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9.1 Solve Equations We are going to learn four different methods for solving equations. METHOD 1: Solving One-Step Equations by Inspection We will use inspection to solve for the variable (unknown). a) If we are given x + 3 = What number added to 3 is equal to 9? That number is ______ So we can say x = _______ b) If we are given x - 2 = 7 What number, less 2, will be equal to 7? That number is ______ c) If we are given 4x = -20 So we can say x = _______ ***REMEMBER 4x is “4 times x” What number multiplied by 4 equals -20? That number is ______ So we can say x = _______ d) If we are given x ÷ 3 = 5 What number divided by 3 gives 5? That number is ______ So we can say x = _______ METHOD 2: Solve Using Systematic Trial What number multiplied by 7, less 2, equals 166? Value (guess) 20 30 25 24 Equation 7x – 2 = 166 True or False Therefore, the solution is x = _________ Now let’s try the cover-up method. This method is a more efficient way to help you solve by inspection. METHOD 3: Cover-Up Method Solve: 2x + 4 = 10 Step 1: Cover up the 2x. 2x + 4 = 10 What value for 2x makes this true? + 4 = 10 6 makes the equation true. 6 + 4 = 10 2 times a number gives 6. What is the number? This means that 2x must be equal to 6. Step 2: Now cover up the x only 3 makes the equation true. Therefore, x = 3 is the solution. 2x = 6 2 =6 METHOD 4: Solve Using Models Refer to Example 4 on Page 376. Work through examples a, b, and c Homework: Solving Equations Worksheet Page 379 # 1, 2, 3, 4, 5, 6, 7, 9 Page 380 # 10, 11, 12, 13, 14, 15, 16 PASS IN #17 Solving Equations Worksheet Answer the following questions in your binders or scribblers. Solve the following by Inspection or the Cover-Up Method: x = 25 2 1) a + 4 = 12 2) x – 3 = 10 3) – 2 = x + 1 4) 5) 60 = -6z 6) x – 6 = 10 7) x – 2 = - 2 8) 2x = 80 9) -22 = 11y 10) x + 16 = 2 11) X + 2 = 12 12) 14) -99 + x = 1 15) x – 21 = 36 16) 4x = 48 13) 6 = y 6 Solve the following by Modeling: 1) 2) 3) 4) 5) 6) 7) 8) 9) y = 16 2 Solving One-Step Equations Use the inverse operation (opposite operation) that is displayed to determine what value the variable has. DON”T FORGET: Any operation performed to one side must be performed to the other side of the equal sign. REMEMBER: ZERO PRINCIPAL: -8+8=0 +3 + (-3) = 0 Example 1: x+6=8 x+6– 6=8– 6 x=2 The variable “x” is not alone on one side of the equal sign, 6 is being added to it. Perform the inverse operation to both sides, which means subtract 6 from both sides. Example 2: x-8=4 The variable “x” is not alone on one side of x - 8 + 8 = 4 + 8 the equal sign, x = 12 8 is being subtracted from it. Perform the inverse operation to both sides, which means add 8 from both sides. Example 3: 5x = -35 5x = -35 5 5 The variable “x” is not alone on one side of the equal sign, It is being multiplied by 5. Perform the inverse operation to both sides to isolate the variable. Divide both sides of the equation by 5. Example 4: The variable “x” is not alone on one side of the equal sign, It is being divided by 3. Perform the inverse operation to both sides to isolate the variable. Multiply both sides of the equation by 3. Solving Equations Worksheet Part A: Solve the following addition equations. SHOW YOUR WORK x + 8 = 13 x + 5 = 15 21 + y = 24 4 + y = 12 12 + y = 16 x + 9 = 15 Part B: Solve the following subtraction equations. SHOW YOUR WORK y - 15 = 25 y – 19 = 15 1-x=4 4 - x = 12 2-y=6 x-3=5 Part C: Solve the following division equations. SHOW YOUR WORK x = 21 7 x = 15 3 y = 10 4 y = 24 6 x = 25 8 y = 72 6 Part D: Solve the following multiplication equations. SHOW YOUR WORK 6x = 42 8y = 72 -2x = 8 -33x = 66 24x = 72 -x = 26 Solving Two-Step Equations Rules: 1. To solve a two-step equation, first undo the addition and subtraction by performing the opposite or inverse operation. 2. Then, undo the multiplication and division by performing the opposite or inverse operation. Example: 4x + 5 = 17 4x + 5 – 5 = 17 – 5 4x = 12 4x = 12 4 4 x=3 Check: 3. To check, substitute or replace the variable with your solution (answer) to see if it makes the equation true. 4x + 5 = 17 4(3) + 5 = 17 12 + 5 = 17 17 = 17 Show all your work in your scribbler or binder: Be sure to SHOW YOUR WORK and SHOW YOUR CHECK!!! 1) 2x + 1 = 7 2) 3n + 1 = 13 3) 4t + 5 = 21 4) 2y + 7 = 11 5) 3p + 2 = 22 6) 4m + 7 = 27 7) 2t + 5 = 15 8) 7c + 2 = 16 9) 2x – 1 = 9 10) 5x – 2 = 13 11) 2x – 5 = 11 12) 4x – 7 = 17 13) 5x – 6 = 24 14) x + 19 = 28 2 18) 2x + 5 = 19 15) 2x – 4 = 11 + 1 16) 6x – 5 = 23 - 4 19) 6p + 14 = 68 20) 12x – 9 = 75 22) 3x – 11 = 10 23) b - 12 = 8 9 27) 5a + 3 = 18 24) 2x + 1 = 7 17) 3y + 6 = 27 21) x + 5 = 10 7 25) a - 2 = 3 4 26) 36 – x = 16 28) 5y – 2.6 = 7.4