Linear equations

Habari Mwanafunzi! Ready to Master Equations?

Ever wondered how a duka owner calculates your change perfectly every time? Or how you can budget your pocket money to buy a new pair of shoes? The secret power they are using is Algebra! And today, we're going to learn the most fundamental part of it: Linear Equations. Don't let the name scare you. By the end of this lesson, you'll be solving them like a pro. Twende kazi!

So, What Exactly is a Linear Equation?

Think of a linear equation like a traditional weighing scale, the one you see at the market for weighing sukuma wiki or maize. The equals sign ( = ) is the balancing point in the middle. For the scale to be balanced, both sides must have the same weight. It's the same with equations!

      LHS (Left Hand Side)  =  RHS (Right Hand Side)
             /|\
            / | \
           /  |  \
   [ Expression 1 ]   |   [ Expression 2 ]
   -------------------|--------------------
                     / \
                    /   \
                   /-----\

The main rule is: Whatever you do to one side of the equation, you MUST do to the other to keep it balanced. This is the golden rule of algebra. Haki!

A linear equation has one "unknown" thing, which we usually call x (or any other letter like y, a, b). Our mission is to figure out the value of this unknown. It's like being a detective!

Image Suggestion: A vibrant, colorful illustration of a traditional Kenyan marketplace scale. On one side, there are several bags labeled 'x'. On the other side, there are numbered weights. The scale is perfectly balanced. The style is like a modern African storybook.

Let's Use a Real Kenyan Example!

Imagine you go to the school canteen with 50 shillings. You buy two mandazis and a cup of tea. The tea costs 10 shillings. After paying, you have no money left. The question is: How much does one mandazi cost?

We can turn this story into a math problem! Let's say the unknown cost of one mandazi is 'm'.

• The cost of two mandazis would be 2 times m, or 2m.
• The cost of the tea is 10.
• The total money you spent is 50.

So, the equation becomes:

2m + 10 = 50

See? We just translated a real-life situation into a linear equation. Sawa?

The Golden Rules for Solving Equations

Our main goal is to find the value of our unknown (like 'm'). To do this, we need to get it all by itself on one side of the equals sign. We do this using two simple rules:

1. Use Opposite Operations: To move something from one side to the other, you do its opposite.
   • The opposite of adding (+) is subtracting (-).
   • The opposite of subtracting (-) is adding (+).
   • The opposite of multiplying (x) is dividing (÷).
   • The opposite of dividing (÷) is multiplying (x).
2. Keep it Balanced: Remember the scale! Whatever you do to one side, you must do to the other.

Let's Solve Our Mandazi Problem!

Okay, detective, let's find the cost of one mandazi. Our equation is:

2m + 10 = 50

Step 1: Get the 'm' term alone.

Right now, '10' is being added to '2m'. To move it, we do the opposite: subtract 10. And we must do it from BOTH sides!

2m + 10 - 10 = 50 - 10
2m = 40

Great! Now we know that two mandazis cost 40 shillings.

Step 2: Find the value of one 'm'.

The term '2m' means '2 times m'. To get 'm' alone, we do the opposite of multiplying by 2, which is dividing by 2. Let's do it to both sides.

2m / 2 = 40 / 2
m = 20

There we have it! One mandazi costs 20 shillings. Easy, right?

Image Suggestion: An encouraging illustration of a Kenyan student in school uniform, standing at a chalkboard and proudly pointing to the solved equation 'm = 20'. The student has a big, confident smile.

What if the Unknown is on Both Sides? Usijali!

Sometimes, you'll see an equation where the unknown (like 'x') appears on both the left and the right side. Don't worry, the rules are still the same. The first step is to gather all the 'x' terms on one side and all the plain numbers on the other side.

Let's solve this one: 5x - 6 = 2x + 9

Problem: 5x - 6 = 2x + 9

Step 1: Collect the 'x' terms on one side. It's usually
         easiest to move the smaller 'x' term. Here, 2x is
         smaller than 5x. To move '2x', we subtract it
         from both sides.

         5x - 2x - 6 = 2x - 2x + 9
         3x - 6 = 9

Step 2: Now it looks like our mandazi problem! Let's collect
         the numbers on the other side. We need to move the '-6'.
         The opposite of subtracting 6 is adding 6.

         3x - 6 + 6 = 9 + 6
         3x = 15

Step 3: Get 'x' completely alone. '3x' means 3 times x.
         The opposite is dividing by 3.

         3x / 3 = 15 / 3
         x = 5

Final Answer: x = 5

Mazoezi Hufanya Ustadi (Practice Makes Perfect!)

