E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals 1.1 E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Linear Equations KFUPM - Prep Year Math Program (c) All Right Reserved  Properties of Equations  Types of Equations  Formulas  Applications of Linear Equation

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Introduction KFUPM - Prep Year Math Program (c) 2009 All Right Reserved An equation in a variable x, is a statement that two expressions involving x are equal. The number z is said to satisfy that equation if, when replace the variable, makes the equation a true statement. Such a number z is called a solution of the equation or a zero or a root of the equation. The set of all the number which satisfy an equation is called a solution set of the equation. To solve an equation means to find its solution set. Two equations are said to be equivalent if they have the same solution set.

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Linear Equations KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Properties Addition: Addition: Adding the same number to both sides of an equation result in an equivalent equation. Multiplication: Multiplication: Multiplying both sides of an equation by the same non-zero number, again result in an equivalent equation.

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Types of Equations KFUPM - Prep Year Math Program (c) 2009 All Right Reserved An equation which is satisfied by every number is called an identity. An equation that is not satisfied by any number is called contradiction (sometimes inconsistent). An equation which is satisfied by same numbers and not by others is called a conditional equation (sometimes consistent).

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 1 The solution set is {0}. The equation is conditional.

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 2 Solve the equations and identify the equation as conditional, identity or contradiction The solution set consists of all real numbers and the equation is an identity. The solution set is the empty set and the equation is a contradiction.

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Formulas KFUPM - Prep Year Math Program (c) 2009 All Right Reserved A formula is an equation that relates several variables. The formula may or may not be linear in all variables, but if it is linear in one of the variables, then we can solve the formula for that variable using the techniques we learned in this section. For example

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 3 Solve each equation for the indicated variables

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 4 Solve each of the following equations for x (a)(b)

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 5 Solve each of the following equations for x (a)(b)

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Equations Involving Absolute Values KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Recall that the absolute value of real numbers is defined by In words, the absolute value is a non-negative real number which is achieved by exactly two real numbers, one positive and one negative.

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 6 Solve each of the following equations for x Solution set is The equation, has no solution because its equivalent to. Which is impossible since the absolute value can never be negative. (a) (b)

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 7 Solve each of the following equations for x Therefore, the solution set is

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 8 Solve each of the following equations for x Checking each solution in the original equation, we find that the solution set to be

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 9 When Saleem bought a new car, he received a discount of 12%. He paid SAR 57,992 for the car. What was the original price of Saleem’s car? Original Price – discount=Sale Price x – 0.12x =57, x =57, 922 x = x =