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.
1) System with Infinite Solutions:
Equation 1: 2x - 3y = 6
Equation 2: 4x - 6y = 12
To solve for infinite solutions, we first simplify Equation 2 by dividing by 2:
2x - 3y = 6
2x - 3y = 6
These two equations are actually the same, so they have infinite solutions.
2) System with No Solution:
Equation 1: 3x + 2y = 10
Equation 2: 3x + 2y = 5
To solve for no solution, we compare the two equations:
3x + 2y = 10
3x + 2y = 5
Since the coefficients on x and y are the same but the constant terms are different, these equations have no solution.
3) System with One Solution:
Equation 1: 5x + 2y = 11
Equation 2: 3x - 4y = 2
Real-world Scenario:
Let's say you are buying donuts and bagels at a bakery. A donut costs $5 and a bagel costs $2. You buy a total of 3 items for $11. Another customer buys a total of 5 items for $27. How many donuts and bagels did each customer buy?
5x + 2y = 11
3x - 4y = 2
Solving this system of equations, we find:
x = 1 (number of donuts)
y = 3 (number of bagels)
Therefore, the first customer bought 1 donut and 3 bagels, and the second customer bought 1 donut and 3 bagels as well, for a total of $11 each.