Solution of equations

Introduction: Finding the Missing Piece

Welcome! In this chapter, we are going to master the art of the solution of equations. At its heart, solving an equation is just like solving a mystery or a puzzle. You are given a set of clues (the equation), and your job is to find the value of the "unknown" (usually \(x\)) that makes the statement true.

Being able to solve equations is a fundamental skill in Pure Mathematics: Algebra. It’s the "engine room" of math—once you can solve equations confidently, you can tackle everything from engineering problems to financial modeling. Don't worry if it seems a bit abstract at first; we will break every process down into simple, logical steps!

1. Linear Equations: Keeping the Balance

A linear equation is one where the unknown (like \(x\)) isn't squared or cubed. It’s a "straight-line" equation. Think of a linear equation like a pair of traditional weighing scales. To keep the scales level, whatever you do to one side, you must do to the other.

Solving with Brackets and Fractions

Sometimes equations look messy because they have brackets or fractions. Here is your game plan:

1. Expand Brackets: Multiply out any terms inside parentheses.
2. Clear Fractions: Multiply the entire equation by the denominator to get rid of fractions.
3. Collect Terms: Get all the \(x\) terms on one side and all the numbers on the other.
4. Solve: Divide to find the final value of \(x\).

Example: Solve \( \frac{2(x + 3)}{3} = 4 \)

Step 1: Multiply both sides by 3 to clear the fraction: \( 2(x + 3) = 12 \)
Step 2: Expand the brackets: \( 2x + 6 = 12 \)
Step 3: Subtract 6 from both sides: \( 2x = 6 \)
Step 4: Divide by 2: \( x = 3 \)

Quick Review: Always aim to "undo" the operations in reverse order (BIDMAS backwards) to isolate your unknown.

2. Rearranging Formulae: Moving the Furniture

Sometimes you aren't looking for a number, but you want to make a different letter the subject of the formula. This is exactly like solving an equation, but with more letters!

When the Subject Appears Twice

If you want to make \(x\) the subject, but \(x\) is on both sides of the equation, follow this trick: Factorise!

Example: Make \(x\) the subject of \( ax - b = cx + d \)

1. Get all terms with \(x\) on one side: \( ax - cx = d + b \)
2. Factorise \(x\) out: \( x(a - c) = d + b \)
3. Divide by the bracket: \( x = \frac{d + b}{a - c} \)

Common Mistake: Forgetting to flip the sign when moving a term across the equals sign. Remember: "Change side, change sign!"

Key Takeaway: If the letter you want is stuck in two places, get them together on one side and factorise it out.

3. Quadratic Equations: The Power of Two

A quadratic equation involves an \(x^2\) term and usually looks like \( ax^2 + bx + c = 0 \). Because of the \(x^2\), these equations can have two different solutions (roots).

Methods of Solving

1. Factorising: The "brackets" method. Great if the numbers are simple. Look for two numbers that multiply to give \(c\) and add to give \(b\).
2. The Quadratic Formula: Your "safety net" for when factorising is too hard:
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
3. Completing the Square: Rewriting the equation in the form \( a(x+p)^2 + q = 0 \). This is very helpful for finding the turning point of a graph later on!
4. Graphically: Looking at where the curve \( y = ax^2 + bx + c \) crosses the x-axis (\(y = 0\)).

Did you know?

The word "Algebra" comes from the Arabic word "al-jabr," meaning "the reunion of broken parts," coined by the mathematician Al-Khwarizmi who developed many of these methods!

4. The Discriminant: The Root Detective

How do we know how many answers a quadratic has without solving the whole thing? We use the discriminant, which is the part inside the square root of the quadratic formula: \( b^2 - 4ac \).

• If \( b^2 - 4ac > 0 \): There are two distinct real roots (the graph crosses the x-axis twice).
• If \( b^2 - 4ac = 0 \): There is one repeated real root (the graph just touches the x-axis).
• If \( b^2 - 4ac < 0 \): There are no real roots (the graph never touches the x-axis).

Memory Aid: "Positive is Plenty (2), Zero is Just One (1), Negative is None (0)."

5. Simultaneous Equations: Crossing Paths

Simultaneous equations are a set of equations that are true at the same time. Graphically, the solution is the point of intersection where the two lines or curves cross.

Linear and Linear

You can use Substitution (replace one variable with an expression) or Elimination (add or subtract equations to "knock out" one variable).

Linear and Quadratic

When you have one linear and one quadratic equation, Substitution is almost always the best way.
Step-by-step:
1. Rearrange the linear equation to get "\(y = ...\)" or "\(x = ...\)".
2. Substitute this into the quadratic equation.
3. Solve the resulting quadratic.
4. Don't forget: Substitute your answers back into the linear equation to find the corresponding values for the second variable!

Quick Review: A linear line and a quadratic curve could cross twice, once (tangent), or not at all!

6. Summary and Final Tips

Solving equations is the foundation of your AS Level journey. Here are the core things to remember:

• Balance: Whatever you do to one side, you must do to the other.
• The Subject: If the subject appears twice in a formula, use factorisation to isolate it.
• Quadratic Check: Always check the discriminant \( b^2 - 4ac \) if you need to know the number of solutions.
• Intersection: Solving simultaneous equations is exactly the same as finding where two graphs cross.

Final Encouragement: Algebra can feel like a lot of steps, but it’s very logical. If you get stuck, go back to the previous line and check your signs—that's where most mistakes hide! You've got this!