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Math help from the Learning Centre

This guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.

Formula Rearrangement

Why should I rearrange equations?

Believe it or not, there are many good reasons to develop your ability to rearrange equations that are important to the your field. It can save time, help you with units and save some brain space! Here are some reasons to develop your equation manipulation skills (in no particular order):

  • Equations are easier to handle before inserting numbers! And, if you can isolate a variable on one side of the equation, it is applicable to every similar problem that asks you to solve for that variable!
  • If you know how to manipulate equations, you only have to remember one equation that has all the variables of question in it - you can manipulate it to solve for any other variable! This means less memorization!
  • Manipulating equations can help you keep track of (or figure out) units on a number. Because units are defined by the equations, if you manipulate, plug in numbers and cancel units, you'll end up with exactly the right units (for a given variable)!

Rules for Rearranging Equations

  1. Apply the same operation to both sides of the equal sign. 
  2. Perform the inverse operation to move or cancel a constant or variable on one side of the equation.

Be aware of the order of operations! Which operation can you perform first?

Examples

Rearrange the following formula for the variable \( m_2 \) 

\[ \frac{K_1}{K_2} = \frac{m_1 + m_2}{m_1} \]

Solution:

Since the goal is to isolate for \( m_2 \), we want to move the variable \( m_1 \).

We need to first move the \( m_1 \) at the bottom of the fraction by multiplying both sides by \( m_1 \)

\[ \frac{K_1}{K_2} \times m_1 = \frac{m_1 + m_2}{\cancel{m_1}} \times \cancel{m_1} \]

This will move \( m_1 \) from the bottom of the right fraction to the other side of the equation

\[ \frac{K_1}{K_2} \times m_1 = m_1 + m_2 \]

Now move we move the remaining \( m_1 \) on the right side by subracting both sides by \( m_1 \) 

\[ \frac{K_1m_1}{K_2} - m_1 = \cancel{m_1} + m_2 - \cancel{m_1} \]

\[ \frac{K_1m_1}{K_2} - m_1 = m_2 \]

Rearrange the following equation for \(k\),

\[ \frac{4k^3}{m} - 3 = 10 + \frac{d}{2} \]

Watch the video for the solution.

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