INTRODUCING LINEAR EQUATIONS IN ONE UNKNOWN

Transcription

1 LESSON 1: INTRODUCING LINEAR EQUATIONS IN ONE UNKNOWN A linear equations in one unknown consists of two expressions that are linked by an = sign. An equation states that two expressions represent the same quantity. Here are some examples of linear equations in one unknown: x + 8 = 45 2(x 5) = = x + 3 x + (3x 2) = 21 (3x + 5) (2x +3) = x = x + 17 Here are some examples that are not linear equations: x(2x 6) = 56 x = 84 16x 12 (x + 2)(4x 1) = = 100 x 3 x 4 An equation is a linear equation in one unknown if it can be written in this form: 0 = kx + b In kx+b = 0 the letters k and b are fixed numbers that are usually given. The letter x stands for an unknown. In a linear equation expressed in terms of x, the power (or exponent) of x is 1. When someone solves a linear equation in one unknown, they find the value of x that makes the right-hand side equal to the left-hand side. In other words, the value(s) of x that makes the equation true, or satisfies it, is the solution. SOLVING LINEAR EQUATIONS IN ONE UNKNOWN PAGE 1

2 LESSON 1 TASKS: WHAT ARE LINEAR EQUATIONS IN ONE UNKNOWN AND WHAT DOES IT MEAN TO SOLVE THEM? 1. Decide if each of the following is or is not a linear equation in one unknown and then complete the sentence to give your reason. The first one has been done for you. 3 + x 2 = 9 10 = x 7 4x 11 3(x + 2) = 12 3x 4 = 7 + 4x 5 10 = 2 25 it contains an x 2 term. 2. Which of the following equations have 3 as their solution? (a) 3x + 6 = 12 (b) 7x 14 = 7 (c) 5(x 2) = 8 x (d) 6(x+2) = Choose the solutions to the following equations from among -1, 0 and 1: (a) 5x + 2 = 10x 3 (b) 12 4x = x 4. Express each of the following relations in an equation: (a) 2 times a number x plus 4 is equal to 2 times the sum of x plus 2. (b) Half of a number x plus 3 is equal to x minus 2. (c) Bus A has twice as many passengers as Bus B. The total passengers in both buses is Express in words the relations in the following equations: (a) 2x + 10 = 90 (b) x + (x + 1) = 41 (c) 7x = 6x + 4 SOLVING LINEAR EQUATIONS IN ONE UNKNOWN PAGE 2

3 LESSON 2: Using mental math to solve linear equation in two unknowns When you solve an equation you find the value of the unknown that makes the equation true. Some linear equations in one unknown can be solved just by looking at them. For example: 4x = 16 2x = 6 x + 15 = = 5 + x Others are more complicated. For example: 6(x + 2) = x 7-2 = y x = -9 In this lesson you will focus on using mental math to solve equations that are not complicated. SOLVING LINEAR EQUATIONS IN ONE UNKNOWN PAGE 3

4 LESSON 2 TASKS: USING MENTAL MATH TO SOLVE LINEAR EQUATIONS IN ONE UNKNOWN 1. (a) 3x = 9 (b) 12x = 24 (c) 6 = 2x 2. x x x (a) = 3 (b) = 12 (c) = (a) x + 9 = 15 (b) 27 x = 12 (c) 3 + x = (a) 21 x = 7 (b) x + 16 = 25 (c) x + x = (a) 3x + 5 = 20 (b) 5x + 15 = 20 (c) 4x + 4 = (a) 22 x = 16 (b) 2x + 6 = 18 (c) 14 = 7 + x 7. (a) 3x + 1 = 10 (b) 24 + x = 29 (c) 2x = (a) 23 = 14 + x (b) 4x 4 = 16 (c) 10 = 2x 2 SOLVING LINEAR EQUATIONS IN ONE UNKNOWN PAGE 4

