| Classwise Concept with Examples | ||||||
|---|---|---|---|---|---|---|
| 6th | 7th | 8th | 9th | 10th | 11th | 12th |
| Content On This Page | ||
|---|---|---|
| Algebraic Expression and Terms Related to it | Addition and Subtraction of Algebraic Expressions | Value of an Algebraic Expression |
| Uses of Algebraic Expressions | ||
Chapter 12 Algebraic Expressions (Concepts)
Welcome to this essential chapter where we delve deeper into the fascinating world of algebra, moving beyond simple equations to explore the structure and manipulation of Algebraic Expressions. Having gained familiarity with solving basic equations involving unknowns, we now focus on the expressions themselves – the building blocks that form the sides of those equations. An algebraic expression is formally defined as a meaningful combination of constants (fixed numerical values like $5$, $-10$, or $\frac{3}{4}$) and variables (symbols, usually letters like $x$, $y$, $a$, $b$, representing quantities that can change or are unknown), interconnected through fundamental mathematical operations: addition ($+$), subtraction ($-$), multiplication ($\times$), and division ($\div$). Think of expressions like $3x + 7$, $5a^2 - 2ab + b^2$, or $\frac{y}{4} - 9z$ as phrases in the language of algebra.
To effectively work with these expressions, we must first understand their anatomy and associated terminology. The distinct parts of an algebraic expression separated by addition ($+$) or subtraction ($-$) signs are called Terms. For example, in the expression $7xy - 5x + 9$, the terms are $7xy$, $-5x$, and $9$. Each term, in turn, is formed by the product of its Factors. Factors are the numbers and/or variables that are multiplied together to create the term. Considering the term $7xy$, its factors are $7$, $x$, and $y$. The numerical factor within a term is given a special name: the Coefficient. In $7xy$, the coefficient is $7$; in $-5x$, the coefficient is $-5$. Understanding how to identify terms, their factors, and their coefficients is a foundational skill for manipulating expressions.
A critical distinction we must master is differentiating between Like Terms and Unlike Terms. Like terms are terms that possess the exact same variable parts, including the same exponents for corresponding variables. The coefficients can be different. For example, $5x^2y$ and $-2x^2y$ are like terms because both involve $x^2$ and $y$. Similarly, $8p$ and $-p$ (which is $-1p$) are like terms. In contrast, Unlike terms are terms whose variable parts differ in some way – either different variables or different exponents on the same variable. For instance, $3xy$ and $3x^2y$ are unlike terms (due to the exponent on $x$), and $7a^2$ and $7b^2$ are unlike terms (different variables). This distinction is paramount because the fundamental rule for simplifying expressions is that only like terms can be combined through addition or subtraction.
Algebraic expressions are also classified based on the number of terms they contain:
- Monomial: An expression consisting of only one term (e.g., $4x^2$, $-9pq$, $7$).
- Binomial: An expression containing exactly two unlike terms (e.g., $a + 5$, $3x^2 - y$).
- Trinomial: An expression containing exactly three unlike terms (e.g., $p^2 + 2pq - q^2$).
- Polynomial: A general term for an expression with one or more terms, where variables typically have non-negative integer exponents. Monomials, binomials, and trinomials are all specific types of polynomials.
The core operational focus of this chapter lies in mastering the addition and subtraction of algebraic expressions. The process hinges entirely on identifying and combining like terms. To add or subtract expressions, we group the like terms together and then perform the required operation (addition or subtraction) on their respective coefficients, keeping the variable part identical. For example, to add $(3x + 5y)$ and $(2x - 3y)$, we group like terms: $(3x + 2x) + (5y - 3y) = (3+2)x + (5-3)y = 5x + 2y$. These operations can be performed either horizontally, as shown, or vertically by aligning like terms under each other, similar to arithmetic calculations.
Another crucial skill is evaluating an algebraic expression. This involves finding the numerical value of the expression when specific numerical values are assigned to its variables. The process requires careful substitution of the given values for their corresponding variables, followed by simplification of the resulting numerical expression using the standard order of operations (often remembered by acronyms like PEMDAS/BODMAS). Furthermore, this chapter reinforces how algebraic expressions serve as a powerful tool for representing patterns, encapsulating formulas (like the perimeter $2(l+b)$ or area $\pi r^2$ learned previously), and stating general mathematical rules concisely, laying essential groundwork for subsequent studies involving multiplication and division of these expressions.
