Introduction to Simultaneous Equations

In algebra, we usually deal with one equation to find one mystery number, like \(x + 5 = 10\). But what happens when we have two mystery numbers (like \(x\) and \(y\)) and two different clues? These are called simultaneous equations because they must both be true at the same time.

Think of it like a detective story:
Clue 1: The sum of two numbers is 10.
Clue 2: One number is 4 more than the other.
By looking at both clues together, you can figure out the numbers are 7 and 3!

In this chapter, we will learn three ways to solve these: Substitution, Elimination, and Graphical methods. We will also look at what happens when one equation is a quadratic.

1. The Substitution Method

The substitution method is like a "Swap Shop." You use one equation to find an expression for one variable, and then "swap" it into the other equation.

When to use it:

This is easiest when one equation already starts with \(y = \dots\) or \(x = \dots\).

Step-by-Step Guide:

  1. Pick the simplest equation and rearrange it to get one variable on its own (e.g., \(y = \dots\)).
  2. Substitute this expression into the other equation.
  3. Solve this new equation to find the value of the first variable.
  4. Put that value back into your first equation to find the second variable.

Example: Solve \(y = 2x + 1\) and \(x + y = 7\).

Step 1: We already have \(y = 2x + 1\).
Step 2: Put this into the second equation: \(x + (2x + 1) = 7\).
Step 3: Simplify: \(3x + 1 = 7\) \(\implies 3x = 6\) \(\implies x = 2\).
Step 4: Find \(y\): \(y = 2(2) + 1 = 5\).
The solution is: \(x = 2, y = 5\).

Quick Tip: Always write your final answer as a pair of values!

2. The Elimination Method

This method involves "eliminating" (getting rid of) one variable by adding or subtracting the two equations together. It’s often the fastest method for linear equations.

The Golden Rule:

To eliminate a variable, the numbers (coefficients) in front of that variable must be the same size.

Step-by-Step Guide:

  1. Line up the equations so \(x\) and \(y\) are on the same side.
  2. If needed, multiply one or both equations by a number to make the coefficients of one variable match.
  3. Add or Subtract the equations:
    • If the signs are the Same, Subtract (SSS).
    • If the signs are Different, Add (DA).
  4. Solve for the remaining variable and then substitute back to find the other.

Example: Solve \(3x + 2y = 18\) and \(x - 2y = 2\).

Notice the \(y\) terms have the same number (2) but different signs (+2 and -2).
Step: Add the equations together.
\((3x + x) + (2y + -2y) = 18 + 2\)
\(4x = 20 \implies x = 5\).
Substitute back: \(5 - 2y = 2 \implies 3 = 2y \implies y = 1.5\).

Key Takeaway: If you find subtraction confusing, remember SSS: Same Sign Subtract!

3. Graphical Solutions

Every linear equation can be drawn as a straight line on a graph. The solution to a pair of simultaneous equations is simply the point where the two lines cross.

  • The \(x\)-coordinate of the crossing point is your \(x\) value.
  • The \(y\)-coordinate of the crossing point is your \(y\) value.

Did you know? If two lines are parallel, they will never cross. This means those simultaneous equations have no solution!

4. One Linear and One Quadratic Equation

Sometimes you will face a "mix" where one equation is a straight line (linear) and the other is a curve (quadratic, involving \(x^2\)).

Important: These usually have two pairs of solutions because a line can cross a curve in two places.

How to solve them:

You must almost always use Substitution for these.

  1. Rearrange the linear equation to get \(x = \dots\) or \(y = \dots\).
  2. Substitute this into the quadratic equation. This will give you a quadratic in one variable.
  3. Rearrange the quadratic to equal zero (\(ax^2 + bx + c = 0\)).
  4. Solve the quadratic (by factorising or using the quadratic formula).
  5. Find the corresponding \(y\) values for each \(x\) you found.

Example: Solve \(y = x^2 - 3\) and \(y = 2x\).

Step 1: Set them equal: \(x^2 - 3 = 2x\).
Step 2: Rearrange: \(x^2 - 2x - 3 = 0\).
Step 3: Factorise: \((x - 3)(x + 1) = 0\).
Step 4: Solutions for \(x\): \(x = 3\) or \(x = -1\).
Step 5: Find \(y\) for each:
If \(x = 3, y = 2(3) = 6\).
If \(x = -1, y = 2(-1) = -2\).
Final Answers: \((3, 6)\) and \((-1, -2)\).

Common Mistakes to Avoid

  • Forgetting the second half: Students often find \(x\) and get so excited they forget to find \(y\). Always find both!
  • Signs in subtraction: When subtracting equations in the elimination method, be very careful with negative numbers. Example: \(5 - (-2) = 7\), not 3.
  • Mixing up pairs: When solving linear/quadratic equations, make sure you pair the correct \(x\) with the correct \(y\).

Quick Review

Linear + Linear: Use Elimination or Substitution. You get one set of coordinates \((x, y)\).

Linear + Quadratic: Always use Substitution. You usually get two sets of coordinates.

Graphical: Look for where the lines intersect.

Note: For more practice on solving the quadratic part of these equations, see the chapter on "Linear, quadratic and cubic equations".