Expressions Using Letter-Numbers

Chapter Overview

Letters can stand for numbers that may vary or are not yet known. This concise language of algebra helps describe patterns, relationships and rules that apply to many cases at once.

This chapter belongs to Ganita Prakash, Grade 7. Focus on explaining each step, checking whether an answer is reasonable, and comparing more than one solution method.

Learning Objectives

After studying this chapter, you should be able to:

• explain and apply variables and constants;
• explain and apply terms and coefficients;
• explain and apply building expressions;
• explain and apply substitution;
• communicate the reasoning behind a solution clearly;
• check answers using estimation, substitution, or a second method.

Key Concepts

| Concept | Meaning | |---|---| | Variables and constants | A variable such as n can take different values; a constant has a fixed numerical value. | | Terms and coefficients | In 5x + 3, the terms are 5x and 3; 5 is the coefficient of x. | | Building expressions | Translate relationships carefully: five more than n is n + 5, while five times n is 5n. | | Substitution | Replace a variable by a given value and then use the order of operations to evaluate the expression. |

Detailed Explanation

Variables and constants

A variable such as n can take different values; a constant has a fixed numerical value.

Terms and coefficients

In 5x + 3, the terms are 5x and 3; 5 is the coefficient of x.

Building expressions

Translate relationships carefully: five more than n is n + 5, while five times n is 5n.

Substitution

Replace a variable by a given value and then use the order of operations to evaluate the expression.

Important Rules and Formulae

• Write multiplication as 4x, not 4 + x.
• Only like terms can be combined: 3x + 2x = 5x, but 3x + 2 cannot be simplified further.
• Brackets show when an entire expression is multiplied.

Worked Examples

Example 1

Problem: Write the perimeter of a rectangle of length l and breadth b.

Solution: l + b + l + b = 2l + 2b = 2(l + b).

Example 2

Problem: Evaluate 3n + 7 when n = 5.

Solution: 3 × 5 + 7 = 22.

Common Mistakes

• Treating a letter as a label rather than a number.
• Combining unlike terms.
• Forgetting that 3(x + 2) multiplies both terms inside the brackets.

Quick Revision

• Variables and constants: A variable such as n can take different values; a constant has a fixed numerical value.
• Terms and coefficients: In 5x + 3, the terms are 5x and 3; 5 is the coefficient of x.
• Building expressions: Translate relationships carefully: five more than n is n + 5, while five times n is 5n.
• Substitution: Replace a variable by a given value and then use the order of operations to evaluate the expression.

Practice Questions

1. Write an expression for a number m decreased by 9.
2. Simplify 7a + 2a − 4.
3. Find 2p + 5 when p = −3.
4. A matchstick pattern uses 4n + 1 sticks. How many for n = 12?

Answers and Explanations

1. m − 9.
2. 9a − 4.
3. −1.
4. 49 sticks.

Self-Check

Explain one rule from this chapter in your own words, create a fresh example, solve it, and verify the answer. If the explanation and verification agree, the concept is understood rather than merely memorised.