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You have learned by now that math is all about problem solving.

    If we have an unknown we 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 "substitution"

To substitute means to replace with a comparable item.  In math, when we say we substitute we mean that replace with an equivalent. 

When we substitute, we want to get to one unknown from the two we started with.

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 substitution: ( I'll write this in several short steps)

1)  Let's try it by substituting an equal value for N. Rewrite the equation N + D = 40   (subtract D from both sides) to say N = 40 – D. 

2)  Now where N occurs in the other equation 5N + 10D = 280,

we will replace (substitute) it with N = 40 – D.

3) Now we have 5(40 – D) + 10 D = 280.                             4) This becomes 200 - 5D + 10D = 280.                                    

Rounded Rectangular Callout: For instance, two quarters would be the same as 5 dimes or as 10 nickels.          5)  Then we get 200 + 5D = 280.                                                                                                               6)     Then we get 5D = 80 and we can see that D = 16.

7)  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 substituting for D. Rewrite the equation N + D = 40 to say D = 40 – N. 

2)  Now where D occurs in the other equation 5N + 10D = 280, we will replace (substitute) it with D = 40 – N.

3) Now we have 5N + 10(40 - N) = 280.                                      4) This becomes  5N + 400 - 10N = 280.                                    

Rectangular Callout: try another substitution problem here substitution2.htm 5)  Then we get 400 - 5N = 280.                                   6)     Then we get -5D = -120 and we can see that D = 24.

7)  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.