Introduce variables for the number of quarters and dimes.
10 quarters and 15 dimes
Practice makes perfect
Let's introduce variables for the number of quarters and dimes.
Description
Variable
Number of quarters Kevin had.
q
Number of dimes Kevin had.
d
Let's use these variables to write equations expressing the information given in the question.
Description
Equation
Kevin had 25 coins.
q+d=25
The total value of the coins was $4.
0.25q+0.10d=4
We now have two equations involving the two unknowns, q and d.
Let's write this system of equations as a matrix equation.
q+d=25 0.25q+0.10d=4
⇒
q+d 0.25q+0.10d
=
25 4
Now, we will write the left-hand side of the above matrix equation as the product of the coefficient matrix and the variable matrix.
q+d 0.25q+0.10d
=
25 4
⇕
1& 1 0.25& 0.10
*
q d
=
25 4
We will solve the matrix equation by multiplying both sides by the inverse of the coefficient matrix to isolate the q and d variables. When multiplying a matrix by its inverse, we obtain the identity matrix.
This result means that Kevin had q=10 quarters and d=15 dimes.
Checking Our Answer
How to use the calculator.
Consider the structure of the above solution.
We set up an equation system for the unknowns q and d and rewrote it as a matrix equation.
1& 1 0.25& 0.10
*
q d
=
25 4
Let's name the matrices involved.
A= 1& 1 0.25& 0.10 ,
X= q d ,
B= 25 4
We then used the inverse of matrix A to find the solution.
AX=B ⟹ X=A^(-1)B
Graphing calculators can do the calculation for us.
First press 2ND, followed by x^(-1) to access the matrix menu. Then, scroll to the right until you reach the EDIT menu, choose to edit matrix A and enter the coefficient matrix.
Go back to the matrix menu and this time choose to edit matrix B and enter the constant matrix.
Once you have the matrices stored, you can use their names to find X=A^(-1)B.
