Key Concepts

Habari Mwanafunzi! Welcome to the World of Algebra!

Ever felt like a detective solving a mystery? Well, get ready, because Algebra is your new detective kit for solving mathematical puzzles! Forget thinking of math as just numbers. Algebra introduces a new character to the story: letters! But don't worry, we'll start with the basics, and I promise you, by the end of this lesson, you'll see Algebra everywhere – from buying a soda at the duka to calculating your bus fare.

Tuko pamoja? Let's begin!

The Building Blocks: Variables and Constants

Imagine you are sent to the shop with 100 shillings. You need to buy a loaf of bread and some sweets. The bread has a fixed price, but the price of one sweet might be unknown to you.

• Constants: A constant is a value that does not change. It's fixed, like a rock! In our shopping trip, the price of bread is Ksh 60. That price is constant. Other examples are the number of counties in Kenya (47) or the number of minutes in an hour (60).
• Variables: A variable is a symbol (usually a letter like x, y, or a) that represents a value we do not know yet. It can change or vary. The unknown price of one sweet is our variable. We can decide to call it s.

A Mystery Box!

  +-------+
  |       |
  |   ?   |  <-- This is our VARIABLE. It holds an unknown value.
  |       |
  +-------+

A Labeled Box!

  +-------+
  |       |
  |  Ksh  |  <-- This is our CONSTANT. We know its value.
  |   60  |
  +-------+

Putting them Together: Terms, Coefficients, and Expressions

Now that we have our building blocks, let's construct something! Let's say you decide to buy 4 sweets.

• Coefficient: This is the number that "hugs" or multiplies a variable. It tells you "how many" of the variable you have. Since you are buying 4 sweets, and the price of one sweet is s, the total cost for the sweets is 4s. Here, 4 is the coefficient.
• Term: A term is a single part of a mathematical "sentence". It can be a constant (like 60), a variable (like s), or a variable with a coefficient (like 4s).
• Expression: An algebraic expression is a combination of terms, constants, variables, and operators (+, -, *, /) but with NO equals sign. It's like a mathematical phrase. The total cost of your shopping is the price of the 4 sweets PLUS the price of the bread.
  So, the expression for your total cost is: 4s + 60

Real-World Example: Let's think about M-Pesa charges. Imagine the transaction fee for sending money is a fixed Ksh 22, but you also want to buy airtime for an unknown amount, which we'll call a. The total amount that will leave your account can be represented by the expression: a + 22.

The Full Story: Equations

An expression is just a phrase, but an equation tells the whole story! An equation is a statement that two expressions are equal. It always has an equals sign ( = ). It's all about balance.

Let's go back to our duka trip. The expression for your cost is 4s + 60. If you give the shopkeeper your 100 shillings and get no change, it means your total cost was exactly Ksh 100. Now we can write an equation!

    4s + 60 = 100

This equation tells a complete story: "The cost of 4 sweets plus the cost of bread is equal to 100 shillings." Now we have a puzzle we can solve to find the mystery value of s!

Think of an equation as a balancing scale. Whatever is on the left side must be perfectly balanced with what is on the right side.

      LEFT SIDE         RIGHT SIDE
   +--------------+   +--------------+
   |    4s + 60   | = |      100     |
   +--------------+   +--------------+
          / \               / \
         / _ \             / _ \
      <--======= Balance =======-->

Let's Recap!

Wow, you've learned the fundamental language of Algebra. Kazi nzuri! Here's a quick summary:

• Constant: A number that stays the same (e.g., 50, -12, 47).
• Variable: A letter representing an unknown value (e.g., x, y, p).
• Coefficient: The number multiplying a variable (e.g., the 3 in 3x).
• Term: A single piece of the puzzle (e.g., 3x, or -10, or y).
• Expression: A mathematical phrase with no equals sign (e.g., 3x - 10).
• Equation: A mathematical statement that two things are equal, containing an equals sign (e.g., 3x - 10 = 5).

Weka Bidii! (Put in Effort!)

Time to test your new detective skills. Read the scenario below and try to answer the questions.

Amina is a farmer. She has a certain number of chickens in her farm; let's call this number c. On Monday, she sells 10 chickens. On Tuesday, her neighbour gives her 15 new chickens.

Challenge 1: Write an algebraic expression for the new number of chickens Amina has.

Challenge 2: If Amina counts her chickens on Wednesday and finds she has a total of 72, write an equation to represent this situation.

Think about it... don't rush! When you are ready, check the solution below.

--- SOLUTION ---

1. The Expression:
   - She starts with 'c' chickens.
   - She sells 10, so we subtract: c - 10
   - She gets 15 more, so we add: c - 10 + 15
   - The final expression is: c + 5

2. The Equation:
   - We know her final number of chickens is represented by the expression c + 5.
   - We are told this final number is equal to 72.
   - So, the equation is: c + 5 = 72

Hongera! You have successfully translated a real-world problem into the language of Algebra. This is a very powerful skill. Keep practicing, stay curious, and remember that every complex problem is just a collection of simple steps. You can do this!