5 LESSON 3: EQUIVALENT EQUATIONS Equivalent equations are equations that have the same solution. Imagine starting with this equation 2x = 8 for short we will call it equation (1). Show that its solution is 4. Now imagine adding 3 to both sides of equation (1): 2x + 3 = or, 2x + 3 = 11 (2) Now, verify that solution to equation (2) is also 4. Now imagine taking equation (2) and multiplying each term by 4: 4 2x = 4 11 or 8x + 12 = 44 (3) Again, verify that the solution to equation (3) is still 4. Equations (1), (2), and (3) are some examples of equivalent equations You can generate equivalent equations by doing the following: add or subtract a number or term to each side of the equation multiply or divide each term by a non-zero number apply properties such as the distributive property Now start with an equation that is too complicated to solve using mental math alone: 5x 2(x 1) = 14 (1) Use the distributive property to generate an equivalent equation: 5x 2x + 2 = 14 (2) Generate another equivalent equation by subtracting the monomial 2x from the monomial 5x: 5x 2x + 2 = 14 (2) 3x + 2 = 14 (3) If you can see the solution to equation (3) you can stop generating equivalent equations and just write the solution. But if you cannot see the solution you can generate another equivalent equation by subtracting the number 2 from both sides of the equation: 3x = 14 2 to give 3x = 12 (4) Now, use mental math to solve equation 4. Or continue on your own to generate another equivalent equation that will allow you to see the solution No matter how complex a linear equation in one unknown may seem, it is important to realize that it can be solved by writing as many equivalent equations as needed to reveal the solution. SOLVING LINEAR EQUATIONS IN ONE UNKNOWN PAGE 5

6 LESSON 3 TASKS: USING EQUIVALENT EQUATIONS TO SOLVE EQUATIONS Solve each of the following equations by generating equivalent equations: 1. (a) 3x + 4 = 25 (b) 7x + 4 = (a) 28 = 2x 14 (b) 100 = 4x (a) 2(x + 4) = 12 (b) 3(4 + x) = (a) 52 = x (b) x = (a) 3x + x = 2 + 3x (b) 6x = 4(3 + x) SOLVING LINEAR EQUATIONS IN ONE UNKNOWN PAGE 6

7 LESSON 4 TASKS USING EQUIVALENT EQUATIONS TO SOLVE EQUATIONS Solve each of the following equations by generating equivalent equations: 1. (a) 5x 16 = 14 (b) -20 = x 7 2. (a) 26 = 5x 14 x (b) 80 5x = 5 3. (a) 2x = 10 3x (b) 4x = 45 5x (c) 42 x = 45 2x (d) 4x + 1 = 7 7x 4. (a) 16(x + 2) = 64 (b) 15 = x + (7x 1) 5. (a) 6x (7x + 12) = -7x + 60 (b) 10x (x + 14) = 56 x (c) x = 2(8 x) (d) 3(2x + 1) 3x = 3(13 x) (e) 8(x 1) = -2(14 2x) (f) 3(2x 1) (4 x) = 14 SOLVING LINEAR EQUATIONS IN ONE UNKNOWN PAGE 7

8 LESSON 5: HOW MANY SOLUTIONS? How many values of x will satisfy this equation? 3x + 9 = 27 Now, how many values of x will satisfy this equation? 3(x + 2) = 3x + 6 How many values of x will satisfy this equation? x = x + 6 An equation that has an infinite number of solutions is called an identity. Give one example of an identity. SOLVING LINEAR EQUATIONS IN ONE UNKNOWN PAGE 8

9 LESSON 5 TASKS: FINDING THE NUMBER OF SOLUTIONS 1. Identify the number of solutions for each of these equations: Equation Number of solutions 3x + 10 = 2 2x + 4 = 2(x 5) x 14 = (50 + 7x) 4 12x = 6(2x 8) 9x + 4 = (2 + 3x) 2. Give an example of a linear equation in one unknown that has exactly one solution: 3. Give an example of a linear equation in one unknown that has no solutions: 4. Give an example of a linear equation in one unknown that has an infinite number of solutions: SOLVING LINEAR EQUATIONS IN ONE UNKNOWN PAGE 9

10 LESSON 8 TASKS: WRAPPING UP! Solve each of the following equations: 1. (a) 8x = 64 (b) 3x + 6 = (a) 17 x = 7 (b) 9x + 13 = (a) 2(x + 9) = 30 (b) 2(x + 12) = (a) 2(x + 6) = 4 + (12 + x) (b) 17 x = (a) 4(x + 10) = 3(15 + x) (b) 42 = 4x x 6. (a) 6x 7 = 2x + 17 (b) 10x 12 = 2(25 x) 2 SOLVING LINEAR EQUATIONS IN ONE UNKNOWN PAGE 10