Writing a system of equations and solving using elimination or substitution

(1.)   In writing a system of equations, first note what you are looking to solve for. In this problem, you are asked to find how many dimes and how many quarters are needed to make up a total amount of $6.35 given that you have a total of 35 coins.

Let D represent the number of dimes and Q represent the number of quarters. From this, you can generate the following equation:

                                                D + Q = 35

Since the value of a dime is $0.10 and that of a quarter is $0.25, then the sum of the # of dimes times the value of a dime (0.10) and the # of quarters times the value of a quarter (0.25) equals the total amount of $6.35. That is, 

                        (0.10)D + (0.25)Q = 6.35

Thus, the system of equations that will solve for the problem contains the following equations:

     D + Q = 35

    (0.10)D + (0.25)Q = 6.35

(2.)   When solving a system of equations with 2 variables (i.e., D and Q), your goal is to eliminate one of the variables in order to solve for the other variable. Once you've solved for one of these variables, you can use it to solve for the other variable by plugging it into one of the original equations.

First, pick a variable to eliminate. I will choose to eliminate Q to demonstrate. If I multiply the second equation by -4 and add the 2 equations together, I will have eliminated Q and, therefore, be able to solve for D:

   -4*[(0.10)D + (0.25)Q = 6.35]   ==>  (-0.40)D + (-1) Q = -25.4

                                                  ==>     (-0.40)D - Q = -25.4

Add the new equation to the first equation:

               D + Q = 35

+    (-0.40)D - Q = -25.4

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   D - (0.40)D + Q - Q = 9.60      ==>      (0.60)D = 9.60

Divide both sides by 0.60 to solve for D:

      (0.60)D / (0.60) = 9.60 / 0.60     ==>     D = 16

Plug 16 in for D in the first equation to solve for Q:

      D + Q = 35     ==>     16 + Q = 35

Subtract 16 from both sides of the equation:    

      16 + Q = 35

     -16         -16

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              Q = 19

You can check your work by plugging in these values for D and Q into the second equation:

      (0.10)D + (0.25)Q = (0.10)16 + (0.25)19 = 1.60 + 4.75 = 6.35

Thus, Pam has 16 dimes and 19 quarters.

Label the dimes as d and the quarters as q. Since Pam has 35 coins in all, and 635 cents in dimes worth 10 cents and quarters  worth 25 cents, we can set up two equations:

10d + 25q = 635

    d  +  q   =   35

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Now multiply the second equation all the way across by -10, and add that result to the first equation.

10d + 25q = 635

-10d -10q = -350

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Add the two equations to eliminate the d variable.

15q = 385

Divide by 15; q = 19.

Substitute 19 back into the first equation to solve for d:

d + 19 = 35

Thus d= 16.

Pam has 19 quarters and 16 dimes.

Hi Tracy,

The first step when converting a word problem to equations is to assign variables to the unknown quantities.  In this case, we don't know how many of EACH type of coin she has, although we do know that she has 35 coins total.

So let's assign some variables - we'll call "d" the number of dimes she has, and "q" the number of quarters she has. 

Since we know that she has 35 coins, we can write:

d + q = 35

Then, since we know that if she has "d" dimes, then the value of those dimes is 0.1 * d (because each dime is worth $0.10).  Also, if she has "q" quarters, then the value of those quarters is 0.25 * q (because each quarter is worth $0.25).  Then we can write a second equation, because we know that all of her coins put together are worth $6.35:

0.1d + 0.25q = 6.35

So our two equations are:

    d +       q = 35
0.1d + 0.25q = 6.35

We can now solve.  I like using the elimination method, so I will multiply the second equation by -10.  Why -10?  Because I see that 10 * 0.1 = 1, which is the coefficient of d in the first equation, so if we multiply by -10, then it will be -1, and then we can add the two equations to eliminate "d":

   d +     q = 35
-1d - 2.5q = -63.5
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     - 1.5 q = 28.5
             q = 19

Then we plug q back in to our d+q=35 equation to solve for d:

d + 19 = 35
d = 16

Thus, she has 16 dimes and 19 quarters.