Welcome to Algebra: Solving Equations and Inequalities!

Hello future Mathematicians! This chapter is the backbone of Algebra. Don't worry if equations sometimes look scary—they are really just puzzles! By the end of this section, you will be a master at finding the hidden values that make mathematical statements true.

Why is this important? Equations and inequalities help us model real-world situations, from calculating costs in business to predicting trajectories in physics. Mastering these skills will unlock many doors in your future studies!

1. Solving Linear Equations

What is a Linear Equation?

A linear equation is the simplest type of equation. It involves a variable (like \(x\)) raised only to the power of one, and it results in a straight line when graphed. Our goal is to isolate the variable.

The Balancing Scales Rule

Think of an equation as a perfectly balanced set of scales. Whatever you do to one side of the equation, you must do to the other side to keep it balanced. This is done using inverse operations (the "undo" button!).

  • Adding undoes Subtracting.
  • Multiplying undoes Dividing.

Step-by-Step Example: Two-Step Equation

Solve: \( 4x - 7 = 13 \)

  1. Handle the addition/subtraction first (The lonely number):
    We need to get rid of the \(-7\). Do the opposite: Add 7 to both sides.
    \( 4x - 7 + 7 = 13 + 7 \)
    \( 4x = 20 \)
  2. Handle the multiplication/division (The number stuck to the variable):
    \( 4x \) means 4 multiplied by \(x\). Do the opposite: Divide both sides by 4.
    \( \frac{4x}{4} = \frac{20}{4} \)
    \( x = 5 \)

Quick Review: Always aim to get the variable term alone first, then divide or multiply to find the final value of \(x\).


2. Solving Simultaneous Linear Equations

What are Simultaneous Equations?

Simultaneous equations are a set of two (or more) equations involving the same variables (usually \(x\) and \(y\)). You are looking for one specific pair of \(x\) and \(y\) values that works perfectly in both equations. Graphically, this is the point where the two lines intersect.

We have two main methods for solving these:

Method A: Elimination (The Cancelling Method)

This method is great when the coefficients (the numbers in front) of one variable are the same or easily made the same. Our goal is to eliminate one variable by adding or subtracting the equations.

Example:

Equation (1): \( 2x + 3y = 13 \)
Equation (2): \( 2x + y = 7 \)

  1. Check coefficients: Notice that the \(x\) coefficients are both 2.
  2. Eliminate: Since the signs of the \(x\) terms are the same (both positive), we subtract the equations.
    (1) - (2): \((2x - 2x) + (3y - y) = (13 - 7)\)
    \( 0x + 2y = 6 \)
  3. Solve for the remaining variable:
    \( 2y = 6 \implies y = 3 \)
  4. Substitute back: Put \(y=3\) into either original equation (Equation 2 is simpler):
    \( 2x + (3) = 7 \)
    \( 2x = 4 \implies x = 2 \)

Solution: \( x=2, y=3 \). (Always check your answer by substituting both values into the other equation.)

Method B: Substitution (The Swapping Method)

This method is perfect when one variable is already isolated or very easy to isolate (i.e., it has a coefficient of 1).

Example:

Equation (1): \( y = 2x - 1 \)
Equation (2): \( 3x + y = 9 \)

  1. Isolate (if necessary): \(y\) is already isolated in (1).
  2. Substitute: Replace the \(y\) in Equation (2) with the expression from Equation (1).
    \( 3x + (2x - 1) = 9 \)
  3. Solve for \(x\): This is now a simple linear equation!
    \( 5x - 1 = 9 \)
    \( 5x = 10 \implies x = 2 \)
  4. Substitute back: Put \(x=2\) into Equation (1):
    \( y = 2(2) - 1 \)
    \( y = 4 - 1 \implies y = 3 \)

Key Takeaway for Simultaneous Equations: You MUST find values for all variables. Don't stop after finding just \(x\) or just \(y\)!

3. Solving Quadratic Equations

What is a Quadratic Equation?

A quadratic equation contains a variable squared (\(x^2\)). When graphed, it forms a parabola (a U-shape). Crucially, quadratic equations usually have two solutions (or roots).

The standard form is: \( ax^2 + bx + c = 0 \), where \(a\) cannot be zero.

Method A: Solving by Factorising

This method relies on the principle that if two things multiply together to give zero, then one (or both) of them must be zero. If \((x+p)(x+q) = 0\), then either \(x+p=0\) or \(x+q=0\).

