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## Solving Linear Equations

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**Bellwork**Solve for x: • x – 5 = 57 • 3 – x = 10 • 4x + 28 = 68 • -3x – 7 = 14 • 4x – 9 + x = 16**Problem No. 1**1. Mica bought a CD for P25 and 8 blank videotapes. The total cost was P265. Find the cost of each blank videotape.**Equation: 25 + 8x = 265**Solution: 25 + 8x = 265 8x = 265 – 25 8x = 240 8 8 Answer: P30 each videotape**Problem No. 2**2. Juan bought 5 orchids in pots and P200 rose plant at a fund raiser. He spent a total of P750. Find the cost of each orchid.**Equation: 5x + 200 = 750**Solution: 5x + 200 = 750 5x = 750 – 200 5x = 550 5 5 x = 110 Answer: The price of each orchid is P110.**Error Analysis**10x + 4 = -2 10x = 2 10x = 2 10 10 x = 2 or 1 10 5 Correct Solution: 10x + 4 = -2 10x = -2 – 4 10x = -6 10x = -6 10 10 x = -3/5**Topic: Solving One Step Equations**Addition Property of Equality – A property that states that we must add the same number on both sides of the equation to make it equal or balance. Example: x – 4 = 7 x – 4 + 4 = 7 + 4 x = 11**Subtraction Property of Equality – A property that states**that we must subtract the same number on both sides of the equation to make it equal or balance. x + 2 = 8 x + 2 – 2 = 8 – 2 x = 6**Multiplication Property of Equality – A property that**states that we must multiply the same number on both sides of the equation to make it equal or balance. ½ x = 8 2 (1/2 x) = 8 (2) x = 16**Division Property of Equality – A property that states**that we must divide the same number on both sides of the equation to make it equal or balance. 6x = 36 6x = 36 6 6 x = 6**Topic: Solving Two-Step Equations**In order to solve two-step linear equations you need to use properties of equality. Example No. 1 2x – 4 = 10 Given 2x = 10 + 4 Isolate term with x 2x = 14 Divide both sides by 2 2 2 x = 7**Example No. 2**3x + 8 = 14 Given 3x = 14 – 8 Isolate the term with x 3x = 6 Divide both sides by 3 3 3 x = 2 Checking: Substitute x = 2 in the equation 3x + 8 = 14 3(2) + 8 = 14 6 + 8 = 14 Therefore it is correct!**Topic: Solving Multi-Step Equations**In order to solve multi-step equations you need to use your knowledge about distributive property and combining like terms. Example No. 1 2x + 3x – 5 = 35 Given 5x – 5 = 35 Add like terms 5x = 35 + 5 Add 5 on both sides 5x = 40 Divide both sides by 5 5 5 x = 8**Example No. 2**3 ( x – 4 ) = 24 Given 3x – 12 = 24 Use Distributive Property 3x = 24 + 12 Add 12 on both sides 3x = 36 Divide both sides by 3 3 3 Checking: Substitute x = 12 in the given eq. 3 ( x – 4 ) = 24 3 (12 – 4 ) = 24 3 (8) = 24 Therefore it is correct! x = 12**Topic: Solving Equations with Variables on Both Sides.**To solve an equation with variables on both sides, you need to put all the variable terms on one side. Example No. 1 9x + 2 = 4x – 18 Given 9x – 4x = -18 – 2 Isolate variable terms 5x = -20 Divide both sides by 5 5 5 x = -4**Example No. 2**5x – 8 = - 2x + 6 Given 5x + 2x = 6 + 8 Isolate variable terms 7x = 14 Divide both sides by 7 7 7 Checking: Substitute x = 2 in the given eq. 5x – 8 = -2x + 6 5(2) – 8 = -2(2) + 6 10 – 8 = -4 + 6 2 = 2 Therefore it is correct! x = 2**Board Drill**4x – 10 + 6x = 100**Board Drill**7 ( x – 5 ) = 21**Board Drill**3x + x – 2 = 3**Board Drill**-6x + 3x – 9 = 18**Board Drill**9x + 10 = 2x + 31**Board Drill**-4x + 7 = 6x - 3**Board Drill**6x - 9 = x + 36**Board Drill**7x + 9 = 3x + 25**Board Drill**5x + 8 = 7x**Topic: Word Problems Involving Linear Equations**A verbal problem or a mathematically-worded problem is a problem of mathematical nature stated in plain words, and which would involve mathematical calculation of some kind before it can be solved.**There are no set of rules or methods**Which enable us to solve all kinds of Problems, because things must be Remembered in relation to different Types of problems. Note:**The following are the general strategies for problem**solving. 1. Read the problem carefully. Be sure that you understand what the problem is all about. 2. Take note of what is asked in the problem.**3. Represent the unknown by any**variable and other unknowns in terms of the same variable according to the conditions of the problem. 4. Formulate the equation.**5. Solve the resulting equation.**6. Check your answer by substituting it to the original equation and check if your answer or answers satisfy the problem.**Examples**Number Relation Problems The sum of two numbers is 36. One number is 3 less than twice the other number. What are the numbers**Representation:**let x = the other number 2x – 3 = one number Equation: x + 2x – 3 = 36 Solution: 3x – 3 = 36 3x = 36 + 3 3x = 39 Answers: 13 and 23**Another Example**Seven more than twice a number is four less than thrice the number. What is the number?**Example**Consecutive Numbers Problems The sum of three consecutive numbers is 135. Find the numbers.**Representation:**let x = 1st number x + 1 = 2nd number x + 2 = 3rd number Equation: x + x + 1 + x + 2 = 135 Solution: 3x = 135 – 3 3x = 132 Answers: 44, 45, and 46**Another Example**Find three consecutive numbers whose sum is 60.**Example**Age Problems Lherry is 3 times as old as Jane. In 4 years time, Lherry will be twice as old as Jane. How old is Jane?**Representation:**Equation: 3x + 4 = 2 ( x + 4)**Solution:**3x + 4 = 2 (x + 4) 3x + 4 = 2x + 8 3x – 2x = 8 – 4 x = 4 Answer: Jane is 4 years old while Lherry is 12 years old.**Another Example**Paulson is four times older than Maria. If the total of their ages is 60. How old is Maria?**Board Drill**Thirty more than thrice a number is 45. What is the number?**Board Drill**Ten more than twice a number is 100. What is the number?**Board Drill**The sum of three consecutive numbers is 93. Find the numbers.**Board Drill**Daisy, 38 years old, is 8 years more than three times as old as her son. How old is her son?**Board Drill**Leah is three times older than Marvin. If the total of their ages is 68. How old is Marvin?**Board Drill**A moving van rents for P1,250 a day plus P5 per kilometer. Mrs. Santos’ bill for a two day rental was P2000. How many kilometers did she drive?