CHAPTER 2-3: WRITING TWO-STEP EQUATIONS
Algebraic expressions just are the ones that have variables
Numeric expressions have only numbers
Equations must have an = sign while expressions do not
STRATEGY #1:TRANSLATE WORD BY WORD
Always try this first.
Just be careful of less THAN and subtracted FROM because these are switched from the order that you read/say them:
A number less THAN 12 is 12 – n but if you say a number less 12, this would be n – 12
12 subtracted FROM a number is n – 12, but 12 subtract a number would be 12 – n
The only other translation to be careful of is when you multiply a SUM or DIFFERENCE by a number or variable:
12 times the SUM of a number and 5 is 12(n + 5), but the sum of 12 times a number and 5 would be 12n + 5
12 times the DIFFERENCE of a number and 5 is 12(n – 5), but the difference of 12 times a number and 5 would be 12n – 5
If you have 2 or more unknowns, use different variables:
The difference of a number and ANOTHER number would be x - y
STRATEGY #2: DRAWING A PICTURE
(When in doubt, draw it out! ;)
I have 5 times the number of quarters as I have dimes.
I translate to: 5Q = D
I check: If I assume that I have 20 quarters, then 5(20) = 100 dimes
Does this make sense? That would mean I have a lot more dimes than quarters.
The original problem says I have a lot more quarters!
My algebra is WRONG! I need to switch the variables.
5D = Q
I check: If I assume that I have 20 quarters, then 5D = 20
D = 4
Does this make sense? YES! I have 20 quarters and only 4 dimes.
Sometimes it helps to make a quick picture.
Imagine 2 piles of coins.
The pile of quarters is 5 times as high as the pile of dimes.
You can clearly see that you would need to multiply the number of dimes
to make that pile the same height as the number of quarters!
STRATEGY #3: MAKE A T-CHART (really and x-y table)
Let’s say you know that every bagel you buy costs the same amount of money, $.65. You buy bagels and spend $4.55. Write the algebraic equation and then solve for the number of bagels that you purchased.
Number of Bagels
|
Price
|
1
|
.65
|
2
|
1.30
|
3
|
1.95
|
b
|
.65b
|
By looking at the pattern from the left column to the right column, you find that the number of bagels TIMES the unit rate of the price/bagel gives you the total purchase price. Now you use a variable like b to come up with the algebraic expression for the purchase of any number of bagels, .65b.
Finally, set up the algebraic equation for the amount of money you spent (given in the problem):
.65b = 4.55
You can also do this for two-step equations. Let’s say you’re joining a gym and there is an initiation fee of $50 and then a monthly fee of $20 a month. Focus on the amount that is happening repeatedly because that is going to be your coefficient of the variable…since that is the amount that will “vary”, as opposed to the $50 one-time fee that will never change.
You can do this with or without a table!
Number of Months
|
Price
|
0
|
50
|
1
|
70
|
2
|
90
|
3
|
110
|
m
|
20m + 50
|
$20m + $50 = your cost
Say the problem asks how many months you’ve been going if you’ve paid $170:
20m + 50 = 170
Solve for the number of months:
20m = 120
m = 6 months
