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Similar to English Sentence, a Mathematical sentence says something.

English Sentence

The sun rises in the East.

Ali is Eating.

Mathematical Sentence

4 + 4 = 8

5 is an odd number.

A sentence can be either open or closed depending on the information provided. Here, we will discuss the meaning of the open sentence in Mathematics.

An open sentence in Mathematics is neither true nor false until the variables have been substituted by specific values. The method of finding the values of variables that result in a true sentence is known as solving the open sentence. The replacement value is known as the solution of the open sentence.

For example, 1 - y = 8 is an open sentence because the value of ‘y’ is unknown and as a result, we can state if it is true or false.

The uncertainty that we have about 1- y = 8 is what makes 1- y = 8 an open statement.

Q. What will be the solution of the equation if the given replacement set is {10, 11, 12, 13, 14, 15}.

4y - 7 = 29

Solution:

We will find the solution of the equation by finding the appropriate value that satisfies the given equation.

Let y = 8

4 (8) - 7 = 29

32 - 7 = 29

25 ≠ 29

So, 8 is not the solution of the given equation

Let y = 9

4 (9) - 7 = 29

36 - 7 = 29

29 = 29

So, 9 is the solution of the given equation.

Till now we have learned an open number system that includes a symbol such as ▭ or a 𐩒. Instead of these symbols, we can use variables to represent missing information. These letters are known as variables because their values can change.

In other words, a variable is a symbol that is used for a number we don't know yet. Variables are usually represented by letters such as x or y.

If x = 2, then x + 4 = 2 + 4 = 6.

If x = 3, then x + 4 = 3 + 4 = 7.

If x = 0, then x + 4 = 0 + 4 = 0.

As you can see, the values of x are changing, and that is why it is known as variables.

Variables and open sentences are a part of Mathematics known as Algebra. Open sentences can use a picture symbol (for example a box or a circle) or a variable (a letter denoting any number). Following are some examples of open sentences using variables.

3 + y = 5

x - 4 = -3

p - 17 = 20

2 × n = 16

a ÷ 4 = 16

In these open sentences, we can determine the correct value for the unknown variables that will make the sentence true. This is known as solving an open sentence.

If the sentence that is required to be solved in straightforward, then you can use mental maths or your knowledge to solve it.

An open sentence can also be solved using inverse operations. The inverse operation does the opposite of another operation. Addition and Subtraction are inverse operations, whereas Multiplication and Division are inverse operations.

For example, 7 - 5 is the same as 7 + (- 5). The numbers 5 and - 5 are known as additive inverse. When we add 5 + (-5), we get the answer 0.

Similarly, 6 ÷ 2/3 is the same as 6 × 3/2. A fraction and its inverse are known as multiplicative inverse. When we multiply, 2/3 × 3/2, we get the answer 1.

Solving Open Sentence Using Inverse Operation Example

Q. Find the value of yin the open sentence:

y + 4 = 16

Solution:

Step 1: We will use the addition and subtraction inverse operation to solve the given open sentence. Subtracting 4 from both the side, we get:

y + 4 - 4 = 16 - 4 (1)

Step 2: Simplify the equation (1) until you get variable = value.

y + 4 - 4 = 16 - 4

y + 0 = 12

y = 12

Hence, the value of y is 12.

Remember: An open is not considered as solved if the variable is negative. For example, - x = 6. The variable should always be positive in the final answer. For example, x = -6.

The open sentence may have one or more than one mathematical operations, To solve such an open sentence, you need to remember the correct order given below:

Bracket

Of

Divide

Multiplication

Addition

Subtraction

Example:

Find the value of y ÷ 2 + 3 = 8

Solution:

Step 1: Decide which inverse operation should be done first.

As per the BODMAS rule, you must calculate y ÷ 2 before 2 + 3. This implies y ÷ 2 must “ stay together”. You can insert brackets to simplify the problem.

We should use inverse addition first to simplify the sentence as shown below:

(y ÷ 2) + 3 - 3 = 8 - 3

Step 2: Simplify the sentence until you left with only one operation.

(y ÷ 2) + 3 - 3 = 8 - 3

y ÷ 2 = 5

Step 3: Simplify using the inverse operation until you get the variable = value.

y ÷ 2 = 5

y ÷ 2 × 2 = 5 × 2

y = 10

We can use the open sentence to solve problems that are described in words. To solve problems, we need to translate the words into mathematical operation as shown below:

Let us now discuss what are open and closed sentences in English.

In English, a sentence is defined as the basic unit of language that expresses using the basic grammatical rules or syntax. A sentence can either be open or closed depending on the information that is provided.

In English, an open sentence is a sentence that contains whereas a closed sentence is an objective statement that is always true or false.

Example:

Open Sentence - She completed her work - Here ' she' is a variable here.

Closed Sentence - The sun rises in the east (always true).

Closed Sentence - The sun sets in the east (always false).

1. Find the value of y in the following open sentence :

y + 6 = 9

Solution:

Step 1: Here, we need to find the value of x that will make the sentence true. To do this, you can ask yourself “ what number should be added to 6 that gives 9?”

As per your knowledge and mental maths, you can add 3 to 6 to get 9.

Step 2: This implies that the value of x is 3. Hence, you can write your answer as x = 3.

2. Find the value of x in the following open number sentence.

y × 4 = 16

Solution:

Step 1: We will use multiplication and division inverse operation to solve the given open sentence. Dividing 4 from both the side, we get,

y × 4 ÷ 4 = 16 ÷ 4 (1)

Step 2: Simplify the equation (1) until you get the format variable = value.

y × 4 ÷ 4 = 16 ÷ 4

y × 1 = 4

y = 4

3. There are 42 students in Class 1. If 20 students are boys, then how many are girls in the class?

Solution:

Step 1: Understand the given problem.

Total number of Students = 40

Number of Boys = 20

Number of Girls = x

Step 2: Devise a Plan.

Total number of students = Total number of girls + Total number of boys

42 = 20 + x

Step 3: Execute a Plan.

42 = x + 20

42 - 20 = x + 20 - 20

22 = x

x = 22

Step 4: Look back at the problem.

22 + 20 = 42

Therefore, there are 22 boys in a class.

FAQ (Frequently Asked Questions)

Q1. What are Open and Closed Sentences in Mathematics?

Ans. In Mathematics, an open number sentence is an expression consisting of variables and results of which become either true or false depending on the values of the variables that are used.

A closed number, on other hand, is a sentence that provides all the information about the numbers and it is always true or false.

Q2. What are the Steps to be Followed While Solving Problems Using an Open Sentence?

Ans. Following are the steps that should be followed while solving problems using an open sentence.

**Understand The Given Problem -**Read the problem carefully a few times. Try to analyze the question that you have. Ensure to know what is asked in the question.**Revise a Plan -**Mention the number sentence that you can use to solve problems.**Execute a Plan -**Solve the number sentence that was asked to solve**Look Back -**Make sure to check your answer once completed. Checking the answer makes sense.

Q3. What are the Variables in an Open Sentence?

Ans. Variables are the values that are unknown in open sentences. Example: x + 2 = 6. Here, x is a variable.