Welcome to the Solution of Equations!

In this chapter, we are going to learn how to find the "unknowns" in various mathematical situations. Think of an equation like a balanced set of scales: whatever is on the left side must equal whatever is on the right. Our job is to use mathematical "detective work" to figure out exactly what value the variables (like \(x\) or \(y\)) must have to keep that balance.

Solving equations is the backbone of almost everything in A Level Maths. Whether you are predicting the path of a rocket or calculating the interest on a bank account, you will always end up needing to solve an equation!

1. Linear Equations and Rearranging Formulas

Linear equations are the simplest type, where the unknown (usually \(x\)) isn't squared or cubed. They often look like this: \(3x + 5 = 11\).

Solving Linear Equations

To solve these, we use the "Reverse Operations" rule. Whatever is being done to \(x\), we do the opposite to both sides of the equation to "get it alone."

  • If it's added, subtract it.
  • If it's multiplied, divide by it.
  • If there are brackets, expand them first.
  • If there are fractions, multiply everything by the denominator to clear them.

Changing the Subject of a Formula

This is just like solving an equation, but instead of getting a number, you get a new formula. For example, making \(x\) the "subject" of \(y = mx + c\) means getting \(x = ...\)

Don't worry if this seems tricky at first: A common challenge is when the new subject appears on both sides of the formula.
Example: To make \(x\) the subject of \(ax + b = cx + d\):
1. Get all the \(x\) terms on one side: \(ax - cx = d - b\)
2. Factorise \(x\) out: \(x(a - c) = d - b\)
3. Divide: \(x = \frac{d - b}{a - c}\)

Quick Review: To change the subject, "move and group" the terms you want, factorise them, and then divide.

2. Solving Quadratic Equations

A Quadratic Equation is one where the highest power is a square, usually written as \(ax^2 + bx + c = 0\). There are three main ways to solve these:

Method A: Factorising

This involves putting the equation into two brackets, like \((x + 2)(x - 3) = 0\). If two things multiply to make zero, one of them must be zero! So, either \(x + 2 = 0\) or \(x - 3 = 0\).

Method B: The Quadratic Formula

This is your mathematical "safety net." If you can't factorise it, use this formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Common Mistake to Avoid: Make sure your equation is equal to zero before you pick out your \(a\), \(b\), and \(c\) values!

Method C: Completing the Square

We rewrite \(x^2 + bx + c\) as \((x + \frac{b}{2})^2 - (\frac{b}{2})^2 + c\). This is very useful for finding the turning point (the "peak" or "valley") of a graph.

Did you know? A quadratic equation can have two solutions, one solution, or even no real solutions at all. We check this using the Discriminant.

Key Takeaway: Factorising is usually fastest, but the Quadratic Formula works every time for any quadratic equation.

3. The Discriminant: Predicting the Roots

Before you even solve a quadratic, you can predict how many solutions (roots) it has by looking at the part of the formula inside the square root: \(b^2 - 4ac\). This is called the Discriminant.

  • If \(b^2 - 4ac > 0\): You have two distinct real roots (the graph crosses the x-axis twice).
  • If \(b^2 - 4ac = 0\): You have one repeated real root (the graph just touches the x-axis).
  • If \(b^2 - 4ac < 0\): You have no real roots (the graph floats above or below the x-axis).

Memory Aid: Think of the discriminant as a "Solution Radar." It tells you what's ahead before you do the heavy work of solving.

4. Simultaneous Equations

Sometimes we have two equations and two unknowns (like \(x\) and \(y\)). To solve them, we need to find where they "agree."

Linear Simultaneous Equations

You can use Elimination (adding or subtracting the equations to kill off one variable) or Substitution (replacing a variable in one equation with an expression from the other).

One Linear and One Quadratic

When you have one of each, Substitution is almost always the best way.
1. Rearrange the linear equation to get \(x = ...\) or \(y = ...\)
2. Plug that into the quadratic equation.
3. Solve the resulting quadratic to find your first variable.
4. Plug those answers back into the linear equation to find the second variable.

Important Point: If you are solving a linear and a quadratic, you will often get two pairs of answers. Don't forget to find the \(y\) value for every \(x\) value you find!

5. Intersection of Graphs

What does it actually mean to "solve" two equations? Graphically, the solutions are the coordinates of the points where the graphs intersect (cross).

Example: If you solve the equations for a line and a circle simultaneously, your answers for \(x\) and \(y\) are the exact spots where the line cuts through the circle.

Quick Review Box:
- Linear: Get \(x\) alone.
- Quadratic: Use \(ax^2 + bx + c = 0\).
- Discriminant: \(b^2 - 4ac\) tells you how many roots exist.
- Simultaneous: Substitution is your best friend for mixed equations.
- Graphs: The solution is the "crossing point."

You've got this! Algebra is just a set of rules. Once you know the "moves," solving equations becomes a satisfying puzzle rather than a chore.