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 $x=8$.