBERS AND EQUATIONS Numbers Classification of numbers in mathema...

DR IBRAHIM DUMBUYA HEAD OF DEPARTMENT INDUSTRIAL TECHNOLOGY NJALA UNIVERSITY MR HASSAN E J TURAY LECTURER COMAHS,USL PRE-PHARMACY AND PRE-MEDICINE 1 MATHEMATICS  PRE – CALCULUS 1. NUMBERS AND EQUATIONS Numbers Classification of numbers in mathematics is a way of grouping numbers into different categories based in their properties and how they are represented. There are many types of numbers but some of the most common ones are: Natural Numbers: These are the positive integers that we use for counting, such as 1,2,3,4, and so on. They are also called counting numbers. Natural numbers are labelled N. Whole Numbers: They are the natural numbers with zero added, such as 0,1,2,3,4 and so on. They are labelled W. Integers: These are the whole numbers with the negative natural numbers added, such as -3, -2, -1, 0, 1, 2, 3 and so on. They are also called singed numbers. They are labelled Z.  Rational Numbers: These are the numbers that can be expressed as a fraction of two integers, such as 1/2 , 3/4 , -5/6 and so on. They are also called fractions or decimals. They are labelled Q.  Irrational Numbers: These are numbers that cannot be expressed as a fraction of two integers, such as , π. Real Numbers: These are the number that are either rational or irrational, such as -2, 0.5, , π and so on. They are also called the complete set of numbers. They are represented by points on a number line. Real numbers are represented by R. Imaginary Numbers: These are numbers that have negative square root such as i or j. Complex Numbers: These are numbers that can be written as a+b¡, where a and b are real numbers and i is an indeterminate satisfying ¡2 = -1. Prime Numbers: These are Numbers that have only two factors, 1 and themselves. Composite Numbers: These are Simultaneous Linear Equations in Two Variables A linear equation in two variables in an equation that can be written in the form ax + by = c, where a, b, and c are constants and x and y are variables. A solution of a linear equation in two variables is a pair of numbers (, y) that makes the equation true when The solution of a system of linear equations in two variables can be found by graphing the equation and finding the point of intersection of the lines, if any. Alternatively, the solution can be found by algebraic methods, such as substitution or elimination. SOLVING LINEAR EQUATIONS BY SUBSTITUTION METHOD The substitution method is a way of solving a system of linear equations in two variables by expressing one variable in terms of the other from one equation, and then substituting into the other equation.  Examples 1. Solve the equation x + y = 5 and 2 – y = 10 Solution + y = 5 -------(i) 2 – y = 1 -----(ii) From (i) : y = 5 – ------(iii)  Examples Substitute (iii) into (ii). 2 – (5 – ) = 1 3 –5=1 3 =6 =2  Examples Substitute = 2 into either equation s to find y. y=5– y=5–2 y=3 ··· The solution of the system is (2, 3).  Examples 2. Solve the following simultaneous equations. (a) (b) Solution (a) Let …………..(i) …………...(ii)  Examples Isolate y from (ii): and substitute in (ii) ⇒ ⇒ ⇒ Now find from  Examples ⇒ = 3 The solution is , Solution (b) Let ………………..(i) ………………(ii)  Examples Isolate y from (ii): and substitute in (i) ⇒ ⇒ ⇒ which gives Now find y from = 6 The solution is , = 6  SOLVING LINEAR EQUATIONS BY ELIMINATION METHOD The elimination method is a way of solving a system of linear equations in two variables by adding or subtracting the equations to eliminate one variable and obtain an equation in one variable.  Examples 1.Solve for and y if and. Solution --------- (1) ----------(2)  Examples To eliminate fractions, we multiply (1) by 4 and (2 ) by 45 which is their LCM respectively. 2 + y = 2 ---------(3) 15 – 9y = 5 --------(4) Multiply (3) by 9  Examples 18 + 9y = 18 + (15 – 9y = 5) 33 = 23 =  Examples Substitute value into (3): 2 + y = 2 2( ) + y = 2 +y=2 y=2- y= The solution of the system is ( , ).  Examples 2. Solve the pair of equations for and y respectively. 2 -1 – 3y-1 = 4 4 -1 – y-1 = 1  Examples Solution 2 -1 – 3y-1 = 4 - = 4 and 4 -1 – y-1 = 1 - = 1. Let p = and q =  Examples Then our new equations are: 2p - 3q = 4 -------(1) 4p-q = 1 -------(2) Multiply (2) by -3  Examples 2p – 3q = 4 -------- (1) -12p + 3q =- 3) ----- (3) -10p = 1 p =-1/10  Examples Reconverting our variables P = and p =. =  Examples Substitute p value into (2). 