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**Table of Contents**Inverse Operations Click on a topic to go to that section. One Step Equations Two Step Equations Multi-Step Equations Variables on Both Sides More Equations Transforming Formulas**Inverse Operations**Return to Table of Contents**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 + 3 = 5 9 – 2 = 7**Equations can also be used to state the equality of two**expressions containing one or more variables. In real numbers we can say, for example, that for any given value of x it is true that 4x + 1 = 14 - 1 If x = 3, then 4(3) + 1 = 14 - 1 12 + 1 = 13 13 = 13**When defining your variables, remember...**Letters from the beginning of the alphabet like a, b, c... often denote constants in the context of the discussion at hand. While letters from end of the alphabet, like x, y, z..., are usually reserved for the variables, a convention initiated by Descartes. Try It! Write an equation with a variable and have a classmate identify the variable and its value.**An equation can be compared to a balanced scale.**Both sides need to contain the same quantity in order for it to be "balanced".**For example, 20 + 30 = 50 represents an equation because**both sides simplify to 50. 20 + 30 = 50 50 = 50 Any of the numerical values in the equation can be represented by a variable. Examples: 20 + c = 50 x + 30 = 50 20 + 30 = y**Why are we Solving Equations?**First we evaluated expressions where we were given the value of the variable and had to find what the expression simplified to. Now, we are told what it simplifies to and we need to find the value of the 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) operations on both sides of the equation. Let's review the inverses of each operation: Addition Subtraction Multiplication Division**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. Multiplication Property If a=b, and c=0, then ac=bc for all real numbers ab, b, and c. Each side of an equation can be multiplied by the same nonzero 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 ab, b, and c. Each side of an equation can be divided by the same nonzero number without changing the solution of the equation.**To solve for "x" in the following equation...**x + 7 = 32 Determine what operation is being shown (in this case, it is addition). Do the inverse to both sides. x + 7 = 32 - 7-7 x = 25 In the original equation, replace x with 25 and see if it makes the equation true. x + 7 = 32 25 + 7 = 32 32 = 32**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 12 12 move move move move**Think about this...**To solve c - 3 = 12 Which method is better? Why? Kendra Added 3 to each side of the equation c - 3 = 12 +3 +3 c = 15 Ted Subtracted 12 from each side, then added 15. c - 3 = 12 -12 -12 c - 15 = 0 +15 +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: ×**What is the inverse operation needed to solve this equation?**7x = 49 1 Addition A Subtraction B Multiplication C D Division**2**What is the inverse operation needed to solve this equation? x - 3 = -12 Addition A Subtraction B Multiplication C D Division**One Step Equations**Return to Table of Contents**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!**Examples:**y + 9 = 16 - 9 -9 The inverse of adding 9 is subtracting 9 y = 7 6m = 72 6 6 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!!! ×**One Step Equations**Solve each equation then click the box to see work & solution. click to show inverse operation click to show inverse operation x - 8 = -2 +8 +8 x = 6 2 = x - 6 +6 +6 8 = x click to show inverse operation click to show inverse operation x + 2 = -14 -2 -2 x = -16 7 = x + 3 -3 -3 4 = x click to show inverse operation click to show inverse operation 15 = x + 17 -17 -17 -2 = x x + 5 = 3 -5 -5 x = -2**One Step Equations**3x = 15 3 3 x = 5 -4x = -12 -4 -4 x = 3 -25 = 5x 5 5 -5 = x click to show inverse operation click to show inverse operation x (2) = 10 (2) 2 x = 20 click to show inverse operation x (-6) click to show inverse operation = 36 (-6) -6 x = -216 click to show inverse operation**Solve.**x - 6 = -11 3**4**Solve. j + 15 = -17**5**Solve. -115 = -5x**6**Solve. = 12 x 9**7**Solve. 51 = 17y**8**Solve. w - 17 = 37**9**Solve. -3 = x 7**10**Solve. 23 + t = 11**11**Solve. 108 = 12r**Two-Step Equations**Return to Table of Contents**Sometimes it takes more than one step to solve an equation.**Remember that to solve equations, you must work backwards through the order of operations to find the value of the variable. This means that you undo in the opposite order (PEMDAS): 1st: Addition & Subtraction 2nd: Multiplication & Division 3rd: Exponents 4th: Parentheses Whatever you do to one side of an equation, you MUST do to the other side!**Examples:**3x + 4 = 10 - 4 - 4 Undo addition first 3x = 6 3 3 Undo multiplication second x = 2 -4y - 11 = -23 + 11 +11 Undo subtraction first -4y = -12 -4___-4 Undo multiplication second y = 3 Remember - whatever you do to one side of an equation, you MUST do to the other!!! ×**Two Step Equations**Solve each equation then click the box to see work & solution. 3x + 10 = 46 - 10 -10 3x = 36 3 3 x = 12 -4x - 3 = 25 +3 +3 -4x = 28 -4 -4 x = -7 6-7x = 83 -6 -6 -7x = 77 -7 -7 x = -11 -2x + 3 = -1 - 3 -3 -2x = -4 -2 -2 x = 2 9 + 2x = 23 -9 -9 2x = 14 2 2 x = 7 8 - 2x = -8 -8 -8 -2x = -16 -2 -2 x = 8**Walter is a waiter at the Towne Diner. He earns a daily wage**of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?**12**Solve the equation. 5x - 6 = -56**13**Solve the equation. 16 = 3m - 8**Solve the equation.**x 2 14 - 6 = 30**15**Solve the equation. 5r - 2 = -12**16**Solve the equation. 12 = -2n - 4**17**Solve the equation. - 7 = 13 x 4**18**Solve the equation. + 3 = -12 x 5 -**19**What is the value of n in the equation 0.6(n + 10) = 3.6? -0.4 A 5 B -4 C D 4**20**In the equation , n is equal to 8 A 2 B 1/2 C D 1/8**Which value of x is the solution of the equation**21 1/2 A 2 B 2/3 C D 3/2**22**Two angles are complementary. One angle has a measure that is five times the measure of the other angle. What is the measure, in degrees, of the larger angle?**Multi-Step Equations**Return to Table of Contents**Steps for Solving Multiple Step Equations**As equations become more complex, you should: 1. Simplify each side of the equation. (Combining like terms and the distributive property) 2. Use inverse operations to solve the equation. Remember, whatever you do to one side of an equation, you MUST do to the other side!**Examples:**-15 = -2x - 9 + 4x -15 = 2x - 9 Combine Like Terms +9 +9 Undo Subtraction first -6 = 2x 2 2 Undo Multiplication second -3 = x 7x - 3x - 8 = 24 4x - 8 = 24 Combine Like Terms + 8 +8 Undo Subtraction first 4x = 32 4___4 Undo Multiplication second x = 8 ×**Now try an example. Each term is infinitely cloned**so you can pull them down as you solve. -7x -7x -7x -7x -7x -7x -7x -7x -7x -7x -7x -7x -7x -7x -7x -7x -7x -7x -7x -7x -7x + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 6x + 6x + 6x + 6x + 6x + 6x + 6x + 6x + 6x + 6x + 6x + 6x + 6x + 6x + 6x + 6x + 6x + 6x + 6x + 6x + 6x = = = = = = = = = = = = = = = = = = = = = -6 -6 -6 -6 -6 -6 -6 -6 -6 -6 -6 -6 -6 -6 -6 -6 -6 -6 -6 -6 -6 x = 9