Grade 9
Course ContentLinear equations
Habari Mwanafunzi! Solving Life's Puzzles with 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, you want to run for the hills. But what if I told you that you already use algebra every single day without even knowing it?
When you calculate how much change you should get from the shopkeeper, or how many more points your favourite team needs to win the league, you are using the basic ideas of algebra. A linear equation is simply a tool, like a panga or a jembe, that helps us find an unknown value. It's a puzzle, and today, we are going to become expert puzzle solvers. Tuko pamoja? Let's begin!
What in the World is a Linear Equation?
At its heart, a linear equation is all about balance. Imagine a weighing scale, the kind you see at the market when buying maize or beans. For the scale to be balanced, both sides must have the exact same weight.
+-------+ +-------+
| | | |
| 5kg | | 5kg |
+-------+ +-------+
| |
_______|_____________|_______
/ \
/ _ \
An equation works the same way. The equals sign ( = ) is the pivot point of our scale. It tells us that whatever is on the left side has the exact same value as whatever is on the right side.
Let's break down the parts:
- Variable: This is the mystery number, the unknown value we are trying to find. We usually represent it with a letter like
x,y, ora. Think of it as the hidden treasure! - Constant: These are the numbers we already know. They are constant, meaning they don't change.
- Coefficient: This is a number that is multiplied by a variable (e.g., in
3x, the coefficient is 3). It tells you 'how many' of the variable you have.
The Golden Rule: Keep it Balanced!
The most important rule in solving equations is simple: Whatever you do to one side of the equation, you MUST do to the other side.
If you add 2kg to the left side of our market scale, you must add 2kg to the right side to keep it balanced. If you take away half the maize from one side, you must take away half from the other. To solve for our unknown 'x', we need to get it all by itself. We do this by using inverse operations (opposites) to remove the other numbers around it.
- The opposite of adding ( + ) is subtracting ( - ).
- The opposite of subtracting ( - ) is adding ( + ).
- The opposite of multiplying ( * ) is dividing ( / ).
- The opposite of dividing ( / ) is multiplying ( * ).
Let's Get Solving: One-Step Equations
These are the perfect place to start. One step, and you have the answer!
Scenario 1: Mangoes for Sale
Juma has a bag with some mangoes. His friend Akinyi gives him 4 more mangoes. Now, Juma has 11 mangoes in total. How many mangoes did he start with?
Let's turn this story into math. Let x be the number of mangoes Juma started with.
The equation is: x + 4 = 11
To find x, we need to get it alone. We need to remove the '+ 4'. The opposite of adding 4 is subtracting 4. Remember the Golden Rule!
x + 4 = 11
// To remove '+ 4', we subtract 4 from BOTH sides.
x + 4 - 4 = 11 - 4
// The '+ 4' and '- 4' on the left cancel each other out.
x = 7
// So, Juma started with 7 mangoes!
See? Not so bad! Let's try another one.
Scenario 2: Buying Exercise Books
The total cost of 5 exercise books is Ksh 100. What is the price of one exercise book?
Let b be the price of one book. The equation is: 5b = 100 (This means 5 times b equals 100).
To get b alone, we need to undo the 'multiply by 5'. The opposite is dividing by 5.
5b = 100
// To remove 'multiply by 5', we divide BOTH sides by 5.
5b / 5 = 100 / 5
// The '5's on the left cancel out.
b = 20
// So, one exercise book costs Ksh 20.
Stepping It Up: 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 (deal with addition/subtraction), and then you open the box (deal with multiplication/division).
Scenario 3: A Matatu Journey
A matatu ride in town costs a flat fee of Ksh 30 to enter, plus an extra Ksh 10 for every kilometre travelled. If Wanjiku's total fare was Ksh 80, how many kilometres did she travel?
Image Suggestion:
A vibrant, colourful illustration of a Kenyan matatu driving down a city street like Tom Mboya Street in Nairobi. The matatu is decorated with graffiti art. Wanjiku is looking out the window, looking thoughtful. A text bubble shows the equation "10k + 30 = 80". The style is cheerful and cartoonish.