Algebraic Expression and Terms Related to it
In arithmetic, we work with numbers that have fixed values, such as 5, -10, or $\frac{1}{2}$. Algebra introduces the concept of quantities that can change or vary. These quantities are represented by letters, called variables. This chapter explores how variables and constants are combined using arithmetic operations to form algebraic expressions, and introduces the terminology associated with these expressions.
Variables
A Variable is a symbol, usually a letter of the English alphabet (like $x, y, z, a, b, c, l, m, n$), that represents a quantity that can take on different numerical values. The value of a variable is not fixed; it can change depending on the situation or the problem.
Example: If you buy apples, the number of apples you buy can vary (you can buy 1, 2, 5, etc.). If 'n' represents the number of apples, then 'n' is a variable. The total cost of apples would depend on 'n' and the price per apple.
Example: The length of the side of a square can be different for different squares. If we let 's' represent the side length, 's' is a variable. The perimeter of the square is $4 \times s$. As 's' changes, the perimeter $4s$ also changes.
Constants
A Constant is a quantity that has a fixed numerical value. Its value does not change in a given context or expression.
Example: In the expression $4 \times s$ (for the perimeter of a square), the number 4 is a constant because its value is always 4, regardless of the size of the square.
Any specific number like $7, -3.5, \frac{1}{2}, 0,$ etc., are constants. Their values are fixed.
In an algebraic expression, a term that consists only of a number (without any variable multiplied by it) is called a constant term.
Algebraic Expression
An Algebraic Expression is formed by combining variables and constants using one or more of the four fundamental arithmetic operations: addition ($+$), subtraction ($-$), multiplication ($\times$), and division ($\div$).
Examples of algebraic expressions:
- $x + 5$: Variable $x$ and constant 5 combined by addition.
- $2y - 7$: Variable $y$ multiplied by constant 2 ($2y$), then constant 7 subtracted.
- $3m$: Constant 3 multiplied by variable $m$. ($3 \times m$).
- $\frac{p}{4}$: Variable $p$ divided by constant 4. ($p \div 4$).
- $x^2 + 2x - 3$: Involves powers of a variable, constants, and operations.
- $ab + c$: Product of variables $a$ and $b$ ($ab$), plus constant $c$.
An algebraic expression represents a value that depends on the values of the variables. It does NOT contain an equals sign ($=$).
Terms
The parts of an algebraic expression that are connected by the addition ($+$) or subtraction ($-$) signs are called the Terms of the expression.
A term is itself a number, a variable, or a product or quotient of numbers and variables.
In the expression $5x + 3$, the terms are $5x$ and $3$. The plus sign separates the two terms.
In the expression $2x^2 - 7xy + 4$, the terms are $2x^2$, $-7xy$, and $4$. The subtraction and addition signs separate these terms.
A term that consists only of a constant (like 4 in $2x^2 - 7xy + 4$) is called a constant term.
Factors
The quantities that are multiplied together to form a term are called the Factors of the term. These factors can be numerical or algebraic (variables).
Example: For the term $5x$, the factors are 5 and $x$. We can also say that 5 is a factor, $x$ is a factor.
Example: For the term $-7xy$, the factors are $-7$, $x$, and $y$. Other factors are $-7x, -7y, xy, -x, -y$, etc.
Example: For the term $2x^2$, the factors are 2, $x$, and $x$. We can write $2x^2$ as $2 \times x \times x$. Other factors include $2x$ and $x^2$.
Coefficient
In a term that is a product of factors, any factor (or group of factors) is called the Coefficient of the remaining part of the term.
The Numerical Coefficient of a term is the numerical factor multiplied by the variable part.
Example: In the term $5x$, 5 is the numerical factor.
- The numerical coefficient of $x$ is 5.
- The coefficient of 5 is $x$.
Example: In the term $-7xy$:
- The numerical coefficient of $xy$ is $-7$.
- The coefficient of $-7$ is $xy$.
- The coefficient of $y$ is $-7x$.
