Solving a System of Equations: A Case Study on Apple Bags

Florence recently bought 7 bags of apples. The red apples were packaged in bags containing 3 apples each, while the green apples were packed in bags of 7. Upon returning home, she found that her total purchase amounted to 37 apples. Now, let's explore how she broke down this purchase using a system of linear equations and algebraic methods.

System of Linear Equations

We define the variables as follows:

R number of bags with red apples G number of bags with green apples

From the problem statement, we have the following two key relations:

Total number of bags: R G 7 Total number of apples: 3R 7G 37

Step-by-Step Solution

Let's solve this problem step-by-step.

Step 1: Express One Variable in Terms of the Other

We can express R in terms of G using the first equation:

R G 7 R 7 - G

Step 2: Substitute and Solve for the Other Variable

Now we substitute this expression for R in the second equation:

3(7 - G) 7G 37 21 - 3G 7G 37 21 4G 37 4G 16 G 4

So, Florence bought 4 bags of green apples.

Step 3: Solve for the First Variable

Now we substitute G 4 back into the first equation to solve for R:

R 4 7 R 3

Therefore, Florence bought 3 bags of red apples.

Verification Using Elimination Method

Alternatively, we can use the elimination method to solve the system of equations:

begin{cases} r G 7 3r 7G 37 end{cases}

Multiplying the first equation by 3:

3r 3G 21

Subtract the modified first equation from the second equation:

(3r 7G) - (3r 3G) 37 - 21 4G 16 G 4

Substituting G 4 back into the first equation:

r 4 7 r 3

Again, we find that Florence bought 3 bags of red apples and 4 bags of green apples.

Conclusion

By applying the method of substitution or elimination, we have proven that the solution to the problem is:

r 3 G 4

This problem showcases the practical application of systems of equations and algebraic methods in solving real-life problems, providing a solid foundation in mathematical reasoning and problem-solving skills.