Equations, Inequalities, and Mathematical Models 1.2 Linear Equations Chapter 1 Equations, Inequalities, and Mathematical Models 1.2 Linear Equations

Objectives At the end of this session, you will be able to: Solve linear equations in one variable. Solve equations with constants in denominator. Solve equations with variables in denominators. Recognize identities. Recognize conditional equations, and inconsistent equations.

Index Linear Equation Solution of a Linear Equation Solving a Linear Equation Solving Linear Equations involving Fractions Solving Linear Equations involving Rational Expressions Types of Equations Summary

1. Linear Equation We have already come across statements of the following types: 3 + 2 = 5 (1) 4 . (5+6) = 4 . 5 + 4 . 6 (2) 5 . (6 - 7) = 5 . 6 – 5 . 7 (3) Now let us consider the following statements: x more than 4 is 9. 7 less than a number x is 6. A number y divided by 6 gives 2. Product of a variable x with itself is 36. We can write the above statements (i) to (iv) respectively, as follows: 4 + x = 9 (4) x – 7 = 6 (5) (6) x . x = x2 = 36 (7) We observe that the symbol ‘ = ‘ (equal to) appears in each of the statements (1) to (7). A statement involving the symbol ‘ = ‘ is called a statement of equality or simply an equality. Thus, each of the statements (1) to (7) is an equality. We observe that statements (1), (2), and (3) do not involve any variable while all the remaining statements (4) to (7) involve one variable. Thus, a statement of equality which involves variables is called an equation. Therefore, we can say that each of the equalities (4) to (7) is an equation.

1. Linear Equation (Cont…) Each equation has has two sides, left side and the right side. For instance, the equation (4), 4 + x = 9, (4 + x) is the left side and 9 is the right side, while in equation (6), , is the left side and 2 is the right side of the equation. Further, we have equations (5): x – 7 = 6 and (7): x2 = 36. Now, in each of the equations (4), (5), and (6), the highest power of the variables involved is one. An equation in which the highest power of the variables involved is one, is called a Linear Equation. Hence, each of the equations (4), (5), and (6) is a linear equation. However, the equations (4), (5), and (6) are linear equations in one variable. Equation (7) is not a linear equation. Thus from the above discussion we arrive at the following definitions: Equation: A statement of equality which involves variables is called an equation. Linear Equation: An equation in which the highest power of the variables involved is one, is called a Linear Equation. Linear Equation in one variable: A linear equation in one variable x is an equation that can be written in the form ax + b = 0 where a and b are real numbers, and a 0.

2. Solution of an Equation Let us consider a linear equation, namely, x – 8 = -4 (1) Left side of (1) is x – 8 and right side is - 4. Let us evaluate the left side of equation (1) for some values of x till the left side becomes equal to the right side of the equation (1). We observe from the above table that the left side of equation (1) becomes equal to the right side, only when 4 is substituted for x. For all other values of x, the left side of equation (1) is not equal to the right side of equation (1). The value of the unknown for which left side of the equation is equal to the right side, is called the solution or root of the equation. We also say that such a value satisfies the equation or results in a true statement. For the given equation (1) we say x = 4 is the solution or root of the equation x – 8 = -4. The set of all such solutions is called the equation’s solution set. For instance, the solution set of the equation x – 8 = -4 is {4}. x Left side of (1) Right side of (1) 1 2 3 4 x – 8 = 0 – 8 = -8 x – 8 = 1 – 8 = -7 x – 8 = 2 – 8 = -6 x – 8 = 3 – 8 = -5 x – 8 = 4 – 8 = -4 -4

3. Solving an Equation Solving a linear equation in one variable means finding the value of the variable that makes the equation true. For example, consider the equation x + 7 = 11 4 + 7 = 11 ( Substituting 4 for x) 11 = 11 (Results in a true statement) The number 4 is said to satisfy the equation. Basically, the operation used in solving equations is to manipulate both members, by addition, subtraction, multiplication, or division until the value of the variable becomes apparent. An equation may be compared with a balance used for weighing. Its sides are two pans and the equality symbol (=) tells us that the two pans are in equilibrium. You must have seen the working of a balance. If we put equal weights in both the pans, then we observe that the two pans remain in balance. Similarly, if we remove equal weights from both the pans, we observe that the pans still remain in balance. Thus, we may add equal weights or amounts to both pans or we may subtract equal weights or amounts from both pans and the balance will remain in balance. Thus, an equation may be compared to a balance. What is done to one side must also be done to the other to maintain a balance. An equation must always be kept in balance or the equality is lost. 11 4 + 7 = Equation compared to a balance scale. 

