Mathematics for science and technology
Mathematics for science and technology

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Mathematics for science and technology

3 Changing the subject of an equation

You may wish to rearrange an equation, to change its subject as some equations may have more than one algebraic symbol that is an unknown. The equation may be expressed in terms of x, say x = y + 4. You may wish to know what the equation would look like if it were expressed in terms of y.

The critical thing with any equation is that you must always carry out the same operations on each side of the equation. If, for example, you subtract x from one side of the equation but not from the other, it changes the equation completely. Since the quantities on each side of an equation are the same, anything done to one side must be done to the other to maintain the equality.

Say you wish to rearrange x = y + 3 so that y becomes the subject of the equation.

You can subtract 3 from both sides of the equation, and simplify.

multiline equation row 1 x equals y plus three row 2 x minus three equals y plus three minus three row 3 x minus three equals y

Which would then be written as:

y = x – 3

to follow the convention to show the subject on the left.

As another example, say you wish to rearrange x equals y divided by five minus two so that y becomes the subject of the equation.

First, multiply both sides by 5: 5x = y − 10

(Note everything on the right has been multiplied by 5 to maintain the equation)

Then add 10 to both sides: 5x + 10 = y

Finally re-write as:

y = 5x + 10

Now try some rearrangement exercises for yourself.

Activity 3 Rearranging equations

Timing: Allow about 10 minutes

In all cases, rearrange the equations to make x the subject of the equation.

  1. x – 5 = y
  2. 3x – 3 = y
  3. equation left hand side x divided by two equals right hand side three times y
  4. equation left hand side x divided by seven minus four equals right hand side two times y
  5. equation left hand side x divided by 2.5 equals right hand side 10 times y
  6. 2x – 12 = 2y

Answer

  1. x – 5 = y add 5 to both sides: x = y + 5
  2. 3x – 3 = y  add 3 to both sides, then divide by 3:  prefix times of multiline equation row 1 x equals open y plus three close divided by three postfix times equals prefix times of y divided by three plus one row 2
  3. equation left hand side x divided by two equals right hand side three times y  multiply both sides by 2:  x = 6y
  4. equation left hand side x divided by seven minus four equals right hand side two times y multiply both sides by 7: x – 28 = 14y  add 28 to both sides : x = 14y + 28

  5. equation left hand side x divided by 2.5 equals right hand side 10 times y multiply both sides by 2.5: x = 25y
  6. 2x – 12 = 2y add 12 to both sides : 2x = 2y + 12 divide both sides by 2 : x = y + 6

Practicing simplifying and rearranging equations and expressions gives you the skills you need to move on to solving them.

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