Mathematics (Specification A) | Simultaneous linear equations

Welcome to Simultaneous Linear Equations!

Hello! In this chapter, we are tackling Simultaneous Linear Equations. Don't worry if the name sounds complicated; it just means we are solving two separate line equations at the same time to find the single point where they cross. Think of it like a detective mission where you have two clues, and you need both clues to identify the single, correct answer for two missing variables (usually \(x\) and \(y\)).

This skill is fundamental in algebra and is used extensively in fields like economics, engineering, and physics to model real-world problems where multiple conditions must be met simultaneously. Let's get started!

What Exactly is a Simultaneous Linear Equation?

A linear equation is an equation that produces a straight line when graphed (like \(y = 2x + 1\)). When you have simultaneous linear equations, you have two (or more) of these equations, and they share two unknown variables (like \(x\) and \(y\)).

• The goal is to find the unique pair of values for \(x\) and \(y\) that satisfies both equations.
• Visually, this solution is the single point where the two straight lines intersect on a graph.

Example Set:
Equation 1: \(x + y = 7\)
Equation 2: \(2x - y = 5\)

Did you know? If the two lines were parallel, they would never intersect, and there would be no solution. If they were the exact same line, there would be infinite solutions! But for IGCSE, we focus primarily on pairs that have a single, unique solution.

Method 1: The Elimination Method

The Elimination Method is often the quickest way to solve simultaneous equations, especially when the equations are set up neatly (like \(ax + by = c\)). The core idea is to manipulate the equations so that when you add or subtract them, one of the variables disappears (is eliminated).

Step-by-Step Guide: Elimination

Step 1: Check the Coefficients
Look at the numbers in front of \(x\) and \(y\) (these are the coefficients). Do any pairs of coefficients already match, or can they be easily made to match?

Step 2: Make Coefficients Match (Scaling)
If no coefficients match, multiply one or both equations entirely by a factor (number) so that the coefficients of one variable become the same size.

Example: To make the \(x\) coefficients match in the system below, we multiply Equation (1) by 2:
(1) \(3x + 4y = 10\)
(2) \(6x - 2y = 4\)

Step 3: Decide to Add or Subtract (The Key Trick!)
Now that one variable has matching coefficients, you must eliminate it by either adding or subtracting the equations.

Memory Aid (SSS DSA):

• Same Sign? Subtract. (If both are \(+4y\) and \(+4y\), you subtract.)
• Different Sign? Add. (If you have \(+4y\) and \(-4y\), you add.)

Step 4: Solve for the First Variable
After eliminating one variable, you are left with a simple equation containing only the other variable. Solve this equation.

Step 5: Find the Second Variable
Substitute the value you just found (from Step 4) back into one of the original, simple equations (Equation 1 or 2). Solve to find the value of the second variable.

Step 6: Check Your Answer
Substitute both values (\(x\) and \(y\)) into the other original equation to make sure they work for both equations simultaneously.

Example of Elimination (Scaling Needed)

Solve the system:
(1) \(2x + y = 8\)
(2) \(3x - 2y = 5\)

1. We need to make the \(y\) coefficients match. Multiply Equation (1) by 2:
   New (1): \(2 \times (2x + y = 8) \Rightarrow 4x + 2y = 16\)
2. We now have \((+2y)\) in the new (1) and \((-2y)\) in (2). They have Different Signs, so we Add the equations:
   \((4x + 2y) + (3x - 2y) = 16 + 5\)
   \(7x + 0y = 21\)
   \(7x = 21\)
   \(x = 3\)
3. Substitute \(x = 3\) into Equation (1):
   \(2(3) + y = 8\)
   \(6 + y = 8\)
   \(y = 2\)
4. Solution: \(x = 3\) and \(y = 2\).
5. Check: Substitute into (2): \(3(3) - 2(2) = 9 - 4 = 5\). (Correct!)

Quick Review: The Elimination Method works by lining up the terms and using addition or subtraction to cancel out one of the variables.

Method 2: The Substitution Method

The Substitution Method is fantastic when one of the variables in one equation is already easy to isolate (get by itself), or if one variable has a coefficient of 1. You solve for one variable and literally 'swap it in' (substitute it) into the second equation.

Analogy: Imagine you know "A is worth 3 Bs." If you see A in another formula, you can just replace it with "3 Bs" to simplify the whole formula.

Step-by-Step Guide: Substitution

Step 1: Isolate a Variable
Choose the easiest equation and rearrange it to make one variable the subject (e.g., \(x = \dots\) or \(y = \dots\)). Aim to avoid fractions if possible!

Step 2: Substitute
Take the expression you found in Step 1 and substitute it into the other original equation. This results in a single equation with only one variable.

Step 3: Solve for the First Variable
Solve this new, simpler equation.

Step 4: Find the Second Variable
Substitute the value found in Step 3 back into the rearranged equation from Step 1. This is usually the quickest way to find the second value.

Step 5: Check Your Answer
Always substitute both values into the second original equation to confirm the solution.

Example of Substitution

Solve the system:
(1) \(y = x + 3\)
(2) \(2x + 3y = 19\)

1. Equation (1) is already perfectly rearranged: \(y = x + 3\).
2. Substitute the expression \((x + 3)\) for \(y\) into Equation (2):
   \(2x + 3 \mathbf{(x + 3)} = 19\)
3. Expand and solve for \(x\):
   \(2x + 3x + 9 = 19\)
   \(5x + 9 = 19\)
   \(5x = 10\)
   \(x = 2\)
4. Substitute \(x = 2\) back into Equation (1) (the easiest option):
   \(y = (2) + 3\)
   \(y = 5\)
5. Solution: \(x = 2\) and \(y = 5\).
6. Check: Substitute into (2): \(2(2) + 3(5) = 4 + 15 = 19\). (Correct!)

A Crucial Note: When substituting an entire expression, use brackets (parentheses) immediately. This reminds you to multiply *all* terms inside the expression, preventing a very common distribution mistake.

Choosing the Best Method

While both methods will always yield the correct answer, choosing the right method can save you time and prevent errors.

Use Substitution When:

• One variable already has a coefficient of 1 or -1 (e.g., \(x - 2y = 5\)).
• One equation is already rearranged (e.g., \(y = 4x\)).

Use Elimination When:

• Both equations are set up in the \(Ax + By = C\) format.
• The coefficients of one variable are easy to match by multiplying by a small integer.

Don't worry if this seems tricky at first—practice identifying the variables, deciding on your method, and carefully tracking your signs. You've got this!

Key Takeaways: Simultaneous Linear Equations

This section taught you how to find the single pair of numbers (\(x\) and \(y\)) that satisfies two linear equations at once.

• Elimination: Line up the equations, make one variable's coefficients match by scaling, then add or subtract based on the signs (SSS DSA).
• Substitution: Isolate a variable in one equation, substitute the resulting expression into the other equation, and solve.
• Always Check: Your final step must be to verify your solution pair in the second original equation.