Question 5: Writing Linear Equations
Now try writing and solving your own systems of linear equations: one with infinite
solutions, one with no solution, and one with one solution. Write a real-world scenario
that goes with at least one system of equations. Show your work.
Infinite Solutions Scenario:
Suppose you are comparing two investment packages: both are offering fixed deposit schemes for 10 years, with a monthly return. The monthly return of investment A is $100, while the monthly return of investment B is $80. A friend tells you that if you invest the same amount in both schemes, after 10 years, you will receive the same amount for both investments. Write a system of linear equations for this scenario, knowing that the rate of interest in investment A differs from that of investment B.
Let x be the amount of money you invest in investment A, and y be the amount of money you invest in investment B. Then the total interest received after 10 years is:
Interest A: 100 * 12 * 10 = 12000x
Interest B: 80 * 12 * 10 + = 9600y
We can write this as a linear equation:
12000x = 9600y
Simplified:
5x = 4y
The system of linear equations will have infinite solutions, as there isn't a unique solution for x and y. You can invest various amounts of money in both investments to receive the same ultimate profit.
One Solution Scenario:
A grocery store sells apples and oranges in two different packages. Package A contains 3 apples and 4 oranges, while package B contains 5 apples and 3 oranges. A customer wants to buy exactly 11 apples and 13 oranges. Write a system of linear equations for this scenario and find how many of each package the customer should buy.
Let x represent the number of package A and y represent the number of package B. We can write this scenario as a system of two linear equations:
3x + 5y = 11 (Equation 1 - for apples)
4x + 3y = 13 (Equation 2 - for oranges)
We'll now solve the system using the substitution or elimination method:
Multiply Equation 1 by 3 and Equation 2 by 5 to eliminate y:
9x + 15y = 33
20x + 15y = 65
Subtract the first equation from the second equation:
11x = 32
x = 32/11
Since the answer for x is not a whole number, there is no solution to this system.
One Solution Scenario:
A taxi service charges a base fee plus a per-mile cost for each trip. You wrote down the prices of two trips you took: one was 10 miles and cost $30, and the other was 15 miles and cost $40. Determine the base fee and the per-mile cost.
Let x be the base fee and y be the cost charged per mile. We can write this scenario as a system of linear equations:
x + 10y = 30 (Equation 1 - for the 10-mile trip)
x + 15y = 40 (Equation 2 - for the 15-mile trip)
We'll now solve the system using the substitution or elimination method:
Subtract Equation 1 from Equation 2 to eliminate x:
15y - 10y = 40 - 30
5y = 10
y = 2
Now that we know the per-mile cost (y), we can find the base fee (x) using Equation 1:
x + 10 * 2 = 30
x + 20 = 30
x = 10
So the base fee is $10, and the per-mile cost is $2.