Let k be the number of kilometres Wanjiku travelled.
The equation is: 10k + 30 = 80
Step 1: Get rid of the constant (the 'ribbon').
The constant here is '+ 30'. The opposite is to subtract 30 from both sides.
10k + 30 = 80
10k + 30 - 30 = 80 - 30
10k = 50
Step 2: Get rid of the coefficient (open the 'box').
Now we have 10k = 50. To get k alone, we need to undo the 'multiply by 10'. The opposite is to divide by 10.
10k = 50
10k / 10 = 50 / 10
k = 5
Answer: Wanjiku travelled 5 kilometres. Hongera! You just solved a two-step equation!
Challenge Mode: Variables on Both Sides!
Sometimes the mystery value 'x' appears on both sides of the equals sign. Don't panic! The goal is the same: get all the variable terms on one side and all the constant terms on the other. It's like sorting beans and maize into two different piles.
Let's solve this equation: 7x - 4 = 3x + 16
Step 1: Move all the 'x' terms to one side.
It's a good habit to move the smaller 'x' term to avoid negative numbers. Here, 3x is smaller than 7x. To move it, we subtract 3x from both sides.
7x - 4 = 3x + 16
7x - 3x - 4 = 3x - 3x + 16
4x - 4 = 16
Look at that! It's now a simple two-step equation, which you already know how to solve.
Step 2: Move all the constants to the other side.
We need to move the '- 4'. The opposite is to add 4 to both sides.
4x - 4 = 16
4x - 4 + 4 = 16 + 4
4x = 20
Step 3: Isolate 'x'.
Divide both sides by 4.
4x / 4 = 20 / 4
x = 5
Amazing! The solution is x = 5.
You've Got This! Wewe ni Msharp!
Congratulations on making it through this lesson! Linear equations are the foundation of all algebra. By understanding how to balance them and solve for the unknown, you have gained a powerful skill for both your exams and for life.
Remember the key ideas:
- An equation is a balanced scale.
- The Golden Rule: Whatever you do to one side, you must do to the other.
- Use inverse operations to isolate the variable.
The secret to mastering mathematics is practice. Don't just read this; grab a pen and paper and try solving some problems yourself. Usijali if you make a mistake, that is how we learn! Keep practising, and soon you will be solving linear equations with your eyes closed. Kazi nzuri!
Habari Mwanafunzi! Solving the Mysteries of Everyday Math with Linear Equations
Welcome to the world of Algebra! I know, I know, the word "Algebra" can sometimes sound a bit scary, full of mysterious 'x's and 'y's. But what if I told you that you already use algebra every single day without even knowing it? When you figure out your change at the duka, calculate your bus fare, or share sweets with your friends, you're using the basic ideas of algebra. Today, we are going to unlock one of its most powerful tools: Linear Equations. Let's begin this adventure!
What Exactly is a Linear Equation?
Let's break down the name. "Equation" simply means a mathematical statement that says two things are equal. It will always have an equals sign ( = ). The most important rule of an equation is that it must always be balanced.
Think of it like a weighing scale at the market. If you have 1kg of sugar on one side, you need 1kg of weights on the other side for it to be balanced. If you add anything to one side, you must add the same amount to the other side to keep it balanced.
+-------+ +-------+
| 2x | | 10 |
+-------+ +-------+
\ /
\ /
\________/
/ \
/________\
|
|
=============+============== <-- The equals sign (=)
A balanced scale means both sides are equal.
The "Linear" part means that if we were to draw this equation on a graph, it would form a perfect straight line. But we'll worry about that later! For now, just remember: An equation is a balanced scale.
Our First Mission: Solving a Duka Puzzle
Let's make this real. Imagine you go to the local shop to buy breakfast.
