Welcome to Solving Equations and Inequalities!
In this chapter, we are going to learn how to find the "missing pieces" of mathematical puzzles. Whether you are figuring out how many goals a striker needs to break a record or calculating the dimensions of a new garden, algebra is the tool that helps you find the answer. Don't worry if this seems tricky at first—algebra is just like a game where we follow a few simple rules to uncover a hidden number!
1. Linear Equations: The Balancing Act
The most important thing to remember about an equation is that the equals sign \( (=) \) acts like a pair of balancing scales. Whatever you do to one side, you must do to the other to keep it balanced.
Simple Linear Equations
To solve an equation, we want to get the unknown (the letter, like \( x \)) all by itself. We do this by using inverse operations (the opposite action).
- The opposite of Addition \( (+) \) is Subtraction \( (-) \)
- The opposite of Subtraction \( (-) \) is Addition \( (+) \)
- The opposite of Multiplication \( (\times) \) is Division \( (\div) \)
- The opposite of Division \( (\div) \) is Multiplication \( (\times) \)
Step-by-Step Example
Solve \( 3x - 5 = 10 \)
- Step 1: We want to get rid of the \( -5 \). The opposite is \( +5 \). Add \( 5 \) to both sides:
\( 3x = 15 \) - Step 2: Now we have \( 3x \) (which means \( 3 \) times \( x \)). The opposite of multiplying by \( 3 \) is dividing by \( 3 \). Divide both sides by \( 3 \):
\( x = 5 \)
Equations with Brackets and Unknowns on Both Sides
Sometimes equations look messier, like \( 2(x + 4) = x + 12 \). Here is the trick:
- Multiply out brackets first: \( 2x + 8 = x + 12 \)
- Get all the letters on one side: Subtract \( x \) from both sides: \( x + 8 = 12 \)
- Solve as normal: Subtract \( 8 \) from both sides: \( x = 4 \)
Quick Review: Always do the same thing to both sides. If you see a bracket, expand it! If you see \( x \) on both sides, "move" the smaller one by adding or subtracting it.
Key Takeaway: Solving an equation is just undoing what has been done to the letter \( x \).
2. Solving Equations Graphically
Did you know you can solve an equation just by looking at a picture? If you have a graph of a function, the solutions (also called roots) are the points where the line crosses the x-axis.
Example: To solve \( 2x - 4 = 0 \) using a graph, you would look at the line \( y = 2x - 4 \) and find the x-coordinate where it hits the horizontal axis.
3. Quadratic Equations
A quadratic equation is one where the highest power is \( x^2 \). These usually have two answers!
Solving by Factorising
We often solve these by putting them into two brackets.
Solve: \( x^2 + 5x + 6 = 0 \)
- We need two numbers that multiply to give \( 6 \) and add to give \( 5 \). These are \( 2 \) and \( 3 \).
- Put them in brackets: \( (x + 2)(x + 3) = 0 \)
- For the answer to be \( 0 \), one of the brackets must be zero. So, \( x = -2 \) or \( x = -3 \).
The Quadratic Formula (Higher Tier)
If an equation won't factorise, we use this "magic" recipe:
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Memory Tip: Think of it as a story! A negative boy \( (-b) \) couldn't decide whether to go to a house party \( (\pm \sqrt{\dots}) \). At the party, there was a boy who was square \( (b^2) \), and he missed out on \( 4 \) awesome chicks \( (4ac) \). The whole thing was over by 2 a.m. \( (2a) \).
Common Mistake: When using the formula, be very careful with negative numbers, especially inside the square root!
4. Simultaneous Equations
These are "two-for-one" puzzles where you have two different equations and need to find the values of \( x \) and \( y \) that work for both.
The Elimination Method
1) \( 2x + y = 10 \)
2) \( 2x - y = 6 \)
If we add these equations together, the \( +y \) and \( -y \) cancel out (they are eliminated!):
\( 4x = 16 \)
\( x = 4 \)
Now, just swap \( x \) for \( 4 \) in the first equation to find \( y \):
\( 2(4) + y = 10 \)
\( 8 + y = 10 \)
\( y = 2 \)
Key Takeaway: Aim to get the same number in front of either the \( x \) or the \( y \), then add or subtract the equations to get rid of one letter.
5. Inequalities
An inequality tells us that one side is bigger or smaller than the other, rather than exactly equal.
- \( < \) Less than
- \( > \) Greater than
- \( \le \) Less than or equal to
- \( \ge \) Greater than or equal to
Solving Inequalities
You solve these exactly like equations!
Example: \( 2x + 3 < 11 \)
Subtract \( 3 \): \( 2x < 8 \)
Divide by \( 2 \): \( x < 4 \)
Showing Answers on a Number Line
- Use an open circle \( (\circ) \) for \( < \) or \( > \) (this means the number itself is not included).
- Use a closed circle \( (\bullet) \) for \( \le \) or \( \ge \) (this means the number is included).
Golden Rule: If you multiply or divide an inequality by a negative number, you must flip the sign! (e.g., \( -2x < 10 \) becomes \( x > -5 \)).
6. Approximate Solutions by Iteration (Higher Tier Only)
Sometimes equations are too hard to solve exactly. Iteration is a way of "guessing, checking, and improving" using a formula over and over again to get closer to the real answer.
Imagine you are trying to find the perfect temperature for your shower. You turn the handle a bit, feel the water, and then adjust it again. Each adjustment is an iteration.
You will usually be given a starting value like \( x_0 = 2 \) and a formula like \( x_{n+1} = \sqrt{x_n + 5} \). You just keep plugging your answer back into the formula until the number stops changing much!
Quick Review:
1. Start with the given value.
2. Calculate the next value.
3. Use that new value to calculate the next one.
4. Stop when the question tells you to or when the decimal places stay the same.
Summary: Your Algebra Toolkit
- Balance: Whatever you do to the left, do to the right.
- Inverse: Use the opposite operation to move numbers.
- Quadratic: Look for two answers; use factorising or the formula.
- Inequalities: Treat like equations, but watch out for negative numbers!
- Graphs: The solution is where the lines cross or where they hit the x-axis.
You've got this! Practice makes perfect, so try a few simple equations today to get the hang of balancing those scales.