Now it's your turn to be the detective! Try solving these equations on your own. Remember the rules!

• 1. x + 7 = 15
• 2. 4y - 5 = 11
• 3. a / 3 = 6 (Hint: The opposite of dividing is...?)
• 4. 8p - 1 = 5p + 11

Mambo Muhimu (Key Takeaways)

• A linear equation is like a balanced scale.
• Your goal is to find the value of the unknown variable (the letter).
• To move terms across the '=' sign, always use the opposite operation.
• To keep the equation true, you must do the same thing to both sides.

You have done a fantastic job today! Linear equations are the foundation of so much in mathematics, science, and even daily life. Keep practicing, and don't be afraid to make mistakes—that's how we learn. Kazi nzuri!

Habari Mwanafunzi! Welcome to the World of Algebra!

Ever tried to figure out the price of one samosa when you know the total cost of your order at the school canteen? Or maybe you've tried to calculate how much pocket money you need to save each week to buy that new pair of shoes? If you have, then congratulations! You've already been using the basic ideas of algebra without even knowing it. Today, we are going to dive into a very important part of algebra called Linear Equations. Don't let the name scare you; it's just a fancy way of talking about finding a missing value in a puzzle. Let's solve some puzzles together!

What Exactly is a Linear Equation?

Think of a traditional weighing scale, the one you see at the market. For it to be balanced, the weight on the left side must be exactly equal to the weight on the right side. A linear equation is just like that!

      LEFT SIDE         RIGHT SIDE
         |=================|
         / \               / \
        /---\             /---\
       |  5  |           |  5  |
        \---/             \---/
          |                 |
          ^=================^
                  / \
                 /___\

It's a mathematical statement that says two things are equal. It contains:

• Variables: These are the unknown values, usually represented by letters like x, y, or any other letter. This is the mystery we need to solve! Think of 'x' as the unknown price of a bundle of sukuma wiki.
• Constants: These are the numbers we already know. For example, 5, 10, -3.
• An equals sign (=): This is the most important part! It's the balancing point of our scale.

The simplest form looks like this: ax + b = c, where 'x' is our variable, and 'a', 'b', and 'c' are constants.

The Golden Rule: Keeping the Balance!

To solve any equation, there is one rule you must NEVER forget. It's the golden rule of algebra:

"Whatever you do to one side of the equation, you MUST do the exact same thing to the other side."

Imagine you and your friend are sharing a plate of chapati. If you take one, to keep things fair, your friend should also be allowed to take one. If you add one to your side, you must add one to their side too. This keeps the equation (and your friendship) balanced!

Solving Linear Equations: Step-by-Step

Our main goal is to find the value of the unknown variable. We do this by getting the variable all by itself on one side of the equals sign. We call this isolating the variable.

Example 1: A Simple Start

Let's say you had some money in your pocket. Your uncle gives you Ksh. 10, and now you have a total of Ksh. 50. How much did you start with? Let's turn this into an equation.

Let x be the money you had at the start.

The money you had (x) + The money from your uncle (10) = Total money (50)

Equation: x + 10 = 50

To find x, we need to get it alone. We need to remove the '+ 10' from the left side. How do we do that? We do the opposite: we subtract 10. And remember the Golden Rule!

Step 1: Write down the equation.
   x + 10 = 50

Step 2: Subtract 10 from the left side to isolate x.
   x + 10 - 10 = 50

Step 3: Whatever you do to the left, you MUST do to the right.
   x + 10 - 10 = 50 - 10

Step 4: Calculate the result.
   x + 0 = 40
       x = 40

So, you started with Ksh. 40! See? Not so hard!

Example 2: Adding a Twist

Now, let's look at a slightly more complex one.

Juma bought 3 new exercise books for his mathematics class. After paying, he used a coupon that gave him a Ksh. 5 discount. The final amount he paid was Ksh. 70. What was the price of one exercise book?

Let y be the price of one exercise book.

The cost of 3 books (3 * y) - The discount (5) = The final price (70)

Equation: 3y - 5 = 70

Our goal is to isolate 'y'. We need to undo the '- 5' and the '3 *'. We always start with the addition or subtraction first!

Step 1: Write down the equation.
   3y - 5 = 70

Step 2: Undo the subtraction by adding 5 to both sides.
   3y - 5 + 5 = 70 + 5
           3y = 75

Step 3: Now, 'y' is being multiplied by 3. Undo this by doing the opposite: divide by 3 on both sides.
   3y / 3 = 75 / 3

Step 4: Calculate the final answer.
        y = 25

The price of one exercise book is Ksh. 25. Well done!