You buy two mandazis and one bottle of soda. The soda costs KSh 30. The shopkeeper tells you the total is KSh 70. You want to know the price of a single mandazi. How do you figure it out?
This is a puzzle we can solve with a linear equation!
First, let's identify our unknown. What is the mystery number we are looking for? It's the price of one mandazi. In algebra, we use a letter to represent an unknown value. Let's use 'm' for mandazi.
Now, let's turn our story into a mathematical sentence:
- The cost of two mandazis is 2 x m, or just 2m.
- The cost of the soda is 30.
- The total cost is 70.
So, the equation is: The cost of two mandazis PLUS the cost of the soda EQUALS the total cost.
2m + 30 = 70Our goal is to find the value of 'm'. To do this, we need to get 'm' all by itself on one side of the equals sign. This is called isolating the variable.
The Golden Rules of Solving Equations
To keep our "scale" balanced, we follow one main rule: Whatever you do to one side of the equation, you MUST do to the other side. We use opposite operations to move numbers away from our variable.
- To get rid of addition, you use subtraction.
- To get rid of subtraction, you use addition.
- To get rid of multiplication, you use division.
- To get rid of division, you use multiplication.
Let's Solve the Mandazi Puzzle!
Here is our equation again:
2m + 30 = 70Step 1: Get rid of the '+ 30'.
The opposite of adding 30 is subtracting 30. So, we subtract 30 from BOTH sides.
2m + 30 - 30 = 70 - 30
// The '+ 30' and '- 30' on the left side cancel each other out.
// On the right side, 70 - 30 is 40.
2m = 40
Step 2: Get rid of the '2' attached to 'm'.
Now we have '2m', which means '2 multiplied by m'. The opposite of multiplying by 2 is dividing by 2. So, we divide BOTH sides by 2.
2m / 2 = 40 / 2
// On the left, 2 divided by 2 is 1, leaving just 'm'.
// On the right, 40 divided by 2 is 20.
m = 20
Answer: We found it! The cost of one mandazi (m) is KSh 20. Kazi nzuri! (Good work!)
Another Real-World Example: Matatu Fares
Image Suggestion: [A vibrant and colourful digital illustration of a Kenyan matatu driving down a city street. The matatu should have graffiti-style art. In the foreground, a student is happily looking at their phone, which displays the equation '30k + 50 = 260'. The style should be fun, modern, and engaging for a young audience.]
Let's try another one. A matatu company charges a flat fee of KSh 50 to book a ride and then KSh 30 for every kilometre travelled. If your total journey cost was KSh 260, how many kilometres did you travel?
Let's use 'k' for kilometres.
- The cost per kilometre is 30k.
- The flat fee is 50.
- The total fare is 260.
Our equation is:
30k + 50 = 260Step 1: Isolate the '30k' term.
We need to remove the '+ 50'. We do this by subtracting 50 from both sides.
30k + 50 - 50 = 260 - 50
30k = 210
Step 2: Isolate 'k'.
Now, we need to remove the '30' that is multiplying 'k'. We do this by dividing both sides by 30.
30k / 30 = 210 / 30
// Hint: To divide 210 by 30, you can cancel the zeros first!
// It becomes 21 / 3, which is 7.
k = 7
Answer: You travelled 7 kilometres. See? You're already becoming an expert!
Challenge Yourself!
Now it's your turn to be the detective. Use the rules we learned to solve these puzzles. Remember to take it step-by-step.
- Solve for y:
5y + 8 = 48 - Solve for a:
a / 3 - 2 = 5(Hint: Add 2 to both sides first!) - Solve for p:
150 - 4p = 70(Hint: Be careful with the signs!) - Word Problem: James is saving up for a football that costs KSh 1,200. He already has KSh 400 in his savings box. If he saves KSh 100 every week, how many weeks will it take him to have enough money? (Let 'w' be the number of weeks).
You've Got This!
Congratulations! You have just learned the fundamentals of solving linear equations. This is a skill that will help you not just in your math class, but in many areas of life. From budgeting your money to understanding science, equations are everywhere.