Example: In the term $p$: Since $p = 1 \times p$, the numerical coefficient of $p$ is 1.
Example: In the term $-q^2$: Since $-q^2 = -1 \times q^2$, the numerical coefficient of $q^2$ is $-1$.
Example: In the term $\frac{m}{4}$: Since $\frac{m}{4} = \frac{1}{4} \times m$, the numerical coefficient of $m$ is $\frac{1}{4}$.
A constant term (like 5 in $5x+3$) is considered a term, but we usually don't talk about its variable coefficient. The numerical coefficient of a constant term is the constant itself.
Tree Diagrams of Expressions
A Tree Diagram is a visual tool that helps to break down an algebraic expression into its component parts: terms and factors. It shows the structure of the expression in a hierarchical way.
The diagram starts with the expression at the top, branches out to its terms, and then each term branches out further to its factors.
Example: Let's create a tree diagram for the expression $5x^2 - 7xy$.
Explanation of the diagram:
- Expression: The entire expression $5x^2 - 7xy$ is at the top.
- Terms: The expression is made up of two terms, $5x^2$ and $-7xy$. These form the first set of branches.
- Factors: Each term is then broken down into its factors.
- The term $5x^2$ is formed by multiplying 5, $x$, and $x$. So, its factors are 5, $x$, and $x$.
- The term $-7xy$ is formed by multiplying -7, $x$, and $y$. So, its factors are -7, $x$, and $y$.
Like and Unlike Terms
Understanding whether terms are 'like' or 'unlike' is essential for adding and subtracting algebraic expressions.
Like Terms are terms that have the exact same variable factors (variables raised to the same powers). Their numerical coefficients can be different.
Examples of like terms:
- $3x$ and $-7x$: Both have the variable $x$ raised to the power 1.
- $5y^2$ and $\frac{1}{2}y^2$: Both have the variable $y$ raised to the power 2.
- $2ab$ and $9ab$: Both have the variables $a$ (power 1) and $b$ (power 1).
- $-4x^2y$ and $11x^2y$: Both have the variable factor $x^2y$.
- $5$, $-10$, $0.75$: All are constant terms (can be thought of as having variable $x^0$).
Unlike Terms are terms that have different variable factors. This can be because the variables are different, or because the same variables are raised to different powers, or the combination of variables is different.
Examples of unlike terms:
- $3x$ and $3y$: Different variables ($x$ vs $y$).
- $5x^2$ and $5x$: Same variable, but different powers ($x^2$ vs $x$).
- $2ab$ and $2a$: Different variable factors ($ab$ vs $a$).
- $-4x^2y$ and $-4xy^2$: Same variables, but powers are different for x and y ($x^2y$ vs $xy^2$).
- $3x$ and $5$: One has a variable, the other is a constant.
Only like terms can be combined by adding or subtracting their numerical coefficients. Unlike terms cannot be combined in this way.
Degree of a Polynomial
The degree is a measure of the highest power of the variables in a polynomial.
Degree of a Term
- The degree of a term with a single variable is the exponent of that variable. For example, the degree of the term $5x^3$ is 3.
- The degree of a term with multiple variables is the sum of the exponents of all the variables in that term. For example, the degree of the term $-7x^2y^3$ is $2+3=5$.
- The degree of a constant term (like 8) is 0, because it can be thought of as $8x^0$.
Degree of a Polynomial
The Degree of a Polynomial is the highest degree among all of its terms.
Example 1 (One Variable): Consider the polynomial $4x^3 - 5x^2 + 2x - 1$.
- Degree of $4x^3$ is 3.
- Degree of $-5x^2$ is 2.
- Degree of $2x$ is 1.
- Degree of $-1$ is 0.
The highest degree is 3. So, the degree of the polynomial $4x^3 - 5x^2 + 2x - 1$ is 3.
Example 2 (Multiple Variables): Consider the polynomial $3x^2y^2 - 2xy^5 + y^3$.
- Degree of $3x^2y^2$ is $2+2=4$.
- Degree of $-2xy^5$ is $1+5=6$.
- Degree of $y^3$ is 3.
The highest degree is 6. So, the degree of the polynomial $3x^2y^2 - 2xy^5 + y^3$ is 6.