4p-q = 1⇒ 4() - q = 1 -0.4 - q = 1 ⇒ -q = 1 +0.4 q = -1.4  Examples Also, q = and q = -1.4 -1.4= y = -5/7  Consistent and inconsistent Systems A system of linear equations in two variables is called consistent if it has at least one solution, and inconsistent if it has no solution. For example, the system +y=5 2 Is consistent, because it has a unique solution (2,3).  Exercise 1.Given that – 5y – 3 = 0 and 2y - 6 + 5 = 0, find the value (, y). 2.Solve the following equations 4 – 3 = 3 + y and 3 +y = 2y + 5 – 12. 3.Given that = 6, find the ratio of : y. 4.Given that = , what is the ratio of to y  Exercise 5. If the equations + by = -4 and b + ay = 1 are satisfied by = 1, y = 2, find the values of a and b. 6. The line whose equation is a + by = 7 passes through the points with coordinates (2, -1) and (-1, -3). Find the values of a and b. 7. Show that the equations a + by = c and p +qy = r will have no value if aq = bp  GRAPHICAL METHOD OF SOLVING LINEAR EQUATIONS The graphical method of solving linear equations involves plotting the equations on a graph and finding their point of intersection. Here are the steps: 1. Draw a graph for each of the given linear equations. Each linear equation will be represented as a straight line on  GRAPHICAL METHOD OF SOLVING LINEAR EQUATIONS 2. Find the coordinates of the point of intersection of the two lines drawn. The coordinates of this point will be the common solution of the given equations. 3. Write the values of x and y. These are the solutions to your pair of linear equations.  x + y = 8 -------(1) y = x + 2 --------(2) Solution x + y = 8 -------(1) ⇒ 4. Check the solution by substituting the values of x and y obtained in both the given equations. Let us take two linear equations and solve them using the graphical method. Table of values of 𝑦= 8− 𝑥 𝑓𝑜𝑟 0 ≤ 𝑥≤ 4 𝑥 0 1 2 3 4 8 8 8 8 8 8 −𝑥 0 −1 −2 −3 −4 𝑦 8 7 6 5 4 Next consider y = x + 2. 𝑥 0 1 2 3 4 +2 +2 +2 +2 +2 +2 𝑦 2 3 4 5 6 Plotting these points on the coordinate plane, we get a graph like this.  GRADIENT OF A LINE JOINING TWO POINTS If and are any two points, then the gradient or slope of the line joining A and B is given by: = tan∅ NOTE: The order of subtraction is very important.  EXAMPLES 1. Find the gradients of the lines passing through the following pairs of points: i) and ii) and iii) and  EXAMPLES Solution i) The gradient of line ii) The gradient of line iii) The gradient of line SLOPE INTERCEPT FORM OF A LINEAR EQUATION The slope intercept form of a linear equation is: , where m is the slope and c is the Y-intercept. Examples 1. (a) Find the gradient of the line. (b) Determine the intercepts on the axes.  EXAMPLES Solution 1. (a) The straight line equation form is , where m is the gradient. Making the coefficient of y unity. becomes ⇒ the gradient is  EXAMPLES (b) Intercept on the axes Intercepts on becomes ⇒  EXAMPLES Intercept on becomes ⇒ ⇒ 2. Find the gradient of the line whose equation is. Solution Divide through the coefficient of y or  POINT⎼ SLOPE FORM Examples 1. Find the equation of a straight line with gradient 2 through the point. Solution Applying the formula gives Cross multiplying gives ⇒⇒  EXAMPLES Example 2: Given the point ((2, 3)) and the slope (m = 4), find the equation of the line. Solution Applying the formula Substitute the given values into the point-slope form: (y - 3 = 4(x - 2)) Simplify to get the equation in slope- intercept form ((y = mx + b)): (y = 4x - 5) So, the equation of the line is (y = 4x - 5).  QUADRATIC EQUATIONS A quadratic equation in one variable is an equation in which the highest power of the variable is 2. Example, 2 - 2 = 8 the standard form for a quadratic equation is a + b + c = 0 where a 0. 2 The equation will always have two solutions or roots but these may be  Methods of Solving Quadratic equations Quadratic equations can be solved (finding their roots) by the following methods: i. iv. Factorization ii. Completing the square iii. Formula iv. Graphical  Solution by Factorisation This method uses the AB = 0 principle. If AB = C then A = 0 or B = 0. Examples 1.Solve the following equations: i. (3 -2) (5 ii. = 1 Solution (3𝑥– 2) (5𝑥– 4) = (3𝑥-2)2 ⇒ 3𝑥(5𝑥– 4) -2(5𝑥– 4) = (3𝑥– 2) (3𝑥– 2) 15𝑥 - 12𝑥- 10𝑥+ 8 = 9𝑥 - 6𝑥- 6𝑥+ 4 2 2 15 - 22x + 8 = - 12x + 4 Collect like terms together 15 - 9 – 22x + 12x + 8 = 0 6 - 10x + 4 = 0 6x(x – 1) – 4(x – 1) = 0 (x – 1) (6x – 4) = 0 x – 1 = 0 or 6x – 4 = 0 x = 1 or 6x - 4 = 0 x = 1 or 6x = 4 x = 1 or 2/3. ii) x - 12/x = 1 Solution x - 12/x = 1 x(x) - x(12/x) = x(1) – 12 = x  – 12 = x – x -12 = 0 – 4x + 3x -12 = 0 x(x + 4) 3(x – 4) = 0 (x + 3) (x – 4) = 0 x = -3 or x = 4 Find the possible values of x/y if a. – xy – 6y = 0 b. – xy – 6 = 0 c) x(x – y) = y (x + 3y) d) – xy – 2 = 0 e) x/y + 20y/x = 9