Linear equations/inequalities; simultaneous equations - Mathematics IGCSE Study Notes
Overview
Imagine you're trying to figure out how many candies you can buy with your pocket money, or how to share pizza fairly with your friends. That's where linear equations and inequalities come in! They're like secret codes that help us solve puzzles about numbers and amounts in everyday life. Sometimes, you have more than one puzzle to solve at the same time. Maybe you need to know how many apples AND how many bananas you can buy, but you have a total budget and a total number of fruits you want. That's when **simultaneous equations** become your superhero tool, helping you find the answers to multiple questions all at once. Learning these topics isn't just for school; it helps you think logically and solve problems, whether you're planning a party budget or figuring out the best deal at the shop!
What Is This? (The Simple Version)
Think of an equation like a balanced seesaw. On one side, you have some numbers and letters (called variables, which are like mystery numbers waiting to be discovered). On the other side, you have more numbers. The equals sign (=) in the middle means both sides must weigh exactly the same.
-
A linear equation is a special kind of seesaw where the mystery numbers (variables) are just plain old numbers, not squared or cubed. So, you might see 'x + 5 = 10', but not 'x² + 5 = 10'. It makes a straight line if you draw it on a graph.
-
An inequality is like a seesaw that isn't perfectly balanced. Instead of an equals sign, it uses symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). For example, 'x < 5' means 'x' can be any number smaller than 5, like 4, 3, 2, or even 4.999!
-
Simultaneous equations are like having two or more balanced seesaws that are connected. The mystery numbers (variables) on one seesaw are the same mystery numbers on the other seesaw. You need to find the values for these mystery numbers that make all the seesaws balance at the same time.
Real-World Example
Let's say you and your friend are buying snacks. You know that a packet of crisps (let's call its price 'c') and a chocolate bar (let's call its price 'h') together cost $3. This is our first equation: c + h = 3.
Then, your friend tells you that two packets of crisps and one chocolate bar cost $5. This is our second equation: 2c + h = 5.
Now, we have two equations that are true at the same time, and we want to find out the individual price of a packet of crisps ('c') and a chocolate bar ('h'). This is a perfect job for simultaneous equations!
- From the first equation (c + h = 3), we know that h = 3 - c.
- We can now 'substitute' this into the second equation. Instead of 'h', we write '3 - c': 2c + (3 - c) = 5.
- Simplify: 2c - c + 3 = 5, which means c + 3 = 5.
- Subtract 3 from both sides: c = 2. So, a packet of crisps costs $2.
- Now that we know c = 2, we can put it back into our first equation: 2 + h = 3.
- Subtract 2 from both sides: h = 1. So, a chocolate bar costs $1.
See? We solved the mystery prices using simultaneous equations!
Solving Linear Equations (Step by Step)
Solving a linear equation means finding the value of the mystery number (variable) that makes the seesaw balance. Our goal is to get the variable all by itself on one side of the equals sign. 1. **Isolate the variable:** Imagine the variable is a treasure you want to dig up. You need to remove eve...
Unlock 4 More Sections
Sign up free to access the complete notes, key concepts, and exam tips for this topic.
No credit card required · Free forever
Key Concepts
- Equation: A mathematical statement showing two expressions are equal, like a balanced seesaw.
- Variable: A letter (like x or y) that represents an unknown number we want to find.
- Linear Equation: An equation where the variable's highest power is 1, meaning it will graph as a straight line.
- Inequality: A mathematical statement showing that two expressions are not equal, using symbols like <, >, ≤, or ≥.
- +6 more (sign up to view)
Exam Tips
- →Always show your working clearly, even for simple steps; marks are often given for method.
- →For inequalities, remember to reverse the sign if you multiply or divide by a negative number.
- +3 more tips (sign up)
More Mathematics Notes