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### Grade 6 - Mathematics12.19 Solving Equations

 Simplifying with Addition and Subtraction: We can use addition and subtraction to get all the terms with variables on one side of an equation, and all the numeric terms on the other. The equations 3x = 17, 21 = y, and z/12 = 24 each have a variable term on one side of the '=' sign, and a number on the other. The equations x + 3 = 12, 21 = 30 - y, and (z + 2) × 4 = 10 do not. We usually do this after simplifying each side using the distributive rules, eliminating parentheses, and combining like terms. Since addition is associative, it can be helpful to add a negative number to each side instead of subtracting. Examples: For the equation 3x + 4 = 12, we can separate the variable term on the left by subtracting a 4 from both sides: 3x + 4 - 4 = 12 - 4 3x = 8. x = 8/3 For the equation 7y - 200 = 10, if we add 200 to both sides of the equation, the 200 and -200 will cancel each other on the left hand side: 7y + (-200) + 200 = 10 + 200 7y = 210 y = 30 Example: For the equation 8 = 20 - z, we can add z to both sides to get 8 + z = 20 - z + z => 8 + z = 20. Now subtracting 8 from both sides, 8 + z - 8 = 20 - 8 Therefore, z = 12 Simplifying by Multiplication When solving for a variable, we want to get a solution like x = 3 or z = 198 etc. When a variable is divided by some number, we can use multiplication on both sides to solve for the variable. Example: Solve for x in the equation x ÷ 12 = 5. Since the x on the left side is being divided by 12, the equation is the same as x × 1/12 = 5. Multiplying both sides by 12 will cancel the 1/12 on the left side: x × 1/12 × 12 = 5 × 12 ==> x × 1 = 60 ==> x = 60. Simplifying by Division When solving for a variable, we want to get a solution like x = 9 or y = 21 etc. When a variable is multiplied by some number, we can use division on both sides to solve for the variable. Example: Solve for x in the equation 7x = 280. Since the x on the left side is being multiplied by 7, we can divide both sides by 7 to solve for x: 7x ÷ 7 = 280 ÷ 7 (7x)/7 = 280 ÷ 7 x/1 = 40 x = 40. Note: Dividing by 7 is the same as multiplying both sides by 1/7. Directions: Solve the following equations and enter the solution. Also write at least 10 examples of your own.
 Q 1: Solve: 5x =45Answer: Q 2: Solve: d - 12 =69Answer: Q 3: Solve: x/7 =4Answer: Q 4: Solve: 6k = 24Answer: Q 5: Solve: x + 8 = 17Answer: Q 6: Solve: n/7 = 8Answer: Q 7: Solve: u + 16 = 18Answer: Q 8: Solve: f + 16 =100Answer: Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!

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