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Question 1196758: I can list all of the properties I need for this big test, but I don't know how to apply my knowledge:( can you explain what I need to do for the following---
division property of equality
subtraction property of equality
addition property of equality
substitution property of equality
reflexive property
symmetric property
transitive property
Please and thank you:)
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52785)   (Show Source):
You can put this solution on YOUR website! .
At least, for each position/question, you should write a relevant mathematical formula.
It would be perfect, if, in addition, you can comment these properties in wording form, where it is applicable and appropriate.
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
The ones that have "of equality" apply to equations. With exception to substitution, we're doing the same thing to both sides of an equation.
The division property of equality has us divide both sides by the same number.
The rule is if a = b, then a/c = b/c where c is not zero.
We need c nonzero to avoid division by zero errors.
Example:
2x = 10
2x/2 = 10/2
x = 5
I divided both sides by 2
The subtraction property of equality is where we subtract the same thing from both sides
The rule is if a = b, then a-c = b-c
Example:
x+1 = 7
x+1-1 = 7-1
x = 6
We subtract 1 from both sides to undo the +1
The additional property of equality is going to follow the same idea, but this time we add the same thing to both sides
If a = b, then a+c = b+c
Example:
w-12 = 20
w-12+12 = 20+12
w = 32
Add 12 to both sides to undo the subtraction applied to w
Not mentioned, but should be, is the multiplicative property of equality
If a = b, then a*c = b*c
Example:
x/3 = 18
3*(x/3) = 3*18 ... multiply both sides by 3
x = 54
The substitution property of equality is where we replace a variable with its stated value (if given or known)
For instance, if we know that x = 54, then x/3 = 18 becomes 54/3 = 18.
Recall that variables are simply placeholders for numbers, so it makes sense to swap out the letters for actual numbers.
Think of the variable as a box that holds the number.
Sometimes we may replace one variable with another algebraic expression
Example: if x = y+10, then 2x+3y = 10 becomes 2(y+10)+3y = 10. I replaced x with (y+10).
It might help to think of a substitute teacher in that they temporarily replace your current teacher. That's one way to remember how the substitution property works.
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Now to the other properties.
The reflexive property is the idea of x = x. Any value is equal to itself. This may seem trivial, but it's useful for proofs later down the road.
The symmetric property is the idea of y = x being the same as x = y. We can swap both sides of any equation. The order of left vs right doesn't matter
Example: 2+3 = 5 is the same as 5 = 2+3
It's up to preference which you think is the better format. Sometimes one format is more convenient than others.
Transitive property:
If a = b and b = c, then a = c
I like to think of this as a bunch of dominoes.
If A knocks down B, and B knocks over C, then A caused C to fall
We can think of the transitive property as a very close cousin of the substitution property
If a = b and b = c, then we can replace 'b' in the first equation with 'c'
So we go from a = b to a = c
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These terms may seem like a lot to memorize. So I recommend using flashcards. Or you can just practice solving equations. When solving equations, be sure to list the property used. Refer to these set of steps as an example
| Equation | Reason | Further Clarification | | 3x+5 = 10 | | | | 3x+5-5 = 10-5 | Subtraction Property of Equality | Subtract 5 from both sides | | 3x = 5 | Simplify | | | 3x/3 = 5/3 | Division Property of Equality | Divide both sides by 3 | | x = 5/3 | Simplify | |
Another example
| Equation | Reason | Further Clarification | | (x/5) - 2 = 12 | | | | (x/5) - 2+2 = 12+2 | Addition Property of Equality | Add 2 to both sides | | x/5 = 14 | Simplify | | | 5*(x/5) = 5*14 | Multiplication Property of Equality | Multiply both sides by 5 | | x = 70 | Simplify | |
How can we check the answer to an equation? By the substitution property.
Let's replace each x in the second example with 70. Then simplify by use of the order of operations (PEMDAS)
(x/5) - 2 = 12
(70/5) - 2 = 12
14 - 2 = 12
12 = 12
We get the same thing on both sides which confirms x = 70 is indeed the solution to that equation.
Side note: When solving equations, we go in reverse of PEMDAS to undo each operation done to the variable we're solving for.
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