The key is to remember the balanced scale. Keep practicing, don't be afraid to make mistakes, and soon you will be solving even more complex problems with confidence. Keep up the amazing work!
Habari Mwanafunzi! Let's Uncover the Mystery of Linear Equations!
Welcome to the exciting world of Algebra! Think of yourself as a detective. Your mission, should you choose to accept it, is to find a hidden number, a secret value we call 'x'. Linear equations are the clues that will help you solve the puzzle. It might sound complicated, but I promise you, by the end of this lesson, you'll be solving them like a pro. Ready? Let's begin!
What Exactly IS a Linear Equation?
At its heart, a linear equation is like a balanced weighing scale. Whatever is on the left side must be exactly equal to what is on the right side. The goal is to keep it balanced while you figure out the value of your unknown variable (usually x).
The main clue? The variable (our 'x') has a power of 1. That means you won't see any scary x2 or x3. It's just plain old 'x'.
A Balanced Scale
===================
[ x + 5 ] ? [ 12 ]
/|\ = /|\
/ | \ / | \
/ | \ / | \
------- -------
LEFT RIGHT
The equation for the scale above is x + 5 = 12. Our job is to find the value of 'x' that keeps the scale perfectly balanced.
Solving the Simplest Puzzles: One-Step Equations
The golden rule of solving equations is simple: Whatever you do to one side of the equals sign, you MUST do to the other side to keep it balanced!
Kenyan Example: Imagine you have some shillings in your pocket (we'll call this 'x'). Your friend gives you 10 shillings, and now you have a total of 50 shillings. How much did you start with?Your equation is:
x + 10 = 50
To solve this, we need to get 'x' by itself. We do the opposite of what's being done to it.
Case 1: Using Subtraction (to undo addition)
To get rid of the '+ 10', we subtract 10. And we must do it from both sides!
x + 10 = 50
// Subtract 10 from both sides
x + 10 - 10 = 50 - 10
// The 10s on the left cancel out
x = 40
See? You started with 40 shillings. Rahisi, sindio? (Easy, right?)
Case 2: Using Division (to undo multiplication)
Kenyan Example: You go to the market and buy 3 mangoes. The total cost is 60 shillings. How much does one mango cost? Let the cost of one mango be 'm'.Your equation is:
3m = 60(This means 3 times 'm' equals 60)
To get 'm' alone, we do the opposite of multiplying by 3, which is dividing by 3.
3m = 60
// Divide both sides by 3
3m / 3 = 60 / 3
// The 3s on the left cancel out
m = 20
So, one mango costs 20 shillings. That's a good price!
Leveling Up: Two-Step Equations
Now things get a little more interesting! A two-step equation just means you have to do two things to find 'x'.
The Strategy: First, deal with any addition or subtraction. Second, deal with any multiplication or division. Think of it like taking off your shoes and then your socks. You do it in a specific order!
Kenyan Example: You pay for a matatu ride with a 100 shilling note. The ride costs 20 shillings, plus you buy two samosas for the journey. You get 40 shillings back in change. How much did each samosa cost? Let the cost of one samosa be 's'.Your total spending is
2s + 20. The money you spent is100 - 40 = 60shillings.So, our equation is:
2s + 20 = 60
Let's solve it step-by-step.
2s + 20 = 60
// Step 1: Undo the addition. Subtract 20 from both sides.
2s + 20 - 20 = 60 - 20
2s = 40
// Step 2: Undo the multiplication. Divide both sides by 2.
2s / 2 = 40 / 2
s = 20
Each samosa cost 20 shillings. You are now a master detective!
Image Suggestion: A vibrant, cartoon-style drawing of a Kenyan market scene. A student is happily buying two samosas from a friendly vendor. A matatu is visible in the background. The scene should feel busy and colourful, with text bubbles showing the equation "2s + 20 = 60".