3. Solving an Equation (Cont…) In case of an equation, the equality symbol (=) will not change if we follow the following rules: Now we shall solve some equations using the above rules: Example 1: Solve x – 3 = 11 To get the value of x from the above equation, we add 3 to both sides of the equation (Rule I). Thus gives RULES EXAMPLE Add the same number to both sides of the equation. Subtract the same number from both sides o the equation. Multiply both sides of the equation by the same non-zero number. Divide both sides of the equation by the same non-zero number. If a = b, then a + c = b + c x + 2 = 5, then x + 2 + 3 = 5 + 3 If a = b, then a - c = b - c x + 2 = 5, then x + 2 - 1 = 5 - 1 If a = b, then a(c) = b(c) If a = b, then a/c = b/c where c is not equal to 0. 5x = 12, then 5x ÷ 5 = 12 ÷ 5

3. Solving an Equation (Cont…) Check: Let us substitute x = 14 in the given equation. For x = 14 Left Side = 14 – 3 = 11 Right Side = 11 That is, for x = 14. Left side = Right side Thus, x = 14 is the root or solution of the given equation. Example 2: Solve 2y = y + 3 Here subtracting y from both sides of the equation (Rule II), we get Check: Let us substitute y = 3 in the given equation. We get Left Side of the given equation = 2(3) = 6 Right Side of the given equation = 3 +3 =6 Hence for y = 3 we get a true statement 6 = 6. Thus y = 3 is the solution of the given equation. Example 3: Solve the equation Multiplying both sides of the equation by 12 (Rule 3), we get

3. Solving an Equation (Cont…) We can check the solution y = 576 by substituting it back into the original equation. Example 4: Solve the equation 15 x = 45 We divide both sides of the equation by 15 (Rule IV). This gives Thus, x = 3 is the required solution. NOTE: In the above examples, we used only one rule at a time to help you understand the different properties that we use to solve equations. However, we often have to use more than one rule to solve a linear equation.  Following steps can be used to solve linear equations that involve more than one rule. STEP 1: Simplify each side, if needed. This would involve removing brackets using distributive property. STEP 2: Collect the variable terms on one side and constant terms on the other side. STEP 3: Isolate the variable and solve. STEP 4: Check the proposed solution in the original equation. 

3. Solving an Equation (Cont…) Example: Solve the equation 2(x - 1) + x = 5(2x + 3) – 2 (x + 3) for x Step 1: Simplify if needed. 2(x - 1) + x = 5(2x + 3) – 2 (x + 3) (This is the given equation) 2x - 2 + x = 10x + 15 – 2x – 6 (Removing brackets by using distributive property) 3x - 2 = 8x + 9 (Combining like terms) Step 2: Collect the variable terms on one side and constant terms on the other side. We collect the variable terms on the left by using rule II, that is, subtracting 8x from both sides of the equation. Then we collect the constant terms on the right. 3x - 2 – 8x = 8x + 9 – 8x (Rule II: Subtracting 8x from both sides) -5x - 2 = 9 (Simplifying) -5x - 2 + 2 = 9 + 2 (Rule I: Adding 2 to both sides) -5x = 11 (Simplifying) Step 3: Isolate the variable and solve. We isolate the variable x by dividing both sides of the equation by –5. Step 4: Check the proposed solution by substituting it in the original equation.

4. Solving Linear Equations involving Fractions Some linear equations may involve fractions for example As fractions are another way to write division, and the inverse of divide is to multiply, we remove fractions by multiplying both sides by the LCD of all the fractions. Recall: Least Common Denominator(LCD) The least common denominator is a polynomial consisting of the product of all prime factors in the denominators, with each factor raised to the greatest power of its occurrence in any denominator. Multiplying every term on both sides of the equation by the least common denominator will eliminate the fractions in the equations. Now, let us solve the above equation by removing fractions: Step 1: Simplify if needed. In this step we simplify both sides of the equation by removing brackets using distributive property. We remove fractions by multiplying both sides of the equation by the LCD of all the fractions.

