Question

Now try writing and solving your own systems of linear equations: one with infinite solutions, one with no solutions, and one with one solution. Write a real-world scenario that goes with at least one system of equations. Show your work.

Answer 1

Sure! Here are three scenarios with their corresponding systems of linear equations:

1. Scenario - Infinite Solutions:
Suppose you decide to open a coffee shop and offer two sizes of coffee cups, small cups priced at $2 and large cups priced at $3. In one day, you sell a total of 100 cups and collect $250 in revenue.

Let's denote the number of small and large cups sold as 's' and 'l' respectively. The system of linear equations representing this scenario is:

s + l = 100 (Equation 1)
2s + 3l = 250 (Equation 2)

To solve this system, we'll use the substitution method:
From Equation 1, we have l = 100 - s.
Substituting this value of l into Equation 2, we get:
2s + 3(100 - s) = 250
2s + 300 - 3s = 250
-s + 300 = 250
-s = -50
s = 50

Now, we substitute the value of s back into Equation 1 to find l:
50 + l = 100
l = 100 - 50
l = 50

The solution of this system is s = 50 and l = 50. This means that there are infinitely many ways to sell a total of 100 cups and collect $250 in revenue by selling different combinations of the two sizes of coffee cups.

2. Scenario - No Solutions:
Suppose you are hosting a baking contest, and the contestants can bake either cakes or pies. Each cake takes 2 hours to make, while each pie takes 3 hours. The total time available for baking is 8 hours, and the contestants should bake a total of 7 desserts.

Let's represent the number of cakes and pies baked by 'c' and 'p' respectively. The system of linear equations representing this scenario is:

2c + 3p = 7 (Equation 1)
2c + 3p = 8 (Equation 2)

To solve this system, we'll subtract Equation 2 from Equation 1:
(2c + 3p) - (2c + 3p) = 7 - 8
0 = -1

We obtain -1 = 0, which is not a true statement. Hence, there are no valid solutions to this system. In the given scenario, it is not possible for the contestants to bake a total of 7 desserts within the time frame of 8 hours.

3. Scenario - One Solution:
Suppose you want to buy some apples and oranges from a grocery store. The cost of each apple is $0.50, and the cost of each orange is $0.75. You spend a total of $5.50 and buy a total of 8 fruits.

Let's represent the number of apples and oranges bought as 'a' and 'o' respectively. The system of linear equations representing this scenario is:

0.50a + 0.75o = 5.50 (Equation 1)
a + o = 8 (Equation 2)

To solve this system, we'll use the elimination method. Multiply Equation 2 by 0.50 to make the coefficients of 'a' in both equations equal:
0.50(a + o) = 0.50(8)
0.50a + 0.50o = 4 (Equation 3)

Now, subtract Equation 3 from Equation 1:
(0.50a + 0.75o) - (0.50a + 0.50o) = 5.50 - 4
0.75o - 0.50o = 1.50
0.25o = 1.50
o = 6

Substitute this value of 'o' back into Equation 2:
a + 6 = 8
a = 8 - 6
a = 2

The solution of this system is a = 2 and o = 6. Therefore, you bought 2 apples and 6 oranges from the grocery store for a total cost of $5.50.