Fundamentals of Algebra (Leaving Cert Mathematics): Revision Notes

📚 Revision Notes
Revision NotesModel AnswersQuizzesFlashcards

Changing the Subject of a Formula

What is the subject of a formula?

When you look at a formula like x=2y−zx = 2y - z, the variable on the left-hand side (in this case, xx) is called the subject of the formula. This means that xx is expressed in terms of the other variables yy and zz.

If you rearrange this same formula to get z=2y−xz = 2y - x, then zz becomes the new subject of the formula. The process of rearranging a formula so that a different variable appears alone on the left-hand side is called changing the subject of a formula.

infoNote

Understanding the subject of a formula is crucial because it tells you which variable is being calculated in terms of the others. In real-world applications, you might need to rearrange formulas to solve for different variables depending on what information you have available.

The key principle

The most important rule to remember when changing the subject of a formula is this:

chatImportant

An equation remains unchanged if the same operation is performed on both sides.

This means you can add, subtract, multiply, or divide both sides of an equation by the same amount without changing the relationship between the variables. This principle is exactly the same as when you solve regular equations.

Step-by-step method

To change the subject of a formula, follow these steps:

  1. Identify the variable you want to make the subject
  2. Isolate this variable by undoing the operations around it
  3. Perform the same operation on both sides of the equation
  4. Work in reverse order - undo the last operation first
  5. Check your final answer makes sense

Think of it like unwrapping a present - you need to remove the outer layers first to get to what's inside.

Worked examples

lightbulbExample

Worked Example 1: Simple Rearrangement

Make cc the subject of the formula: bc−d=abc - d = a

Starting with: bc−d=abc - d = a

Step 1: Add dd to both sides to isolate the term with cc bc=a+dbc = a + d

Step 2: Divide both sides by bb to get cc on its own c=a+dbc = \frac{a + d}{b}

Final answer: c=a+dbc = \frac{a + d}{b}

lightbulbExample

Worked Example 2: Working with Fractions

Make yy the subject of the formula: x=3y2−1x = \frac{3y}{2} - 1

Starting with: x=3y2−1x = \frac{3y}{2} - 1

Step 1: Add 11 to both sides x+1=3y2x + 1 = \frac{3y}{2}

Step 2: Multiply all terms by 22 to clear the fraction 2(x+1)=3y2(x + 1) = 3y 2x+2=3y2x + 2 = 3y

Step 3: Divide both sides by 33 y=2x+23y = \frac{2x + 2}{3} or y=2(x+1)3y = \frac{2(x + 1)}{3}

Final answer: y=2(x+1)3y = \frac{2(x + 1)}{3}

lightbulbExample

Worked Example 3: Complex Algebraic Expressions

Make cc the subject of the formula: a=bcb+ca = \frac{bc}{b + c}

Starting with: a=bcb+ca = \frac{bc}{b + c}

Step 1: Multiply both sides by (b+c)(b + c) to clear the fraction a(b+c)=bca(b + c) = bc

Step 2: Expand the brackets ab+ac=bcab + ac = bc

Step 3: Rearrange to collect terms with cc ac−bc=−abac - bc = -ab

Step 4: Factor out cc on the left side c(a−b)=−abc(a - b) = -ab

Step 5: Divide both sides by (a−b)(a - b) c=−aba−bc = \frac{-ab}{a - b}

Final answer: c=−aba−bc = \frac{-ab}{a - b}

Common techniques and exam tips

When tackling formula rearrangement problems, there are several strategies that will help you work more efficiently and avoid common pitfalls:

  • Work backwards: Think about what operations have been applied to your target variable, then undo them in reverse order
  • Clear fractions early: Multiply through by denominators to make the algebra simpler
  • Factor when possible: Look for opportunities to factor out your target variable
  • Check your work: Substitute your answer back into the original equation to verify it's correct
  • Be careful with signs: Pay close attention to positive and negative signs, especially when moving terms across the equals sign
chatImportant

Common Exam Trap: Don't forget to change the sign when moving terms from one side of the equation to the other. This is one of the most frequent mistakes students make!

bookmarkSummary

Key Points to Remember:

  • The subject of a formula is the variable that appears alone on the left-hand side
  • You can perform the same operation on both sides without changing the equation
  • Work in reverse order - undo the last operation that was applied to your target variable first
  • Clear fractions by multiplying through by the denominator
  • Always check your answer by substituting back into the original formula

Explore Leaving Cert Mathematics Model Answers by Topics

Fundamentals of Algebra

Explore Leaving Cert Mathematics Quizzes by Topics

Fundamentals of Algebra

Explore Leaving Cert Mathematics Flashcards by Topics

Fundamentals of Algebra

Join 100,000+ Leaving Cert students studying Revision Notes with us.

Select your subjects, and get access to A+ resources today.