From Words to Equations: The Real-World Challenge

In your exams, you will often get "word problems". The trick is to carefully read the story and translate it from English (or Kiswahili) into the language of mathematics.

Problem:

The perimeter of a rectangular shamba (farm) is 100 meters. Its length is 10 meters more than its width. Find the length and width of the shamba.

Let's break it down!

1. Identify the unknowns. We don't know the width or the length. Let's call the width 'w'.
2. Relate the unknowns. The problem says the length is "10 meters more than the width". So, length = w + 10.
3. Find the formula. The formula for the perimeter of a rectangle is P = 2(length + width).
4. Build the equation. We know P = 100.

   100 = 2 * ((w + 10) + w)

5. Solve it!

Step 1: Simplify inside the bracket first.
   100 = 2 * (2w + 10)

Step 2: Expand the bracket by multiplying everything inside by 2.
   100 = 4w + 20

Step 3: Now it looks like our previous examples! Isolate the 'w' term. Subtract 20 from both sides.
   100 - 20 = 4w + 20 - 20
         80 = 4w

Step 4: Isolate 'w'. Divide both sides by 4.
   80 / 4 = 4w / 4
       20 = w

So, the width (w) is 20 meters. But we are not finished! The question asks for length AND width.

Length = w + 10 = 20 + 10 = 30 meters.

Answer: The shamba is 30 meters long and 20 meters wide. Always make sure to answer the full question!

A Sneak Peek: Visualizing Equations

Did you know that linear equations can be drawn? When you have two variables (like x and y), the equation represents a straight line on a graph called a Cartesian Plane. It's like a map for numbers!

        | y-axis
        |
      3 +       /
        |      /
      2 +     /
        |    /
      1 +   * (0,1)
  ------*---+---+---+--> x-axis
     -1 |  /|   1   2
     -2 + /
        |/

The line in the diagram above could represent an equation like y = 2x + 1. Every single point on that line is a possible solution to the equation! We will explore this amazing topic in our next lesson on graphing.

Kazi ya Ziada (Extra Work)

You have done an amazing job today! To become a true master, practice is key. Try solving these on your own:

1. Solve for x: `5x + 8 = 43`
2. Solve for k: `k/4 - 2 = 3` (Hint: Get rid of the -2 first!)
3. Akinyi is 5 years older than her brother, Otieno. The sum of their ages is 31. How old is Otieno?

Remember the Golden Rule, take it one step at a time, and don't be afraid to make mistakes. That's how we learn! Keep practicing, and you will find that mathematics is a powerful tool you can use every single day. Safari njema in your studies!

Habari mwanafunzi! Let's Unlock the Secrets of Algebra: Mastering Linear Equations!

Welcome to the exciting world of Algebra! I know, I know... sometimes when you see letters like 'x' and 'y' in a math problem, it can look a bit confusing. But what if I told you that you already use algebra every single day without even knowing it? When you figure out how much change you should get at the duka, or how much M-Pesa fare you need for a trip to town and back, you're using the basic ideas of algebra!

Think of a linear equation as a puzzle or a mystery. Your job is to be the detective and find the value of the hidden number. Ready to put on your detective hat? Let's begin!

What in the World is a Linear Equation?

At its heart, an equation is all about balance. Imagine a weighing scale, the kind you see at the market. For the scale to be balanced, both sides must have the same weight. An equation works the same way. The equals sign ( = ) is the center of our scale.

A linear equation is a special type of equation where the highest power of our unknown value (our mystery number!) is one. It looks something like this: ax + b = c.

• Variable: This is the letter (like x or y) that represents the unknown number we are trying to find.
• Constants: These are the numbers we already know (like b and c).
• Coefficient: This is the number right next to the variable (like a). It tells us how many of the variable we have.

Let's visualize that balance scale:

        x + 2   =     5
       _______     _______
      | x | | |   | | | | |
      |   |o|o|   |o|o|o|o|o|
      -------     -------
          / \         / \
         /   \       /   \
        /_____\     /_____\
           ^           ^
           |___________|

To keep it balanced, whatever we do to one side, we must do to the other. That is the golden rule of algebra!

Image Suggestion: A vibrant, colourful digital illustration of a traditional two-pan balance scale in the middle of a bustling Kenyan market. On the left pan, there is a small, neatly tied sack with a big 'x' painted on it, along with 3 shiny metal coins (representing +3). On the right pan, there are 10 identical shiny coins. The scale is perfectly balanced. In the background, people are buying fruits and vegetables.

Solving the Mystery: One-Step Equations

Let's start with the simplest mysteries. To solve for our unknown (the variable), we need to get it all by itself on one side of the equals sign. We do this using inverse operations, which simply means doing the opposite!