Monomials, Binomials, Trinomials, and Polynomials
Algebraic expressions can be classified in two main ways: by the number of terms they contain, and by the number of variables they use.
1. Classification Based on Number of Terms
- Monomial: An algebraic expression with only one term.
Examples: $5x$, $-3y^2$, $10$.
- Binomial: An algebraic expression with exactly two unlike terms.
Examples: $x + y$, $2a - 5$, $p^2 + q^2$.
- Trinomial: An algebraic expression with exactly three unlike terms.
Examples: $x + y + z$, $a^2 - 2a + 1$.
- Polynomial: A general term for any algebraic expression that contains one or more terms (where variables have non-negative integer powers). Monomials, binomials, and trinomials are all types of polynomials.
2. Classification Based on Number of Variables
- Polynomial in One Variable: An expression that contains only one variable.
Examples: $3x^2 - 5x + 1$ (uses only variable $x$), $4y^3 - 7y$ (uses only variable $y$).
- Polynomial in Two Variables: An expression that contains two different variables.
Examples: $x^2 + 2xy + y^2$, $3a - 5b$.
- Polynomial in Three or More Variables: An expression that contains three or more different variables.
Examples: $a+b+c$, $2xy - 3yz + 4zx$.
Example 1. Identify the terms and their numerical coefficients in the expression $4x^2 - 3xy + 5$.
Answer:
The terms in the expression $4x^2 - 3xy + 5$ are $4x^2$, $-3xy$, and $5$.
| Term | Numerical Coefficient |
|---|---|
| $4x^2$ | 4 |
| $-3xy$ | -3 |
| $5$ | 5 (Constant term) |
Example 2. Classify the following algebraic expressions as monomial, binomial, or trinomial:
(a) $5x - 3$, (b) $10y$, (c) $a + b - 2c$, (d) $p^2q$
Answer:
- (a) $5x - 3$: Contains two unlike terms. It is a Binomial.
- (b) $10y$: Contains one term. It is a Monomial.
- (c) $a + b - 2c$: Contains three unlike terms. It is a Trinomial.
- (d) $p^2q$: Contains one term. It is a Monomial.
Example 3. Identify the like terms in the following collection: $2x^2, 5y, -3x, -y, x^2, 7x, -8y, -4x^2$
Answer:
We group terms with the same variable factors:
- Terms with $x^2$: $(2x^2, x^2, -4x^2)$
- Terms with $y$: $(5y, -y, -8y)$
- Terms with $x$: $(-3x, 7x)$
Example 4. Find the degree of the following polynomials:
(a) $7x^5 - 3x^4 + 2x^2 - 9$
(b) $5x^3y^2 + 8xy^2 - 3x^2y$
Answer:
(a) For the polynomial $7x^5 - 3x^4 + 2x^2 - 9$:
The degrees of the terms are 5, 4, 2, and 0. The highest degree is 5.
Therefore, the degree of the polynomial is 5.
(b) For the polynomial $5x^3y^2 + 8xy^2 - 3x^2y$:
- Degree of the term $5x^3y^2$ is $3+2=5$.
- Degree of the term $8xy^2$ is $1+2=3$.
- Degree of the term $-3x^2y$ is $2+1=3$.
The highest degree among the terms is 5.
Therefore, the degree of the polynomial is 5.
Addition and Subtraction of Algebraic Expressions
We have learned about algebraic expressions, including terms, variables, constants, coefficients, and how to classify expressions based on the number of terms and whether terms are like or unlike. Just as we perform addition and subtraction with numbers, we can also perform these operations with algebraic expressions. The key to adding or subtracting algebraic expressions is combining like terms.
Addition of Algebraic Expressions
To add two or more algebraic expressions, we combine the terms that are alike. Unlike terms cannot be added or subtracted directly; they remain as separate terms in the sum.
Steps to Add Algebraic Expressions:
- Identify all the like terms in the expressions that are being added.
- Group the like terms together. You can do this by rearranging the terms in a horizontal line or by arranging them in columns.
- Add the numerical coefficients of each group of like terms. The variable part remains the same for that group.
