Inverse Operations. What is an equation?

Transcription

1 Inverse Operations What is an equation? An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent). Equations are written with an equal sign, as in 2+=5 9 2=7 1

2 Equations can also be used to state the equality of two expressions containing one or more variables. 4x + 1 = 14 1 If x =, then 4() + 1 = = 1 1 = 1 An equation can be compared to a balanced scale. Both sides need to contain the same quantity in order for it to be "balanced". 2

3 For example, = 50 represents an equation because both sides simplify to 50. We refer to this type of equation as a numerical equation. It consists of numbers and there are no variables = = 50 Any of the numerical values in the equation can be represented by a variable. Examples: 20 + u = 50 x + 0 = = y The equations seen above are called algebraic equations. This is because it contains at least one variable. When solving equations, the goal is to isolate the variable on one side of the equation in order to determine its value (the value that makes the equation true). In order to solve an equation containing a variable, you need to use inverse (opposite/undoing) operation(s) on both sides of the equation. Let's review the inverses of each operation: Addition Multiplication Squaring Subtraction Division Square Root Cubing Cube Root

4 There are four properties of equality that we will use to solve equations. They are as follows: Addition Property If a=b, then a+c=b+c for all real numbers a, b, and c. The same number can be added to each side of the equation without changing the solution of the equation. Subtraction Property If a=b, then a c=b c for all real numbers a, b, and c. The same number can be subtracted from each side of the equation without changing the solution of the equation. There are four properties of equality that we will use to solve equations. They are as follows: Multiplication Property If a=b, then ac=bc for all real numbers a, b, and c. Each side of an equation can be multiplied by the same number without changing the solution of the equation. Division Property If a=b, and c=0, then a c = b c for all real numbers a, b, and c. Each side of an equation can be divided by the same nonzero number without changing the solution of the equation. 4

5 To solve for "x" in the following equation... x + 7 = 2 Determine what operation is being shown (in this case, it is addition). Do the inverse to both sides = 0 x + 7 = x = 25 In the original equation, replace x with 25 and see if it makes the equation true. x + 7 = = 2 2 = 2 Since 25 makes the equation true, 25 is known as a solution. A solution is a number that can replace the variable to make the equation a true statement. For each equation, write the inverse operation needed to solve for the variable. a.) y +7 = 14 subtract 7 b.) a 21 = 10 add 21 c.) 5s = 25 divide by 5 d.) x = 5 multiply by = 1 12 e. 4 x = 5 5

6 Think about this... To solve c = 12 Which method is better? Why? Kendra Added to each side of the equation c = c = 15 Ted Subtracted 12 from each side, then added 15. c = c 15 = c = 15 Think about this... In the expression To which does the " " belong? Does it belong to the x? The 5? Both? The answer is that there is one negative so it is used once with either the variable or the 5. Generally, we assign it to the 5 to avoid creating a negative variable. So: 6

7 One Step Equations The purpose of solving an equation is to find a solution. A solution is a value that can replace the variable in order to make the equation true. To solve equations, you must work backwards through the order of operations to find the value of the variable. Remember to use inverse operations in order to isolate the variable on one side of the equation. Whatever you do to one side of an equation, you MUST do to the other side! 7

8 Examples: y + 9 = The inverse of adding 9 is subtracting 9 y = 25 6m = The inverse of multiplying by 6 is dividing by 6 m = 12 Remember whatever you do to one side of an equation, you MUST do to the other so that the equation stays in balance. x 8 = x = 6 One Step Equations Solve each equation then below to see work & solution. 2 = x = x x + 2 = x = 16 7 = x + 10 = x x + 5 = 5 5 x = 2 15 = x = x 8

9 One Step Equations x = 15 x = 5 x (2) = 10 (2) 2 x = 20 4x = x = 25 = 5x = x ( 6) x = 6 ( 6) 6 x = Example: 5 5 x = 6 x = 6 x = x = Multiply both sides by the reciprocal Click to reveal steps 9

10 Try 2 x = 1 Try 4 x =

11 Try 8 x = 9 1 Solve. x 6 = 11 Answer 11

12 2 Solve. j + 15 = 17 Answer Solve. 115 = 5x Answer 12

13 4 Solve. 21 = x 7 5 Solve. 2 t = 11 1

14 6 Solve. 8 x = 5 8 Justify that x = 2.5 is the solution to the following equation. Show all the work! 4x = 10 14

15 Translating and Solving One Step Equations Return to Table of Contents Sometimes you need to translate a situation or a word problem and then use your skills to solve the equation. 15

16 Key words: Addition: Subtraction: Multiplication: Division: 16

17 Try translating, then solving these. boxes to reveal translation The sum of 5 and a number is x = 8 x = 78 6 is x less than = 20 x x = 14 One half of x is the same as 4. A number divided by is 16. x = 4 x = 68 = 16 x = Translate and solve. The difference of a number and 7 is 49. Answer x 7 = 49 x = 56 [This object is a pull tab] 17

18 78 Translate and solve. A number divided by 5 is forty. Answer x/5 = 40 x = 200 [This object is a pull tab] 98 Translate and solve. One third of a number is eleven. Answer (1/)x = x = 11 [This object is a pull tab] 18