• The opposite of adding (+) is subtracting (-).
• The opposite of subtracting (-) is adding (+).
• The opposite of multiplying (x) is dividing (÷).
• The opposite of dividing (÷) is multiplying (x).

Scenario 1: Addition
You had some credit (x) on your phone. Your mum sends you 50 shillings more, and now you have a total of 80 shillings. How much did you start with?

The equation is: x + 50 = 80

    // Our goal is to get 'x' alone.
    // The opposite of adding 50 is subtracting 50.
    // So, we subtract 50 from BOTH sides to keep it balanced.

    x + 50 - 50 = 80 - 50

    // The +50 and -50 on the left cancel each other out.
    
    x = 30

    // So, you started with 30 shillings of credit. Kazi nzuri!

Scenario 2: Multiplication
You buy 4 exercise books for your new term, and the total cost is 100 shillings. What is the price of one exercise book?

The equation is: 4b = 100 (Let's use 'b' for book)

    // Here, '4b' means 4 times b.
    // The opposite of multiplying by 4 is dividing by 4.
    // We divide BOTH sides by 4.

    4b / 4 = 100 / 4

    // The 4's on the left cancel out.

    b = 25

    // Each book costs 25 shillings. That's a good price!

Level Up! Tackling Two-Step Equations

Now things get a little more interesting! A two-step equation requires two inverse operations. Think of it like unwrapping a present: you first take off the ribbon (addition/subtraction), and then you remove the wrapping paper (multiplication/division).

Scenario: The Matatu Fare
You are taking a matatu from your estate to town. The conductor says the fare has a fixed charge of 20 shillings, plus 10 shillings for every kilometre you travel. Your total M-Pesa payment for the trip was 90 shillings. How many kilometres was your journey?

Let k be the number of kilometres.

The equation is: 10k + 20 = 90

Let's solve it step-by-step!

    // Equation: 10k + 20 = 90

    // Step 1: Deal with the addition first.
    // The opposite of adding 20 is subtracting 20.
    // Do it to both sides!

    10k + 20 - 20 = 90 - 20

    // This simplifies to:
    10k = 70

    // Step 2: Now deal with the multiplication.
    // The opposite of multiplying by 10 is dividing by 10.
    
    10k / 10 = 70 / 10

    // This simplifies to:
    k = 7

    // So, your journey to town was 7 kilometres long. See? You did it!

Image Suggestion: A fun, cartoon-style illustration of a Kenyan student holding a smartphone with an M-Pesa confirmation message for KSh 90. The student has a thoughtful expression. In a thought bubble above their head, the equation '10k + 20 = 90' is shown, with a big question mark over the 'k'. In the background is a colourful matatu with graffiti art.

The Ultimate Challenge: Variables on Both Sides!

Sometimes, the mystery number (our variable) appears on both sides of the equation. Don't panic! The goal is the same. We just need to gather all the variable "family members" on one side of the equals sign and all the constant number "family members" on the other.

Scenario: Comparing Phone Plans
Safaricom has a plan that costs 20 shillings to join plus 2 shillings per minute (2m + 20).
Airtel has a different plan that costs 10 shillings to join plus 3 shillings per minute (3m + 10).
At how many minutes (m) will the cost of both plans be exactly the same?

To find out, we set the two plans equal to each other:

3m + 10 = 2m + 20

    // Equation: 3m + 10 = 2m + 20

    // Step 1: Gather the variables. Let's move them to the left.
    // It's usually easier to move the smaller variable term.
    // We subtract 2m from BOTH sides.

    3m - 2m + 10 = 2m - 2m + 20

    // This simplifies to:
    m + 10 = 20

    // Step 2: Now it's a simple one-step equation!
    // Gather the constants on the right.
    // Subtract 10 from BOTH sides.

    m + 10 - 10 = 20 - 10

    // This simplifies to:
    m = 10

    // Conclusion: At exactly 10 minutes, the cost for both phone plans will be the same!

You're an Algebra Star!

You have done an amazing job! We've seen that linear equations are just puzzles about balance. By using inverse operations and working step-by-step, you can solve for any unknown value.

Remember the key ideas:

• Keep the equation balanced at all times.
• Use inverse operations to isolate the variable.
• For two-step equations, handle addition/subtraction first.
• If variables are on both sides, bring them together on one side first.

Practice is how you become a champion. Try these problems on your own. You've got this. Kazi nzuri!

Practice Problems

• x - 12 = 5
• 5y = 45
• c / 3 = 9
• 4a + 7 = 31
• 8p - 10 = 3p + 15