Building Speed with Equations

Algebra 1 Tutorial

Building Speed with Equations

Intro

There are lots of formal steps that we learn for working with algebraic equations, but after you’ve learned them, there are quicker ways to go about getting an equation in a form you want.

Sample Problem

Solve the following equation for z:

x + b / (z – 23) = a + 125

Does look difficult? Well, it isn’t really, using the techniques described here!

Solution

The variety of techniques for quickly solving such an equation are presented below.

Adding and Subtracting:

Example: Simplifying an equation in order to move a term from one side to the other by ‘adding or subtracting the term to both sides of the equation’. Been there, done that! Hey, just move it and change the sign! This is simple and direct, saving time and brain power!

Formerly,

          x + a = b   ;   x + a - a = b - a  and cancelling,  x = b - a

Instead of doing that, just move the term to the other side of the equation and change the sign:

          x + a = b     or     x = b - a

alternately,

          x - n = m     or     x = m + n

Multiplying:

For multiplication, you change the ‘vertical’ location of the term instead (from numerator to denominator or vice-versa) upon moving a factor from one side of the equation to the other:

         a * b = c     or     a = c / b

Actually, you’ve changed the power, or sign of the exponent of b from ^+1 to ^-1, so more descriptively:

         a * b^1 = c     or     a = c * b^-1 = c / b

[You will learn to do these things by ‘inspection’, but if your teacher balks and insists that you ‘show your work’, or you’re presenting your procedure at a blackboard, you can always revert to the formal procedure:

         a * b = c

dividing both sides of the equation by b,

	(a * b) / b = c / b

And canceling the b’s in the numerator and denominator on the left side of the equation (together they = 1), we get:

         a = c / b

What a waste of time!]

But don’t even think about that if you’re working on your own – just move the term as described.

So…

         x * a = y / c     or     x = y / (c * a)

or…

         x / z = y / n     or     x * n = y * z

The possibilities are endless. But BE CAREFUL. Everything you move in such a manner must be a factor of the entire side of the equation. So, this move:

         1 + x = y * b

         1 = (y * b) / x is WRONG, because x is not a factor of the whole left side of the equation!

Other Operations:

In general, since an equation is, and always must, be balanced, i.e. both sides are equal in value, both sides can be added to, subtracted from, multiplied or divided by the same number or even raised to the same power without unbalancing the equation.

Eventually, you will be able to do this by inspection – it just takes some eye-balling and imagining the terms in a new structure.

Example:

Solve for x:

         123 + z = 3.25 / √x

Answer:

         x = ((3.25 / (123 + z))^2

Here, we mentally exchanged the term (123 + z) with √x and then squared both sides.

Here’s another to work on:

Solve for each term alone in a separate equation.

         x + b / (z – 23) = a + 125

[Note the proper use of parentheses in the solutions. This is important to prevent misinterpretation of the exact order of operations. However, without the parentheses, the generally accepted order would be:

1. exponents and roots
2. multiplication and division
3. addition and subtraction]

Answers:

	x = (a + 125) – b / (z - 23)

	b / (z – 23) = a + 125 – x   ;   b = (a + 125 – x) * (z – 23)

It’s OK to use an intermediate step to keep track of your work and some equations are so complex that even seasoned mathematicians will do that.

	b / (z – 23) = a + 125 – x ; b / (a + 125 – x) = z – 23 ;

             finally, z = b / (a + 125 – x) + 23

	a = x + b / (z – 23) – 125   That one was easy!

Well, good luck and have fun!



About The Author

Math Expert: Basic Math, Algebra, Geometry, Pre-Ca
BSEE, University of Cincinnati, 1979. Well versed in math and the foundation of understanding in both math and science. I have tutored before, while in school and afterwards and have always found it rewarding. I\'m looking forward to sharing knowledge.
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