4. Solving Linear Equations involving Fractions (Cont…) Step 2: Collect the variable terms on one side and constant terms on the other side. We collect the variable terms on the left by using rule II, that is, subtracting 3x from both sides of the equation. Then, we collect the constant terms on the right. 10x + 150 –3x = 3x + 108 – 3x (Rule II: Subtracting 3x from both sides) 7x + 150 = 108 (Combining like terms) 7x + 150 + 150 = 108 - 150 (Rule I: Adding 150 to both sides) 7x = -42 (Simplifying) Step 3: Isolate the variable and solve. We isolate the variable x by dividing both sides of the equation by 7. Step 4: Check the proposed solution by substituting it in the original equation.

5. Solving Linear Equations Involving Rational Expressions We have already solved a linear equation with constants in the denominator. Now we will solve a linear equation involving variables in the denominator. The procedure for solving this type of equation is the same as solving linear equation involving fractions but we must avoid any values of the variable that make the denominator zero. Example: The denominator in this equation will equal zero if x = 2. Therefore, we put a restriction x  2. Now, let us solve this equation keeping this in mind. Step 1: Simplify if needed. The denominators are (x – 2) and 3, so the LCD is 3(x - 2). We multiply each term of the equation by 3(x - 2). We will also put a restriction x  2.

5. Solving Linear Equations Involving Rational Expressions (Cont…) Step 2: Collect the variable terms on one side and constant terms on the other side. 3x + 2x = 10 - 2x + 2x (Rule I: Adding 2x to both sides) 5x = 10 (Combine like terms) Step 3: Isolate the variable and solve. The proposed solution 2, is not a solution because of the restriction x  2. Thus, the given equation has no solution. The solution set for this equation is an empty set .

6. Types of Equations Equations can be placed into different categories depending upon their solution sets. We have the following types of equations: Identity: An equation is classified as an identity when it is true for all real numbers for which both sides of the equation are defined. Example: 7x + 14 = 7(x + 2). We will solve this equation for x. 7x + 14 = 7x + 14 (Using distributive property) 7x + 14 – 7x = 7x + 14 – 7x (Collecting all variable terms on one side) 14 = 14 Observe that the variable x disappears and we end with a true statement, 14 is indeed equal to 14. NOTE: This does not mean that x = 14.  Whenever the variable gets cancelled and results in a true statement, it means that the solution to the equations is a set of all real numbers. That is, if we substitute any real number for x in this equation, the left side of the equation will equal the right side of the equation. So the answer is all real numbers, which means this equation is an identity. Conditional equation: A conditional equation is an equation that is not an identity, but is true for at least one real number. A statement such as 2x - 1 = 0 is an equality only when x has one particular value. Such a statement is called a conditional equation, as it is true only under the condition that x = 1/2. Likewise, the equation y - 7 = 8 holds true only if y = 15.

6. Types of Equations (Cont…) The solution of a conditional equation can be verified by substituting the variable with its value. Example: Solve x + 1 = 5 for x. x + 1 – 1 = 5 – 1 (Subtracting 1 from both sides) x = 4 On substituting 4 for x in the equation, we get a true statement. That means 4 is the solution of the given equation. The given equation is not an identity and is true only if x = 4. Thus, this equation is an example of conditional identity. Inconsistent equation: An inconsistent equation is an equation with one variable that has no solution. NOTE: This type of equation is not true for even one real number. Example: 5x – 2 = 5(x + 1) 5x – 2 = 5x + 5 (Using distributive property) 5x – 5x – 2 = 5x – 5x + 5 (Collecting all variable terms on one side) -2 = 5 (False Statement!) In case of a inconsistent equation, the variable disappears and we end up with a false statement. Thus, an inconsistent equation has no solution.

7. Summary Let us recall what we have learnt so far: Equation: An equation is a statement of equality that involves variables. Linear Equation: A linear equation is an equation in which the highest power of the variables is one. Linear Equation in one variable: A linear equation in one variable x is an equation that can be written in the form ax + b = 0 where a and b are real numbers, and a 0. Solution of a Linear Equation: The value of the variable for which left side of the equation is equal to the right side, is called the solution or root of the equation. If an equation contains fractions, we begin by multiplying both sides by the least common denominator, thereby clearing fractions. If an expression contains rational expressions with variable denominators, avoid any values in the solution set that make the denominator zero. Identity: An equation is classified as an identity when it is true for all real numbers for which both sides of the equation are defined. Conditional equation: A conditional equation is an equation that is not an identity, but is true for at least one real number. Inconsistent equation: An inconsistent equation is an equation with one variable that has no solution.