The Final Challenge: Variables on Both Sides!
Sometimes, the mystery variable 'x' appears on both sides of the equation. Don't panic! The goal is the same: get all the 'x' terms on one side and all the plain numbers on the other.
Let's look at this equation: 5x - 4 = 2x + 11
Strategy: It's usually easiest to move the smaller 'x' term to make sure you're working with positive numbers.
5x - 4 = 2x + 11
// Step 1: Move the 'x' terms to one side.
// The smaller x-term is 2x, so let's subtract 2x from both sides.
5x - 2x - 4 = 2x - 2x + 11
3x - 4 = 11
// Look! It's now a two-step equation we already know how to solve!
// Step 2: Move the numbers to the other side. Add 4 to both sides.
3x - 4 + 4 = 11 + 4
3x = 15
// Step 3: Get 'x' by itself. Divide by 3.
3x / 3 = 15 / 3
x = 5
We found it! The secret value is 5. Sawa?
Summary: Your Detective Toolkit
To solve any linear equation, remember these key rules:
- The Golden Rule: Always keep the equation balanced. What you do to one side, you must do to the other.
- The Goal: Isolate the variable (get 'x' all by itself).
- The Method: Use opposite operations to undo what's being done to the variable (addition undoes subtraction, multiplication undoes division).
- The Order: For multi-step equations, handle addition/subtraction first, then multiplication/division.
Kazi ya Ziada (Extra Work)
Now it's your turn to be the detective! Try solving these mysteries on your own.
a + 7 = 194b = 363c - 5 = 167d + 2 = 4d + 14
Great work today! Remember, mathematics is not about being the fastest; it's about understanding the clues and solving the puzzle. Keep practicing, and you will become unstoppable. Uko na uwezo! (You have the ability!)
Habari Mwanafunzi! Kujeni Tufungue Akili na Algebra!
Karibu sana to our lesson on a very exciting part of Mathematics! Ever felt like a detective trying to solve a mystery? Well, that's exactly what Algebra is all about. It's like a puzzle where you have to find a missing number. Today, we are going to become expert detectives in solving one of the most common puzzles in math: Linear Equations. Usijali (don't worry), by the end of this, you will be solving them like a pro! Tuko pamoja? Let's begin!
Hii 'Linear Equation' ni Nini Hasa?
Okay, let's break down that big name. An equation is simply a mathematical statement that says two things are equal. It always has an equals sign ( = ). Think of it like a traditional balancing scale, or 'mizani'. For it to be balanced, both sides must have the same weight.
LEFT SIDE RIGHT SIDE
/|\ /|\
/ | \ / | \
/ | \ / | \
/___|___\ /___|___\
| |
=========================
^ (Equals Sign)
The word Linear just means that if we were to draw it on a graph, it would make a perfect straight line. We'll get to graphing later in the year, but for now, just know it means our puzzle is a straightforward one!
A linear equation has three main players:
- Variable: This is the mystery number we are looking for! It's usually represented by a letter like x, y, or a. It's the "who" in our detective story.
- Coefficient: This is the number that is right next to our variable, multiplying it. If you see `5x`, the coefficient is 5.
- Constant: This is just a plain number, with no letter attached to it.
For example, in the equation `2x + 5 = 15`, `x` is the variable, `2` is the coefficient, and `5` and `15` are the constants.
Sheria ya Dhahabu: The Golden Rule of Equations
To be a great equation detective, you only need to remember one golden rule:
"WHATEVER YOU DO TO ONE SIDE OF THE EQUATION, YOU MUST DO THE EXACT SAME THING TO THE OTHER SIDE."
Think about our 'mizani'. If you add a 2kg stone to the left side, you must also add a 2kg stone to the right side to keep it balanced. If you remove something from one side, you must remove the same from the other. This rule is your key to solving every single linear equation!
Twende Kazi! Let's Solve Some Equations (Step-by-Step)
The main goal is to get the variable (the letter) by itself on one side of the equals sign. Let's see how with some local examples.
