Algebra

Algebra is a powerful branch of mathematics that introduces us to a world of symbols, equations and the art of problem-solving. It serves as a bridge between the elementary arithmetic of numbers and the complex mysteries of higher mathematics. In this chapter, we'll explore the fundamental concepts of algebra and how it plays a crucial role in various aspects of our lives.

Constants and Variables in Algebra

Constant: A constant is a symbol that represents a fixed or unchanging value. Integers, fractions and decimals are constants.

Examples of constants are −15, −4/5, 0, 6, 9, 23, ?, 5/4, 3 ¼, 0.25, etc.

Variable: A variable is a symbol that can take on different numerical values. It represents an unknown or changing quantity. Variables are represented by letters.

Examples of variables are a, b, c, p, q, r, x, y, z, etc.

Term

In algebra, a term is formed by multiplying or dividing variables and constants.

Examples are 2a, 3abc, 4t, −5xy, 17??z, −4a/3, 2c², −2x²??/11, etc.

Each term can represent a specific part of an algebraic expression.

Algebraic Expressions

An algebraic expression is a combination of one or more terms that are connected by addition and subtraction. These expressions can involve variables and constants.

Example: An algebraic expression 7a² − 3b + 11 has one constant, two variables and three terms.

Algebraic expressions are used to represent mathematical relationships and can be evaluated or simplified to find their values.

Various Types of Algebraic Expression

Monomial: A monomial is an algebraic expression consisting of only one term.

Examples:

Algebraic expressions like 7x, 5x²y, 3a², −11, x/2, 7/2, etc.

Binomial: A binomial is an algebraic expression consisting of two terms.

Examples:

Algebraic expressions like 5 −  x², 3a + 8b,  4x − x²y, x²y + y²z, etc.

Trinomial: A trinomial is an algebraic expression consisting of three terms.

Examples:

Algebraic expressions like y² − 2x²z − 7, x² + x − y, p − q + r, 3d + e + h, etc.

Quadrinomial: A quadrinomial is an algebraic expression consisting of four terms.

Examples: Algebraic expressions like 11 − a² + b² − 2abc, x − 3z + x² + xz, etc.

Polynomial: A polynomial is an algebraic expression consisting of variables, constants, coefficients, exponents and mathematical operators. The terms of polynomials are the parts of the expression that are generally separated by “+” or “−” signs.

Example: All polynomials are algebraic expressions.

Factors and Coefficients

Factors: Each term within an algebraic expression is derived from the multiplication of constants and variables. These constants and variables are the factors of that particular term.

Examples:

In the term 0.5a, the factors are 0.5 and a.

In the term 5x²y, the factors are 5, x² and y.

Coefficient of a Term: Within a term of an algebraic expression, the numerical value that appears in the multiplication alongside the variables including the sign is known as the coefficient of that term.

Examples:

In the term 5xy, the coefficient for the term is 5.

In the term −5x²y, the coefficient for the term is −5.

Like and Unlike Terms

Like Terms: Like terms are terms within algebraic expressions that share the same variable factors.

Examples:

5p and 2p/3 are like terms. Both terms share the same variable factor p.

−5x²y, 12x²y and 6x²y are like terms. All terms share the same variable factors x2 and y.

Unlike Terms: Unlike terms are terms within algebraic expressions that have different variable factors.

Examples:

5p and 2q/3 are like terms. Terms share the different variable factors p and q.

−5x, 5y and 12x² are like terms. Terms share the different variable factors x, y and x2.

Substitution

Substitution is the process of determining the value of an algebraic expression by substituting the variables with specific numerical values.

Example:

If we substitute x = 3 in the expression 2x² − 5x, then the value of the expression is: 

2x² − 5x = 2(3)² − 5(3)
               = 2 × (9) − 5 × (3)
                = (2 × 9) − (5 × 3)
                 = 18 − 15
                  = 3

To perform addition or subtraction with algebraic expressions, it is important to gather like terms into different groups and then calculate the sum or difference of these like terms within each group.

Equation

An equation is a statement of equality between two algebraic expressions.

Examples: a3 = 8, x2 + 7 = 9, x − y = 3/2, 7a + 5b = 1 − 11a, etc.

Linear Equation

A linear equation is an equation in which the highest power of the variables is 1.

Examples:  a + 7 = 9,  x − y = 3/2, 7p + 5 = 1 − 11p, etc.

Solution of an Equation

The solution of an equation is a number that when substituted for the variable in the equation makes the left-hand side (LHS) equal to the right-hand side (RHS).

Solving a Linear Equation by Trial and Error: This method involves making guesses about the possible values of the variable. Test several values of the variable, calculate the LHS and RHS for each and look for a value where LHS equals RHS. When LHS equals RHS for a specific variable value that value is considered the solution to the equation.

Example:

Take the equation 2x − 3 = 1.

Test various values of 'x' to find the solution using the trial and error method:

→ For x = 1,

LHS = 2x − 3
        = 2 × (1) − 3
         = 2 − 3
          = −1

RHS = 1

LHS ≠ RHS

Hence, x = 1 is not the solution of the equation 2x − 3 = 1.

→ For x = −1,

LHS = 2x − 3
        = 2(−1) − 3
         = −2 − 3
          = −5

RHS = 1

LHS ≠ RHS

Hence, x = −1 is not the solution of the equation 2x − 3 = 1.

→ For x = 2,

LHS = 2x − 3
        = 2(2) − 3
         = 4 − 3
          = 1

RHS = 1

LHS ≠ RHS

Hence, x = 2 is the solution of the equation 2x − 3 = 1.

Example 1: What is the value of the algebraic expression

if x = 4?

a) 9
b) 11
c) 13
d) 15

Answer: c) 13

Explanation: If x = 4, the value of the algebraic expression is:

Example 2: The temperature in Fahrenheit scale is represented by F and the temperature in Celsius scale is represented by C. If F = 95 C + 32, What is the value of F when C = − 40°?

a) − 32°
b) − 40°
c) 32°
d) 40°

Answer: b) − 40°

Explanation: F = 95 C + 32

If C = − 40°, then

F = 95 C + 32
    = 95 × (−40) + 32
     = − 72 + 32
      = −40°

The temperature on the Fahrenheit scale and the temperature on the Celsius scale are equal to −40°.