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Solving Linear Equations. 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.

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Solving Linear Equations

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Solving linear equations l.jpg

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


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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.


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Equation: 25 + 8x = 265

Solution:

25 + 8x = 265

8x = 265 – 25

8x = 240

8 8

Answer: P30 each videotape


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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.


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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.


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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


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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


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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


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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


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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


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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 + 4Isolate term with x

2x = 14 Divide both sides by 2

2 2

x = 7


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Example No. 2

3x + 8 = 14Given

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!


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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


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Example No. 2

3 ( x – 4 ) = 24Given

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


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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


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Example No. 2

5x – 8 = - 2x + 6Given

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


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Board Drill

4x – 10 + 6x = 100


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Board Drill

7 ( x – 5 ) = 21


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Board Drill

3x + x – 2 = 3


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Board Drill

-6x + 3x – 9 = 18


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Board Drill

9x + 10 = 2x + 31


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Board Drill

-4x + 7 = 6x - 3


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Board Drill

6x - 9 = x + 36


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Board Drill

7x + 9 = 3x + 25


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Board Drill

5x + 8 = 7x


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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.


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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:


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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.


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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.


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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.


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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


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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


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Another Example

Seven more than twice a number is four less than thrice the number. What is the number?


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Example

Consecutive Numbers Problems

The sum of three consecutive numbers is 135. Find the numbers.


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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


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Another Example

Find three consecutive numbers whose sum is 60.


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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?


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Representation:

Equation:

3x + 4 = 2 ( x + 4)


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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.


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Another Example

Paulson is four times older than Maria. If the total of their ages is 60. How old is Maria?


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Board Drill

Thirty more than thrice a number is 45. What is the number?


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Board Drill

Ten more than twice a number is 100. What is the number?


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Board Drill

The sum of three consecutive numbers is 93. Find the numbers.


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Board Drill

Daisy, 38 years old, is 8 years more than three times as old as her son. How old is her son?


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Board Drill

Leah is three times older than Marvin. If the total of their ages is 68. How old is Marvin?


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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?


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