Lesson 1 Solving of Linear Equations
An equation in one variable is made of of two expressions that include the variable and numbers and the two expressions are equal.
We solve the equation by using the properties of equality. If we add (subract or multiply or divide) to one expression then we must add (subract or multiply or divide) the same to the other expression.
Since the two expressions are separated by the = I will refer to them as the "left side" and the " right side". Meaning that, the "left side" is the expression to the left of the = symbol. And the "right side" is the expression to the right of the = symbol.
In the first example below the 5x10 is the left side and the 2(x4) is the rght side.
Example 1a
5x  10 = 2(x4) First you must multiply the x  4 by 2 before you begin your steps of solving.
5x  10 = 2x  8 Now begin solving: Subtract 2x from both sides. I prefer the variable on the left side.
5x 10 2x= 2x  8  2x This is how we move the x term to the left side of the equation.
3x  10 =  8 Now ADD 10 to both sides. I prefer the number terms on the right side.
Did you notice that the 10 is negative and that it is on the left side of the equation?
This is how we move the 10 term to the right side of the equation.
3x  10 + 10 = 8 + 10
3x = 2 To finish with 1x instead of 3x, divide by 3 or multiply by 1/3.
3x/3 = 2/3 divide by 3 or multiply by 1/3.
1x = 2/3 We always want to finish with 1x.
Example 1b
3(x8) = 2x  4 +3x LOOK ; I simplified each side before solving.
3x  24 = 5x  4 do you see where the 5x came from?
Now subtract 5x from both sides
3x 5x 24 = 5x 5x 4
2x 24 =  4
2x 24 + 24 =  4 +24
2x = 20 Now divide both sides by the 2.
1x = 10
WHAT about x? Can we finish with x= rather than x= ?
Sometimes you may have x at the end of the problem, x = 10 is not a finished problem.
So please know that it is ok to get x = 10 but that you need one more step: x= 10.
We always finish with what 1x or x equals.
If a negative x = positive 10 then isn't it logical
that positive x = negative 10?
If we solve an equation for x and we have x = 14 at the end of a problem then that means +x = +14.
Be sure to give that as you final answer; x=14. (We divided both sides by 1.)
Please stop and do ASSIGNMENT 1AB .
Example 1c
2x/3  x/6 = 1/2 This is x/6 subtracted from 2x/3 to equal 1/2.
Please note that 2x/3 means 2 thirds of x. It is not the same as 2/3x.
We need to subtract the 1x/6 from the 2x/3 that means LCD is needed.
The LCD is 6 so the 2x/3 becomes 4x/6. (2/3 is same as 4/6) 4x/6  1x/6 = 1/2
3x/6 = 1/2 To find what 1x is instead of the 3sixth of x, we multiply both sides by 6.
HOPE you know that 6 times 3/6 equals 3 the whole number.
Right? so 6 times (3/6 of x) is 3x.
(6)3x/6 = (6)(1/2) Multiply both sides by 6.
3x = 3
x= 1
Example 1d
2x/3 + 6 = 4 This is the sum of 2thirds of a number and 6 to = 4.
Let's begin solving by subtracting 6.
AND remember , a little arithmetic lesson thrown in here,
(3halves) times (2thirds) of x is 6x/6 =1x. We MUST finish with 1x on left side.
And (3 halves) times (2) is6/2 = 3.
1x = 3 or simply x = 3
NOW do ASSIGNMENT 1CD then continue below.*********************
Need a cool math example that is easy!!!
Try this out of our classroom example. It is cool.
http://www.coolmath.com/algebra/Algebra2/01AlgebraReview/02_fracdec.htm
Example 1e
Example 1f
I used blue and red text below; be sure you can see it.
(3x2)/12  x/9 = 2x OR
All terms must be multiplied by the LCM,
even if they are not fractional.

1. LCM for the denominators 12 and 9 is 36.
Hope you know that 36 is the smallest number that both 12 and 9 divide into evenly.
1. Multiply all terms by the 36. JUST write it do not multiply yet!!
36*(3x  2)



36*x

=

36*2x

12


9



2. Divide the 36 by each of the denominators.
BE SURE YOU are writing this in your notebook. COPY all of these steps.
3*(3x  2)



4*x

=

36*2x

1


1



3. Now multiply the numerators. And yes, all the denominators should be 1 now and thus do not have to be written.
3*(3x2)  4x = 72x
9x  6  4x = 72x Again, be sure you understand the simplifying.
5x  6 = 72x
6 = 67x Note the x is on the right side. We divide by the 67.
6/67 = x or x =  6/67
You really should write each of the example problems down on your paper and see if you can solve them yourself.
Then come back and check them.
Just reading math is not going to get it,
unless you have a photogenic mind and an exceptional long term memory.
Do Assignment 1EF .