GRADE 7 MATH LEARNING GUIDE Lesson 26: Solving Linear ...

嚜澶RADE 7 MATH LEARNING GUIDE

Lesson 26: Solving Linear Equations and Inequalities in One Variable Using

Guess and Check

Time: 1 hour

Prerequisite Concepts: Evaluation of algebraic expressions given values of the

variables

About the Lesson: This lesson will deal with finding the unknown value of a variable

that will make an equation true (or false). You will try to prove if the value/s from a

replacement set is/are solution/s to an equation or inequality. In addition, this lesson

will help you think logically via guess and check even if rules for solving equations

are not yet introduced.

Objective:

In this lesson, you are expected to:

1. Differentiate between mathematical expressions and mathematical equations.

2. Differentiate between equations and inequalities.

3. Find the solution of an equation and inequality involving one variable from a

given replacement set by guess and check.

Lesson Proper:

I. Activity

A mathematical expression may contain variables that can take on many values.

However, when a variable is known to have a specific value, we can substitute this

value in the expression. This process is called evaluating a mathematical expression.

Instructions: Evaluate each expression under Column A if x = 2. Match it to its value

under Column B and write the corresponding letter on the space before each item. A

passage will be revealed if answered correctly.

COLUMN A

_____ 1.

_____ 2.

_____ 3.

_____ 4.

3+x

3x 每 2

x每1

2x 每 9

_____ 5.

1

x?3

2

_____ 6.

_____ 7.

_____ 8.

_____ 9.

_____ 10.

_____ 11.

_____ 12.

_____ 13.

5x

x每5

1每x

每4+x

3x

14 每 5x

每x + 1

1 每 3x

COLUMN B

A.

C.

E.

F.

H.

I.

L.

O.

S.

每3

每1

每5

1

每2

4

5

6

10

PASSAGE: ※_________________________________________§

1

II. Activity

Mental Arithmetic: How many can you do orally?

1)

2(5) + 2

2)

3(2 每 5)

3)

6(4 + 1)

4)

每(2 每 3)

5)

3 + 2(1 + 1)

6)

7)

8)

9)

10)

5(4)

2(5 + 1)

每9+1

3 + (每1)

2 每 (每4)

III. Activity

Directions: The table below shows two columns, A and B. Column A contains

mathematical expressions while Column B contains mathematical equations.

Observe the items under each column and compare. Answer the questions that

follow.

Column A

Column B

Mathematical Expressions

x+2

2x 每 5

x

7

___________

___________

Mathematical Equations

x+2=5

4 = 2x 每 5

x=2

7=3每x

___________

___________

1) How are items in Column B different from Column A?

2) What symbol is common in all items of Column B?

3) Write your own examples (at least 2) on the blanks provided below each

column.

Directions: In the table below, the first column contains a mathematical expression,

and a corresponding mathematical equation is provided in the third column. Answer

the questions that follow.

Mathematical Verbal Translation

Mathematical Verbal Translation

Expression

Equation

2x

five added to a

2x = x + 5

Doubling a number gives

number

the same value as adding

five to the number.

2x 每 1

twice a number

decreased by 1

1 = 2x 每 1

1 is obtained when twice a

number is decreased by 1.

7+x

seven increased by a

number

7 + x = 2x + 3

Seven increased by a

number is equal to twice

the same number

increased by 3.

3x

thrice a number

3x = 15

Thrice a number x gives

15.

x每2

two less than a

number

x每2=3

Two less than a number x

results to 3.

2

1) What is the difference between the verbal translation of a mathematical

expression from that of a mathematical equation?

2) What verbal translations for the ※=§ sign do you see in the table? What other

words can you use?

3) Can we evaluate the first mathematical expression (x + 5) in the table when x

= 3? What happens if we substitute x = 3 in the corresponding mathematical

equation (x + 5 = 2x)?

4) Can a mathematical equation be true or false? What about a mathematical

expression?

5) Write your own example of a mathematical expression and equation (with

verbal translations) in the last row of the table.

