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Changing the subject of a formula


In this entry, we looked at manipulating equations for different pronumerals.  The cases we looked at were mostly linear in that the pronumeral we were changing for appeared as a single power of x, and the other terms were fairly simple to deal with as well.  

This set takes these ideas just a little step further as we look at manipulating equations that might also be quadratic of exponential in appearance.  


The subject of an equation is the pronumeral that is by itself on one side on the equals sign and it usually is at the start of the formula.

For example, in the formula $A=pb+y$A=pb+y, $A$A is the subject because it is by itself on the left hand side of the equals sign.

When we're changing the subject of a formula, we often have more than one pronumeral but we use a process similar to when we solve equations. 

  • Group any like terms
  • Simplify using the inverse of addition or subtraction.
  • Simplify further by using the inverse of multiplication or division.
  • Other inverse operations might be needed like square and square root



Question 1

Make $m$m the subject of the following equation:


Question 2

Rearrange the formula $r=\frac{k}{k-9}$r=kk9 to make $k$k the subject.

Further Examples

Question 3

The surface area of a sphere is given by the formula $A=4\pi r^2$A=4πr2. Make the radius, $r$r, the subject of the equation.

Question 4

The kinetic energy of an object is given by the formula $E=\frac{1}{2}mv^2$E=12mv2. Make the speed, $v$v, the subject of the equation.

(Note that speed is a positive quantity.)

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