Changing the Subject 1 (where Subject Appears Once)

Introduction

This tutorial will guide you through the process of changing the subject of a formula where the subject appears only once. This skill is essential in algebra, particularly when working with equations in different contexts, such as converting between temperature scales or solving for variables in mathematical formulas.

Step 1: Understanding the Subject of a Formula

• The subject of a formula is the variable that is isolated on one side of the equation.
• For example, in the formula for converting Celsius to Fahrenheit, Fahrenheit (F) is the subject: [ F = \frac{9}{5}C + 32 ]
• When changing the subject, our goal is to isolate a different variable (e.g., Celsius, C) in terms of Fahrenheit (F).

Step 2: Rearranging Simple Formulas

1. Example: Change ( y = 3x + 1 ) to make ( x ) the subject.
   • Subtract 1 from both sides: [ y - 1 = 3x ]
   • Divide both sides by 3: [ x = \frac{y - 1}{3} ]

Step 3: Handling Brackets in Formulas

1. Example: Change ( y = 3(x + 1) ) to make ( x ) the subject.
   • Expand the equation: [ y = 3x + 3 ]
   • Subtract 3 from both sides: [ y - 3 = 3x ]
   • Divide by 3: [ x = \frac{y - 3}{3} ]

Step 4: Dealing with Fractions

1. Example: Change ( y = \frac{1}{3}x - 1 ) to make ( x ) the subject.
   • Multiply both sides by 3: [ 3y = x - 3 ]
   • Add 3 to both sides: [ x = 3y + 3 ]

Step 5: Using Square Roots

1. Example: Change ( y = 3x^2 ) to make ( x ) the subject.
   • Divide by 3: [ \frac{y}{3} = x^2 ]
   • Take the square root of both sides: [ x = \sqrt{\frac{y}{3}} \quad \text{(consider both positive and negative roots)} ]

Step 6: More Complex Rearrangements

1. Example: Change ( y = 3\sqrt{x} + 1 ) to make ( x ) the subject.
   • Subtract 1: [ y - 1 = 3\sqrt{x} ]
   • Divide by 3: [ \frac{y - 1}{3} = \sqrt{x} ]
   • Square both sides: [ x = \left(\frac{y - 1}{3}\right)^2 ]

Step 7: Swapping Terms in Subtraction and Division

1. Example: Change ( y = a - b/x ) to make ( x ) the subject.
   • Use the swap trick: [ b/x = a - y ]
   • Rearranging gives: [ x = \frac{b}{a - y} ]

Conclusion

In this tutorial, you learned how to change the subject of formulas through various methods, including basic algebraic manipulation, handling brackets, working with square roots, and using the swapping technique for subtraction and division. Mastering these techniques will improve your problem-solving skills in algebra and practical applications. Next steps could include practicing with more complex formulas or exploring applications in physics or finance.