- Write the final expression by combining the results of adding the coefficients for each type of like term and including the unlike terms if any.
We can use either the horizontal method or the column method for addition.
Horizontal Method:
Write all the given expressions in a single row, joined by addition signs. Then, rearrange the terms within the row to group the like terms together. Finally, add the coefficients within each group of like terms.
Column Method:
Write each expression in a separate row, one below the other, such that like terms are placed directly under each other in vertical columns. Then, add the terms column-wise by adding the numerical coefficients in each column. If a term is missing in an expression, you can leave that space blank or write +0 times the variable part (e.g., +0y).
Example 1. Add the following expressions:
$3x + 5y - 7z$, $9x - 2y + 4z$, and $-2x - y + 3z$
Answer:
Horizontal Method:
Write the expressions in a row with addition signs:
$(3x + 5y - 7z) + (9x - 2y + 4z) + (-2x - y + 3z)$
Remove the parentheses (since we are adding, signs inside do not change):
$3x + 5y - 7z + 9x - 2y + 4z - 2x - y + 3z$
Group the like terms (terms with $x$, terms with $y$, terms with $z$):
$(3x + 9x - 2x) + (5y - 2y - y) + (-7z + 4z + 3z)$
Add the numerical coefficients within each group of like terms:
For $x$ terms: $(3 + 9 - 2)x = (12 - 2)x = 10x$.
For $y$ terms: $(5 - 2 - 1)y = (3 - 1)y = 2y$. (Remember $-y$ means $-1y$).
For $z$ terms: $(-7 + 4 + 3)z = (-3 + 3)z = 0z = 0$.
Combine the results to get the final expression:
Sum $= 10x + 2y + 0 = 10x + 2y $.
Column Method:
Write each expression in a row, aligning like terms vertically in columns:
Add the coefficients in each column. For the $y$ column, $-y$ is $-1y$.
The sum is $10x + 2y + 0z$, which simplifies to $10x + 2y$.
The sum of the expressions is $10x + 2y$.
Subtraction of Algebraic Expressions
To subtract one algebraic expression from another, we also combine like terms. The key step in subtraction is to handle the signs correctly.
Rule: To subtract an expression, add its additive inverse. The additive inverse of an expression is obtained by changing the sign of each term in the expression.
Steps to Subtract Algebraic Expressions:
- Identify the expression that is being subtracted (the subtrahend). This is the expression that comes after the word "from" or immediately after a minus sign outside parentheses.
- Change the sign of each term in the subtrahend expression. If a term is positive, make it negative; if a term is negative, make it positive.
- Add the resulting expression (with changed signs) to the other expression (the minuend).
- Group the like terms together and add their coefficients.
- Write the final expression.
We can use either the horizontal method or the column method for subtraction.
Horizontal Method:
Write the expression to be subtracted in parentheses preceded by a minus sign. Then remove the parentheses by multiplying the entire expression inside by $-1$, which effectively changes the sign of every term inside. Finally, group like terms and add their coefficients.
Column Method:
Write the expression to be subtracted below the expression it is subtracted from, aligning like terms in columns. Before adding, mentally or physically change the sign of each term in the lower expression. Then, add the terms column-wise using the new signs.
Example 2. Subtract $4a - 7b + 3c$ from $5a + 2b - c$.
Answer:
We need to subtract the expression $(4a - 7b + 3c)$ from the expression $(5a + 2b - c)$.
Horizontal Method:
Write the subtraction problem:
$(5a + 2b - c) - (4a - 7b + 3c)$
Remove the parentheses. The signs of the terms in the first expression remain unchanged. For the second expression, change the sign of each term because it's preceded by a minus sign:
$5a + 2b - c - 4a + 7b - 3c$
Group the like terms:
$(5a - 4a) + (2b + 7b) + (-c - 3c)$
Combine the numerical coefficients within each group of like terms. (Remember $-c$ is $-1c$).
For $a$ terms: $(5 - 4)a = 1a = a$.
For $b$ terms: $(2 + 7)b = 9b$.
For $c$ terms: $(-1 - 3)c = -4c$.
Combine the results to get the final expression:
Difference $= a + 9b - 4c $.