Step-by-Step Example:

Solve: \( x^2 + 5x + 6 = 0 \)

  1. Ensure it equals zero: (It already does.)
  2. Factorise: Find two numbers that multiply to give \(c\) (6) and add to give \(b\) (5).
    (The numbers are 2 and 3).
    \( (x + 2)(x + 3) = 0 \)
  3. Set each factor equal to zero:
    \( x + 2 = 0 \implies x = -2 \)
    \( x + 3 = 0 \implies x = -3 \)

Solutions: \( x = -2 \) and \( x = -3 \).

Common Mistake: Forgetting to flip the signs! If the factor is \((x+2)\), the solution is \(x=-2\).

Method B: Using the Quadratic Formula (The Superhero Method!)

The quadratic formula works for every single quadratic equation, even if it can't be factorised easily.

The Formula:

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

Steps:

  1. Identify a, b, and c: Make sure your equation is in the form \( ax^2 + bx + c = 0 \).

    Example: \( 2x^2 - 5x - 3 = 0 \)

    \( a = 2 \), \( b = -5 \) (Watch the negative sign!), \( c = -3 \).

  2. Substitute into the formula:

    $$ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-3)}}{2(2)} $$

  3. Simplify carefully: Pay extra attention to the part under the square root, called the discriminant.

    \( x = \frac{5 \pm \sqrt{25 - (-24)}}{4} \)
    \( x = \frac{5 \pm \sqrt{49}}{4} \)

  4. Find the two solutions (\(+\) and \(-\)):

    Solution 1 (using +): \( x = \frac{5 + 7}{4} = \frac{12}{4} = 3 \)
    Solution 2 (using -): \( x = \frac{5 - 7}{4} = \frac{-2}{4} = -0.5 \)

Key Takeaway for Quadratics: Always start by rearranging the equation so that one side equals zero.

4. Solving Linear Inequalities

What is an Inequality?

An inequality is similar to an equation, but instead of finding a single value, we find a range of values that satisfy the condition.

The Four Key Symbols:

  • \( < \) : Less than (e.g., \(x\) is smaller than 5)
  • \( > \) : Greater than (e.g., \(x\) is larger than 5)
  • \( \le \) : Less than or equal to
  • \( \ge \) : Greater than or equal to
Solving Inequalities

You solve inequalities almost exactly the same way you solve linear equations: using inverse operations and the balancing scales rule.

Example: Solve \( 5x - 3 \ge 12 \)

  1. Add 3 to both sides: \( 5x \ge 15 \)
  2. Divide both sides by 5: \( x \ge 3 \)
The ONE Crucial Difference: Reversing the Sign!

This is the most important rule in solving inequalities, and it is where students often make mistakes.

RULE: If you multiply or divide both sides of an inequality by a negative number, you MUST reverse the direction of the inequality sign.

Example: Solve \( -2x + 1 < 9 \)

  1. Subtract 1 from both sides: \( -2x < 8 \)
  2. Divide both sides by -2. (Since we are dividing by a negative number, we must reverse the sign!)
    \( \frac{-2x}{-2} > \frac{8}{-2} \)
    \( x > -4 \)
Representing Inequalities on a Number Line

A number line is a visual way to show the range of values that satisfy the inequality.

  • Open Circle (Hollow): Used for \( < \) or \( > \). This means the number itself is not included.
  • Closed Circle (Solid): Used for \( \le \) or \( \ge \). This means the number itself is included.

Example: \( x \le 5 \). We draw a closed circle at 5 and draw the arrow pointing to the left (the "less than" direction).

Did You Know? Inequalities are used constantly in computing. Every time a program checks if a number is "too high" or "too low," it's using an inequality!

Key Takeaway for Inequalities: Treat them like equations, but remember the sign reversal rule when multiplying or dividing by a negative.

Chapter Summary: Your Toolkit

You have learned four powerful techniques for solving mathematical puzzles:

  • Linear Equations: Use inverse operations to isolate the variable. (The Balancing Act).
  • Simultaneous Equations: Use Elimination or Substitution to find the single point \((x, y)\) where both conditions are met.
  • Quadratic Equations: Solve by Factorising or using the reliable Quadratic Formula. Remember: usually two solutions!
  • Inequalities: Solve like linear equations, but be very careful to reverse the inequality sign if you multiply or divide by a negative number.

Keep practising these steps! Solving equations is all about consistency and careful application of the rules. You've got this!