3.6 Solving equations

Solving by inspection

Solving by inspection is the use of guessing to solve a maths problem.

For example, to solve x+7=11, to find x, if you think about it long enough, the answer might POP out at you as being x must be 4. You’ll notice you’ve worked this out by thinking “what added to 7 will give me 11”, so you start with 11 fingers (represented by the 1), and count down 7, obtaining 4. Essentially, to work out the answer, you have used x=11-7=4.

The opposite of subtract is addition.

The opposite of multiply is divide. For example, x*4=8, therefore x=8/4=2.

The opposite of square is square root. For example, if x^2=64, therefore x=\sqrt(64)=8.

Solving equations using algebra

Realizing what we’ve learnt by solving by inspection (which is not systematic enough to be used every time when solving for a question, since an answer rarely POPS out at you!), the steps to solve equations with algebra are (applying to the problem 4x-10=-x):

  1. Apply distributive law to remove brackets, if necessary
  2. Add numbers to (or subtract numbers from) both sides of the equation to simplify
    For example, 4x-10+\boldsymbol{10}=-x+\boldsymbol{10}, therefore 4x=-x+10
  3. Group variables (x) on one side and constants on the other
    For example, 4x+\boldsymbol{x}=-x+10+\boldsymbol{x}, therefore 5x=10
  4. Divide both sides through by numbers to solve for x
    For example,  through-dividing by 5, 5x/\boldsymbol{5}=10/\boldsymbol{5}, therefore x=2

Another example where the distributive law is first necessary is when solving 4(x+3)=28, in this example, you need to in-multiply 4 into the (x+3), giving you 4*x+4*3=4x+12. Then, you can solve it, with the above steps, to give you x=8.


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