Column Method:
Write the expression being subtracted $(4a - 7b + 3c)$ below the expression it is subtracted from $(5a + 2b - c)$, aligning like terms in columns.
Now, change the sign of each term in the lower expression (4a becomes -4a, -7b becomes +7b, +3c becomes -3c). Then add the terms column-wise using the new signs.
The result is $1a + 9b - 4c$, which simplifies to $a + 9b - 4c$.
The difference is $a + 9b - 4c$.
Value of an Algebraic Expression
As we've learned, an algebraic expression contains variables, and the values of these variables can change. Because of this, an algebraic expression does not have a single fixed numerical value. Instead, its value depends on the specific numerical value(s) assigned to the variable(s) within it. To find the value of an expression, we substitute the given numbers for the variables and then perform the calculations following the order of operations (BODMAS).
Finding the Value of an Algebraic Expression
To calculate the numerical value of an algebraic expression for particular given values of its variable(s):
- Take the given algebraic expression.
- Replace each variable in the expression with the specific numerical value assigned to that variable. Be careful when substituting negative numbers, often using parentheses.
- Perform all the arithmetic operations (addition, subtraction, multiplication, division, powers) in the correct order according to the BODMAS rule. Remember that multiplication ($2x$ means $2 \times x$) and powers ($x^2$ means $x \times x$) need to be handled correctly.
- The final result obtained after completing all the calculations is the numerical value of the expression for the given variable value(s).
Example 1. Find the value of the expression $2x + 5$ if $x = 3$.
Answer:
Given expression: $2x + 5$.
Given value of the variable: $x = 3$.
Substitute the value $x=3$ into the expression:
$2x + 5 = 2 \times (3) + 5 $.
Perform the multiplication first (BODMAS): $2 \times 3 = 6$.
$= 6 + 5 $.
Perform the addition:
$= 11 $.
The value of the expression $2x + 5$ when $x = 3$ is 11.
Example 2. Find the value of the expression $p^2 - 2p + 1$ if $p = -2$.
Answer:
Given expression: $p^2 - 2p + 1$.
Given value of the variable: $p = -2$.
Substitute $p = -2$ into the expression. Use parentheses when substituting a negative value, especially when there are powers or multiplications.
$p^2 - 2p + 1 = (-2)^2 - 2 \times (-2) + 1 $.
Perform calculations following BODMAS: Powers first, then multiplication, then addition/subtraction.
Calculate the power: $(-2)^2 = (-2) \times (-2)$. Negative $\times$ Negative $=$ Positive. $2 \times 2 = 4$. So, $(-2)^2 = 4$.
Value $= 4 - 2 \times (-2) + 1$.
Calculate the multiplication: $-2 \times (-2)$. Negative $\times$ Negative $=$ Positive. $2 \times 2 = 4$. So, $-2 \times (-2) = 4$.
Value $= 4 - (-4) + 1 $. (Note: $4 - (-4)$ means 4 minus negative 4).
Convert subtraction of negative to addition of positive: $4 - (-4) = 4 + 4 = 8$.
Value $= 8 + 1$.
Perform the addition:
$= 9 $.
The value of the expression $p^2 - 2p + 1$ when $p = -2$ is 9.
Uses of Algebraic Expressions
Algebraic expressions are not just mathematical constructs; they are incredibly useful tools that allow us to describe relationships, generalise patterns, and translate real-world problems into mathematical language. They form the foundation for equations and help us solve a wide variety of problems.
Representing Formulas
Many formulas in various fields, especially in mathematics and science, are expressed using algebraic expressions. Variables are used to represent quantities that can change, allowing the formula to be applied to different specific situations.
Examples:
- The formula for the perimeter of a rectangle is $P = 2(l+b)$, where $l$ represents the length and $b$ represents the breadth. $2(l+b)$ is an algebraic expression for the perimeter in terms of $l$ and $b$.
- The formula for the area of a square is $A = s^2$, where $s$ is the side length. $s^2$ is an algebraic expression for the area in terms of $s$.
- The formula for Simple Interest is S.I. $= \frac{P \times R \times T}{100}$, where $P$ is the principal amount, $R$ is the rate of interest, and $T$ is the time period. $\frac{P \times R \times T}{100}$ is an algebraic expression for the interest amount in terms of $P, R,$ and $T$.
