More on Solving Equations
2 unique types of equations:
Identity equation: You solve it and you get the same thing on both sides...if you solve until you cannot do anything more, you get 0 = 0.
What this means is that you can pick any number and the equation will work!
The Distributive Property is the simplest example of an Identity Equation:
3(x + 7) = 3x + 21
Distribute on the left side and you'll get:
3x + 21 = 3x + 21
At this point, you should already know this is an Identity!
If you keep solving, you would subtract 3x from each side and you'll get:
21 = 21
and you know again that this is an Identity.
If you now subtract 21 from each side:
0 = 0
BUT I WOULDN'T GO THIS FAR! AS SOON AS YOU HAVE THE SAME THING ON BOTH SIDES, YOU CAN STOP AND SAY IT'S AN IDENTITY EQUATION!!!
Null set equation:
You solve it and you get an impossible answer:
3(x + 7) = 3x + 10
3x + 21 = 3x + 10
21 = 10
WHEN WILL THAT HAPPEN??? NEVER!!! SO THERE IS NO POSSIBLE SOLUTION TO THIS! The answer is the null set.
Wednesday, October 6, 2010
Math 6H (Period 6 & 7)
Solving Equations & Inequalities 2-4 & 2-5
GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation
Here are the steps and justifications (reasons)
1. focus on the side where the variable is and focus specifically on what is in the way of the variable being by itself ( isolated)
2. What is the operation the variable is doing with that number in its way?
3. Get rid of that number by using the opposite (inverse) operation
*Use + if there is a subtraction problem
*Use - if there is an addition problem
*Use x if there is a division problem
*Use ÷ if there is a multiplication problem
GOLDEN RULE OF EQUATIONS; DO UNTO ONE SIDE OF THE EQUATION WHATEVER YOU DO TO THE OTHER!!
4. Justification: You have just used one of the PROPERTIES OF EQUALITY
which one?
that's easy-- Whatever operation YOU USED to balance both sides that's the property of equality
We used:
" +prop= " to represent Addition Property of Equality
" -prop= " to represent Subtraction Property of Equality
" xprop= " to represent Multiplication Property of Equality
" ÷prop= " to represent Division Property of Equality
5. You should now have the variable all alone (isolated) on one side of the equal sign.
6. Justification: Why is the variable alone?
For + and - equations you used the Identity Property of Addition (ID+) which simply means that you don't bring down the ZERO because you add zero to anything-- it doesn't change anything... [Note: there is no ID of subtraction]
For x and ÷ equations, you used the Identity Property of Multiplication (IDx) which simply means that you don't bring down the ONE because when you multiply by one it doesn't change anything [NOTE: there is no ID of division]
7. Put answer in the final form of x = ____and box this in.
GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation
Here are the steps and justifications (reasons)
1. focus on the side where the variable is and focus specifically on what is in the way of the variable being by itself ( isolated)
2. What is the operation the variable is doing with that number in its way?
3. Get rid of that number by using the opposite (inverse) operation
*Use + if there is a subtraction problem
*Use - if there is an addition problem
*Use x if there is a division problem
*Use ÷ if there is a multiplication problem
GOLDEN RULE OF EQUATIONS; DO UNTO ONE SIDE OF THE EQUATION WHATEVER YOU DO TO THE OTHER!!
4. Justification: You have just used one of the PROPERTIES OF EQUALITY
which one?
that's easy-- Whatever operation YOU USED to balance both sides that's the property of equality
We used:
" +prop= " to represent Addition Property of Equality
" -prop= " to represent Subtraction Property of Equality
" xprop= " to represent Multiplication Property of Equality
" ÷prop= " to represent Division Property of Equality
5. You should now have the variable all alone (isolated) on one side of the equal sign.
6. Justification: Why is the variable alone?
For + and - equations you used the Identity Property of Addition (ID+) which simply means that you don't bring down the ZERO because you add zero to anything-- it doesn't change anything... [Note: there is no ID of subtraction]
For x and ÷ equations, you used the Identity Property of Multiplication (IDx) which simply means that you don't bring down the ONE because when you multiply by one it doesn't change anything [NOTE: there is no ID of division]
7. Put answer in the final form of x = ____and box this in.
Tuesday, October 5, 2010
Algebra (Period 1)
Formulas 3-7
Formulas are special equations that have known relationships.
d = rt
we can solve for r or t
r = d/t
t = d/r
What about solving for l in the following
p = 2l + 2w
first we need to isolate the variable l so let's subtract 2w first from both sides of the equation
p = 2l + 2w
-2w -2w
p-2w = 2l
now we need to divide BOTH sides by 2
p-2w = 2l
2 -----> 2
(p-2w)/2 = l
and we solved for l
What if the variable is in the denominator
3z/x = 4/y
You could solve using cross products
3zy = 4x
and then divide both sides by 4
3zy/4 = 4x/4
which becomes
3zy/4 = x or written better as 3yz/4 = x
Solve for a in the following formula for solving for the average (mean) of three items
A = (a + b+ c)/3
you could clear it of the fraction by multiplying both sides by 3
3A = a + b+ c then subtract b and subtract c froom both sides
-b -c -b -c
3A - b- c = a
Solve for r
C = 2∏r divide both sides by 2∏
C/2∏ = 2∏r/2∏
and get
C/2∏ = r
A = ½bh Solve for h
You could use cross products
2A = bh
then divide both sides by b
and get
2A/b = h
What about A = 1/R solving for R
using cross products you get AR = 1
Then divide both sides by A and get R = 1/A
Isolate that variable!!
