Rearranging & Changing the Subject of Equations

What is Rearranging Equations?

Rearranging equations is like reorganizing your bookshelf: you want one specific variable (like $x$ or $y$) to stand out by being on its own. Rearranging doesn't solve the equation—it's about rewriting it so that a chosen variable becomes the "subject."

For example, rearrange $a = b + c$ to make $c$ the subject: $$c = a - b$$

 

General Steps for Rearranging

1. Identify the Variable to Isolate
   Decide which variable (the "subject") you want to isolate.

2. Undo Operations Step by Step
   Work backward using the inverse of operations:

   • Addition $\leftrightarrow$ Subtraction
   • Multiplication $\leftrightarrow$ Division
   • Squaring $\leftrightarrow$ Square Rooting
3. Keep the Equation Balanced
   Whatever you do to one side, do the same to the other.

 

Key Techniques

Addition and Subtraction

• Rearrange $a = b + c$ to make $c$ the subject: $$c = a - b$$

Multiplication and Division

• Rearrange $a = bc$ to make $c$ the subject: $$c = \frac{a}{b}$$

Square and Square Root

• Rearrange $a = x^2$ to make $x$ the subject: $$x = \pm \sqrt{a}$$

 

Rearranging with More Complexity

Example 1: Formula with Square Roots

Rearrange $a = \sqrt{x} + 5$ to make $x$ the subject.

Step 1: Subtract 5 from both sides: $$a - 5 = \sqrt{x}$$

Step 2: Square both sides to cancel the square root: $$x = (a - 5)^2$$

 

Example 2: Formula with Fractions

Rearrange $a = \frac{b}{c}$ to make $b$ the subject.

Step 1: Multiply both sides by $c$ to cancel the denominator: $$b = ac$$

 

Example 3: Rearranging with Squares

Rearrange $y = 3x^2 + 7$ to make $x$ the subject.

Step 1: Subtract 7 from both sides: $$y - 7 = 3x^2$$

Step 2: Divide by 3: $$x^2 = \frac{y - 7}{3}$$

Step 3: Take the square root (don’t forget the $\pm$): $$x = \pm \sqrt{\frac{y - 7}{3}}$$

 

 

 

 

Example 4: Nested Fractions

Rearrange $z = \frac{a}{b+c}$ to make $c$ the subject.

Step 1: Multiply both sides by $b + c$: $$z(b + c) = a$$

Step 2: Expand: $$zb + zc = a$$

Step 3: Subtract $zb$ from both sides: $$zc = a - zb$$

Step 4: Divide by $z$: $$c = \frac{a - zb}{z}$$