Final answer:
Amy used 1 46-cent stamp and 21 7-cent stamps to make the total of $1.94 in postage. This was determined by solving the linear equation representing the total postage cost and finding the combination of stamps that equals 194 cents.
Step-by-step explanation:
The problem involves solving a system of linear equations to determine how many 46-cent stamps and 7-cent stamps Amy used to pay $1.94 in postage. Let's define two variables: let x be the number of 46-cent stamps and y be the number of 7-cent stamps.
The total value of the stamps used is $1.94, which can be converted to cents (194 cents). The total cost can be represented by the equation 46x + 7y = 194, which is our first equation. From this equation, we want to find positive integer solutions for x and y.
The second equation represents the physical count of the stamps, which is not given explicitly in this problem. However, based on the cost, we can infer that any solution must consist of a whole number of stamps, meaning both x and y must be non-negative integers.
To solve the system, we can use substitution or linear combination methods. One way is to express y in terms of x from the first equation and then look for integer solutions that satisfy the total amount. This would give us a manageable number of possible combinations to test since the value of every single 46-cent stamp would significantly reduce the amount that must be covered by the 7-cent stamps.
After solving, we find that Amy could have used 2 46-cent stamps (x = 2) and 20 7-cent stamps (y = 20). This is because 2(46) + 20(7) equals 92 + 140, which is 232 cents or $2.32.
However, since the total cost needs to be $1.94 (or 194 cents), we need to adjust the values to get the correct total. Subtracting one 46-cent stamp from the assumed two, we get 1(46) + 20(7), which is 46 + 140, totalling 186 cents. Now, we are 8 cents short, which can be covered by adding one more 7-cent stamp. This gives us 1 46-cent stamp and 21 7-cent stamps (1*46 + 21*7 = 194 cents).