These formulas are essentially rules written in a compact algebraic form. By substituting specific numerical values for the variables, we can calculate the value of the quantity represented by the expression.
Translating Word Problems
Algebraic expressions provide a powerful way to translate statements from everyday language into mathematical language. This translation is often the first crucial step in solving word problems using algebra. By identifying the unknown quantities and expressing the relationships between them using variables and operations, we can set up equations or simply represent quantities algebraically.
Examples of translating statements into expressions:
- "5 more than a number $x$": This means adding 5 to the number $x$. Expression: $x + 5$.
- "Twice a number $y$ decreased by 3": "Twice a number $y$" is $2y$. "decreased by 3" means subtract 3. Expression: $2y - 3$.
- "The cost of $m$ pens if each pen costs $\textsf{₹} 10$": The total cost is the cost per pen multiplied by the number of pens. Expression: $10m$.
- "A number $p$ divided by 4": Expression: $\frac{p}{4}$.
- "7 less than the product of $a$ and $b$": The product of $a$ and $b$ is $ab$. 7 less than that is $ab - 7$. Expression: $ab - 7$.
Generalizing Patterns
Algebraic expressions are excellent tools for describing patterns or rules that apply to a sequence of numbers or a series of geometric figures. They allow us to represent the rule for finding any term in a pattern without having to list all the preceding terms.
Example: Consider the sequence of even numbers: $2, 4, 6, 8, 10, \dots$
The first term is $2 \times 1$. The second term is $2 \times 2$. The third term is $2 \times 3$. The fourth term is $2 \times 4$.
We can see a pattern: each term is 2 multiplied by its position in the sequence. If we let 'n' be the position of the term (where $n$ is a natural number 1, 2, 3, ...), then the rule for finding the $n^{th}$ term of this sequence is $2 \times n$, or $2n$. The expression $2n$ generalises the pattern of even numbers.
Example: Consider the number of matchsticks needed to make a pattern of squares in a row.
1 square needs 4 matchsticks.
2 squares in a row need $4 + 3 = 7$ matchsticks.
3 squares in a row need $7 + 3 = 10$ matchsticks.
We can see that after the first square (4 sticks), each additional square in the row adds 3 sticks. If there are 's' squares in a row, the number of matchsticks is $4 + (s-1) \times 3$. Simplifying this expression: $4 + 3s - 3 = 3s + 1$. The algebraic expression $3s + 1$ gives the number of matchsticks needed for 's' squares in a row. This generalises the pattern.
Example 1. Write an algebraic expression for the statement: "The sum of a number $x$ and 7".
Answer:
We are asked to write an algebraic expression for "The sum of a number $x$ and 7".
Let the number be represented by the variable $x$.
"The sum of..." means we need to add. The quantities to be added are the number $x$ and the constant 7.
So, the sum is $x + 7$.
The algebraic expression is $x + 7$.
Example 2. The length of a rectangle is 5 units more than its breadth. If the breadth is $b$ units, write an expression for its length and its perimeter.
Answer:
Given: The breadth of the rectangle is $b$ units.
To Find: Expressions for its length and its perimeter.
The length is described as "5 units more than its breadth". This means we add 5 to the breadth.
Length = Breadth + 5
Substitute the variable for breadth:
Length $= b + 5$ units.
So, the algebraic expression for the length is $b + 5$.
Now, find the expression for the perimeter of the rectangle. The formula for the perimeter of a rectangle is $P = 2 \times (\text{Length} + \text{Breadth})$.
Substitute the expressions for length ($b+5$) and breadth ($b$) into the formula:
Perimeter $= 2 \times ((b + 5) + b) $.
Simplify the expression inside the parenthesis by combining like terms ($b$ and $b$): $b + b = 2b$.
Perimeter $= 2 \times (2b + 5) $.
Apply the distributive property (multiply 2 by each term inside the parenthesis): $2 \times 2b = 4b$, and $2 \times 5 = 10$.
Perimeter $= 4b + 10 $ units.
So, the algebraic expression for the perimeter is $4b + 10$.
The expression for the length is $b + 5$, and the expression for the perimeter is $4b + 10$.