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You have learned by now that algebraic math is all about problem solving.
If we have an unknown we are looking for, we also need a piece of information that allows us to write an equation to solve.
For example, if we have 12 dimes then we know we have $1.20 by 10 * 12 = 120 and 120 ÷ 10 = 1.2
This kind of problem has become easy for us. Many other problems, however, involve more than one unknown.
Suppose we have two unknowns (variables) and we want to find out what they are or what their relationship is to each other. If we can get the equation reduced to one unknown then we can solve it.
One method for solving this type of problem is called "elimination"
Before you start cheering, to say elimination does not mean we just erase the problem! It does mean that start with more than one variable and we eliminate down to one.
Example: we have some nickels
and some
dimes
in a
pile. We know that the total number of coins is 40 and we know that the total
money is $2.80. How many nickels and how many dimes do we have?
Solution: We can solve this by calling the number of nickels “N” and the number of dimes “D” and then we can say that N + D = 40. Since we know that each nickel is worth 5 cents and each dime is worth 10 cents we can say that 5N + 10D = 280. (Note: writing all the money amounts in cents)
Now we have the problem set up algebraically but we do not have a solution. We need either to solve for N or for D. This can be done by elimination: ( I'll write this in several short steps)
1) Let's start by writing the equations together (and I'll name them #1 and #2 for identification) #1 N + D = 40
#2 5N + 10D = 280 if we can eliminate one variable then only one remains
2) Now we pick either variable and match the coefficients: in the first equation N has a coefficient of 1. If we multiply it (and the other parts also) by 5 then it will be the same as in #2. Remember that what we do to one side of the equation we must do to both.
#1 5N + 5D =
200
#2 5N + 10D = 280
3) Now we will subtract one equation from the other - remember the = sign tells us that the left side and the right side are equal (they look different but have the same value)
#1 5N + 5D = 200
#2 5N
+ 10D = 280 confused?
this is done by looking at like parts 5N - 5N = 0 and 5D
- 10D = -5D and 200 - 280 = -80
now
0 - 5D = -80
4)
This we can solve: - 5D = -80 becomes D = 16.
5) Since we know that there are 16 dimes and there were 40 coins, we know that there are 24 nickels.
That checks since 16 dimes is $1.60 and 24 nickels is $1.20 and the total is $2.80.
OR We could have done it this way:
1) Let's try it by eliminating D instead of N.
#1 N + D = 40
#2 5N + 10D = 280
2) We will multiply both sides by 10
#1 10N + 10D = 400
#2 5N + 10D = 280
3) Now we will subtract one equation from the other - remember the = sign tells us that the left side and the right side are equal (they look different but have the same value)
#1 10N + 10D = 400
#2 5N + 10D = 280 this is done by looking at like parts 10N - 5N = 5N and 10D - 10D = 0 and 400 - 280 = 120
now 5N + 0 = 120
4)
This we can solve: 5N = 120 becomes N = 24.
5) Since we know that there are 24 nickels and there were 40 coins, we know that there are 16 dimes.
That checks since 16 dimes is $1.60 and 24 nickels is $1.20 and the total is $2.80.
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