Formulas are special equations that have known relationships.
d = rt
we can solve for r or t
r = d/t
t = d/r
What about solving for l in the following
p = 2l + 2w
first we need to isolate the variable l so let's subtract 2w first from both sides of the equation
p = 2l + 2w
-2w -2w
p-2w = 2l
now we need to divide BOTH sides by 2
p-2w = 2l
2 -----> 2
(p-2w)/2 = l
and we solved for l
What if the variable is in the denominator
3z/x = 4/y
You could solve using cross products
3zy = 4x
and then divide both sides by 4
3zy/4 = 4x/4
which becomes
3zy/4 = x or written better as 3yz/4 = x
Solve for a in the following formula for solving for the average (mean) of three items
A = (a + b+ c)/3
you could clear it of the fraction by multiplying both sides by 3
3A = a + b+ c then subtract b and subtract c froom both sides
-b -c -b -c
3A - b- c = a
Solve for r
C = 2∏r divide both sides by 2∏
C/2∏ = 2∏r/2∏
and get
C/2∏ = r
A = ½bh Solve for h
You could use cross products
2A = bh
then divide both sides by b
and get
2A/b = h
What about A = 1/R solving for R
using cross products you get AR = 1
Then divide both sides by A and get R = 1/A
Isolate that variable!!
Math 6H (Period 6 & 7)
Solving Equations & Inequalities 2-4
Before we began today's lesson, we reviewed some of the difficult mathematical expressions and equations found on Page 448. We discussed the importance of a well placed comma... in MATH as well as in Language Arts!!
Our math example was
the sum of three and a number b, times even
(3 + b)7
and
the sum of three and a number b times seven
3 + 7b
Our Language Arts example... is one of my favorites.
Where does the comma belong in the following:
A woman without her man is nothing.
I insist it is...
A woman, without her, man is nothing.
However, I acknowledge that some men would differ and insist it is
A woman, without her man, is nothing.
So you see, the comma makes all the difference... in math expressions and equations as well as in language arts!!
We solve by "undoing" the operations in each equation.
We use inverse operation & undo Aunt Sally
It's the reverse of PEMDAS!!
GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation
Its exactly like unwrapping a present... which I did in class today!! :)
GOLDEN RULE OF MATHEMATICS
What you do to one side of the equation you MUST do to the other side!!
w + 18 = 64 we must undo addition using the inverse of + (that is, subtraction)
- 18 -18
w + 0 = 46
and then we write
w = 46
What property allows us to subtract 18 from both sides of the equation?
The SUBTRACTION property of equality which we abbreviate with
-prop=
What property allows us to write w instead of w + 0
w + 0 = w
That is the Identity Property of Addition.
How about
p - 84 = 102 we must undo subtraction using the inverse of - (which is ADDITION)
p - 84 = 102
+84 +84
p + 0 = 186
and then we write
p = 186
What property allows us to add 84 to both sides of the equation?
the addition property of equality, which we abbreviate as
+prop=
What property allows us to write p instead of p + 0
The Identity property of Addition.
Why is it called the Identity Property of Addition?
The number never changes its identity
a + 0 = a for all numbers!!
What happens if we have
b + 7 > 8
This is an inequality but we solve this as we would an equation
b + 7 > 8
- 7 -7
b + 0 > 1
and then we write b > 1
h - 7 > 7
+ 7 +7
h + 0 > 14
and then we write
h > 14
Our textbook gives the answer as "greater than 14" but I want you to put it in math symbols. That is, please answer with
h > 14.
Before we began today's lesson, we reviewed some of the difficult mathematical expressions and equations found on Page 448. We discussed the importance of a well placed comma... in MATH as well as in Language Arts!!
Our math example was
the sum of three and a number b, times even
(3 + b)7
and
the sum of three and a number b times seven
3 + 7b
Our Language Arts example... is one of my favorites.
Where does the comma belong in the following:
A woman without her man is nothing.
I insist it is...
A woman, without her, man is nothing.
However, I acknowledge that some men would differ and insist it is
A woman, without her man, is nothing.
So you see, the comma makes all the difference... in math expressions and equations as well as in language arts!!
We solve by "undoing" the operations in each equation.
We use inverse operation & undo Aunt Sally
It's the reverse of PEMDAS!!
GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation
Its exactly like unwrapping a present... which I did in class today!! :)
GOLDEN RULE OF MATHEMATICS
What you do to one side of the equation you MUST do to the other side!!
w + 18 = 64 we must undo addition using the inverse of + (that is, subtraction)
- 18 -18
w + 0 = 46
and then we write
w = 46
What property allows us to subtract 18 from both sides of the equation?
The SUBTRACTION property of equality which we abbreviate with
-prop=
What property allows us to write w instead of w + 0
w + 0 = w
That is the Identity Property of Addition.
How about
p - 84 = 102 we must undo subtraction using the inverse of - (which is ADDITION)
p - 84 = 102
+84 +84
p + 0 = 186
and then we write
p = 186
What property allows us to add 84 to both sides of the equation?
the addition property of equality, which we abbreviate as
+prop=
What property allows us to write p instead of p + 0
The Identity property of Addition.
Why is it called the Identity Property of Addition?
The number never changes its identity
a + 0 = a for all numbers!!
What happens if we have
b + 7 > 8
This is an inequality but we solve this as we would an equation
b + 7 > 8
- 7 -7
b + 0 > 1
and then we write b > 1
h - 7 > 7
+ 7 +7
h + 0 > 14
and then we write
h > 14
Our textbook gives the answer as "greater than 14" but I want you to put it in math symbols. That is, please answer with
h > 14.
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