Example 1: A Trip to the Duka
Scenario: You go to the shop with some money. You buy a packet of milk for KSh 60 and you are left with KSh 140. How much money did you have initially?
Let's call the money you had initially 'm'. The equation is:
m - 60 = 140To find 'm', we need to get it by itself. Right now, 60 is being subtracted from it. To undo the subtraction, we do the opposite: we add 60.
Step 1: Write down the equation.
m - 60 = 140
Step 2: Add 60 to BOTH sides to keep it balanced (The Golden Rule!).
m - 60 + 60 = 140 + 60
Step 3: Simplify both sides.
m + 0 = 200
m = 200
So, you started with KSh 200! See? Not so hard!
Image Suggestion: A vibrant, colourful illustration of a Kenyan duka (small shop). A teenager is handing over a KSh 200 note to the shopkeeper, with items like milk, bread, and sugar on the counter. The style is cheerful and educational.
Example 2: The Two-Step Problem (Ongeza Spidi!)
Scenario: You are buying textbooks. The delivery fee is a flat KSh 200. Each textbook costs KSh 500. If your total bill is KSh 1700, how many textbooks did you buy?
Let 't' be the number of textbooks.
500t + 200 = 1700Here we have two steps. Always deal with the addition or subtraction first, before the multiplication or division.
Step 1: Undo the addition. Subtract 200 from both sides.
500t + 200 - 200 = 1700 - 200
500t = 1500
Step 2: Now 't' is being multiplied by 500. Undo this by dividing both sides by 500.
500t / 500 = 1500 / 500
t = 3
You bought 3 textbooks. You are a genius!
Example 3: When the Variable is on Both Sides!
Sometimes, the mystery letter appears on both sides of the equation. Don't panic! The goal is just to group all the 'letter' terms on one side and all the 'plain number' terms on the other.
Scenario: A farmer has two plots of land. Plot A has `5x` bags of maize. Plot B has `2x + 12` bags. If both plots have the same number of bags, how many bags does each plot have?
5x = 2x + 12Our mission is to get all the 'x' terms together.
Step 1: Move the smaller 'x' term. Let's subtract '2x' from both sides.
5x - 2x = 2x - 2x + 12
3x = 12
Step 2: Now it looks like an easy one-step problem! Divide by 3.
3x / 3 = 12 / 3
x = 4
The question asks how many bags each plot has.
Plot A = 5x = 5 * 4 = 20 bags.
Plot B = 2x + 12 = (2 * 4) + 12 = 8 + 12 = 20 bags.
The answer is 20 bags. It works!
Image Suggestion: A digital painting of a smiling Kenyan farmer standing proudly between two plots of green maize. On one side, there are sacks of maize neatly stacked. The sun is shining, and the mood is optimistic and prosperous.
Mazoezi Hufanya Ubingwa (Practice Makes Perfect)
Now it's your turn to be the detective! Grab a pen and paper and try to solve these. The answers are at the bottom, but don't peek until you've tried your best!
- You have 15 sweets. After your friend gives you some more, you now have 27. How many sweets did your friend give you? (Equation: `15 + s = 27`)
- A taxi charges a flat fee of KSh 100, plus KSh 50 per kilometre. If a trip cost KSh 600, how many kilometres was it? (Equation: `50k + 100 = 600`)
- Find the value of 'y' in: `7y - 5 = 3y + 15`
Hongera! You've Mastered the Basics!
Congratulations! You have successfully learned the fundamentals of solving linear equations. Remember the Golden Rule: keep your equation balanced! This skill is a foundation for almost everything else you will do in Algebra, all the way to KCSE and beyond. Keep practicing, and don't be afraid to make mistakes—that's how we learn.
Keep that brilliant mind working!
Answers to Practice Questions: 1. s = 12, 2. k = 10, 3. y = 5
Pro Tip
Take your own short notes while going through the topics.
