In this article, we will review the methods for solving single-variable equations and take a look at two "special cases" that can happen these "special cases" many cause students a great deal of confusion.
Remember: "isolate the variable" or "un-do" and reverse PEMDAS.
Example 1: 3 x - 5 = 4 First, we need to add 5 to both sides.
3 x - 5 + 5 = 4 + 5 Now, simplify.
3x = 9 Divide both sides by 3.
(3x) / 3 = 9/3 Simplify.
x = 3
Substitute this "answer" into the original equation.
3 (3) - 5? 4
9-5? 4
4 = 4 It checks. Our solution is x = 3 and its graph is a solid dot at 3 on a number line labeled x.
Example 2: a / 3 - a / 4 = 8 We decide what number all the denominators will divide into evenly. For 3 and 4, that number is 12. We now need to multiply every term by 12 .
12 (a / 3) - 12 (a / 4) = 12 (8) The denominators each divide are even into 12, so they are gone.
4a - 3a = 96 Combine like terms.
a = 96
Now check: 96/3 - 96/4? 8
32-24? 8
8 = 8 It checks. So the solution is a = 96 and its graph is a solid dot at 96 on a number line labeled a.
Now, let & # 39; s look at at one "special case"
Example 3: 8 (2x - 3) = 4 (4 x - 8) We need to eliminate the parentheses using the distributive property.
16x - 24 = 16x - 32 Now, move the x & # 39; s to the left.
16x - 16x - 24 = 16x - 16x - 32 Simplify.
This condition is FALSE Did we make a mistake? No.
This is the first of those "special cases." When terms drop away and the remaining statement is FALSE , it means that the original "equation" is not true. There is NO value that can ever make this "equation" TRUE. So this is a NO solution situation.
There are times when inequality symbols are I have we have not studied yet. We are bringing problematic inequalities can be problematic, but this is not one of those. = symbol.
Example 4: -3 (x - 3)> 5 - 3 x Eliminate the parentheses (Distribute)
-3 x + 9> 5 - 3 x Now, take x & # 39; s to the left.
-3 x + 3 x + 9> 5 - 3 - 3 x + 3 x Simplify
9> 5 (This is read as "9 is greater than 5.")
Well, it happened again except for one thing. The x terms canceled out, but the remaining statement is TRUE It will be true true no matter what value of x you check. Try some. Everything works! Really weird! Unlike the other "special case" that had NO solution, this time we are many solutions. INFINITE solution situation.
To summarize: When solving single-variable equations, one of three possibilities will happen:
1) You will get ONE solution. x = some number
The original equation is called a conditional equation Because it is true for only a certain value.
2) You will get NO solutions as evidenced by a final FALSE statement.
This original equation is a call a contradiction Because it can not not be true.
3) You will get INFINITE solutions as evidenced by a final TRUE statement.
This original equation is called an identity Because it is always TRUE.
I think you can understand why so many high school students get confused. These special cases just seem really weird!
