Expert: Josh - 12/21/2007

Question
Hi i use to be really good at math but i have been slacking on my studies and im worried i wont mount to anything so yea i need your help in math...im dealing wid questions like 2m plus 12 equals 3m minus 31....HELP!
lol im so confused and please help fast!

Answer
No worries. You use the same tools you have previously learned. When you have an equation like 2m+12=3m-31, take a moment to think about what it really means?

UNDERSTANDING THE PURPOSE OF SOLVING EQUATIONS

First of all, what is a mathematical equation?
Well, for a given problem, it basically describes two underlying conditions which amount to the same thing.

Example: Paul bought two ipods (let's call this x) and two Wii (let's call this y) for Christmas, and it costed him $700. Sarah bought 3 ipods and one Wii from the same shop and it costed her $450. How much does each item cost?

This problem is translated into mathematical equations drawing from two different scenarios. For paul, 2x+2y=700. For Sarah, 3x+y=450. The task in this problem is to work out x (the cost of each ipod) and y (the cost of wii). Anyway, we'll return to this problem later...

The word "equation" alludes to the fact that the left hand side expression is equal to the right hand side expression.

Next, identify the "known"'s and "unknown"'s in your question. Obviously, the alphabets such as "x", "y", or "m" in your example, represent the unknown quantity that we need to determine.

What are the obstacles in finding "m" when we look at something like 2m+12=3m-31?

We have the unknown variable "m" on both sides of the equation. Our objective is to ultimately turn it into m=....

Arithmetic operations (any combination of addition, subtraction, multiplication and division) provide the path to reaching this end goal. REMEMBER THIS.

In going from 2m+12=3m-31 to m=something, we are NOT allowed to bend the truth. At all times equality must hold. So, whatever we add (or subtract) from one side of the equation, we must do the SAME to the other side of the equation. The same applies to multiplication and division.

Beginning with 2m+12=3m-31, we can subtract 2m from both sides of the equation. The motivation is to isolate "m" on one side of the equation. Doing this, we get
2m+12 - 2m = 3m-31-2m, normally we don't write this, i'm doing this just to make it clear what steps we are taking. Simplifying, we get 12 = m-31. We are now one step away from getting m=something. Adding 31 to both sides of the equation, we get m=43. We have determined the unknown value of m!

Returning to the example i gave earlier, we have two unknowns "x" and "y". We need to solve two independent equations simultaneously.

Let's call 2x+2y=700 Equation [1]
Let's call 3x+y=450  Equation [2]

We are allowed to multiply Equation [2] by 2 on BOTH sides without bending the truth. This gives 6x+2y=450...call this Equation [3].

We now have
2x+2y=700 from [1] and
6x+2y=900 from [3].

We can work out x, or y one at a time by eliminating the other unknown from the equation. Observe that we have 2y from [1] and [3], we don't bend the truth if we subtract the left hand side of Equation [1] from the left hand side of Equation [3]; and likewise, doing the same to the right hand side of Equation [1] and [3].

[3] - [1] produces
6x+2y-(2x+2y)=900-700
Simplify this,
4x = 200,
dividing both sides by 4
x = 50.

So, the ipod costed $50.
It's then a simple matter to find "y", the cost of Wii.
From equation [1], [2] or [3], we substitute the value of "x=50" and work out y.

e.g., putting x=50 into 2x+2y=700, we get 100+2y=700, subtracting 200 from both sides gives 2y=600. Finally, dividing both sides by 2 gives y=300. This means the Wii costs $300.

Obviously, i just made up the prices on the way....but hopefully you get the idea.