IV. Activity

From the previous activities, we know that a mathematical equation with one variable

is similar to a complete sentence. For example, the equation x 每 3 = 11 can be

expressed as, ※Three less than a number is eleven.§ This equation or statement may

or may not be true, depending on the value of x. In our example, the statement x 每 3

= 11 is true if x = 14, but not if x = 7. We call x = 14 a solution to the mathematical

equation x 每 3 = 11.

In this activity, we will work with mathematical inequalities which, like a mathematical

equation, may either be true or false. For example, x 每 3 < 11 is true when x = 5 or

when x = 0 but not when x = 20 or when x = 28. We call all possible x values (such

as 5 and 0) that make the inequality true solutions to the inequality.

Complete the following table by placing a check mark on the cells that correspond to

x values that make the given equation or inequality true.

x = 每4

x = 每1

x=0

x=2

x=3

x=8

0 = 2x + 2

3x + 1 < 0

每1 ? 2 每 x

(x 每 1) =

每1

1) In the table, are there any examples of linear equations that have more than

one solution?

2) Do you think that there can be more than one solution to a linear inequality in

one variable? Why or why not?

3

V. Questions/Points to Ponder

In the previous activity, we saw that linear equations in one variable may have a

unique solution, but linear inequalities in one variable may have many solutions. The

following examples further illustrate this idea.

Example 1. Given, x + 5 = 13, prove that only one of the elements of the replacement

set

{每8, 每3, 5, 8, 11} satisfies the equation.

x + 5 = 13

For x = 每8:

For x = 每3:

For x = 5:

For x = 8:

For x = 11:

每8 + 5 = 每3

每3 + 5 = 2

5 + 5 = 10

8 + 5 = 13

11 + 5 = 16

13 = 13

每3 ? 13

2 ? 13

10 ? 13

16 ? 13

Therefore 8 is

Therefore 每8 Therefore 每3 is Therefore 5 is

Therefore 11 is

is not a

not a solution. not a solution. a solution.

not a solution.

solution.

Based on the evaluation, only x = 8 satisfied the equation while the rest did

not. Therefore, we proved that only one element in the replacement set satisfies the

equation.

We can also use a similar procedure to find solutions to a mathematical inequality,

as the following example shows.

Example 2. Given, x 每 3 < 5, determine the element/s of the replacement set {每8,每3,

5, 8, 11} that satisfy the inequality.

x每3 2.5

4

Solve for the value of x to make the mathematical sentence true. You may try several

values for x until you reach a correct solution.

1) x + 6 = 10

2) x 每 4 = 11

3) 2x = 8

6) 4 + x = 9

7) 每4x = 每16

8) ?

1

4)

x?3

5

2

x?6

3

9) 2x + 3 = 13

10) 3x 每 1 = 14

5) 5 每 x = 3

VII. Activity

Match the solutions under Column B to each equation or inequality in one variable

under Column A. Remember that in inequalities there can be more than one solution.

COLUMN A

_____ 1.

3+x=4

_____ 2.

3x 每 2 = 4

_____ 3.

x 每 1 < 10

_____ 4.

2x 每 9 ? 每7

_____ 5.

1

x ? 3 ? ?3

2

_____ 6.

2x > 每10

_____ 7.

x 每 5 = 13

_____ 8.

1 每 x = 11

_____ 9.

每3 + x > 1

_____ 10.

每3x = 15

_____ 11.

14 每 5x ? 每1

_____ 12.

每x + 1 = 10

_____ 13.

1 每 3x = 13

COLUMN B

A.

B.

C.

D.

E.

F.

G.

H.

I.

J.

K.

L.

M.

N.

O.

每9

每1

每5

1

每2

4

每4

6

10

2

18

11

每10

3

每12

VIII. Activity

Scavenger Hunt. You will be given only 5-10 minutes to complete this activity. Go

around the room and ask your classmates to solve one task. They should write the

correct answer and place their signature in a box. Each of your classmates can sign

in at most two boxes. You cannot sign on own paper. Also, when signing on your

classmates＊ papers, you cannot always sign in the same box.

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