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 the no solution equation
Show your work.
1. Infinite solutions:
Equation 1: 2x + 3y = 10
Equation 2: 4x + 6y = 20
To show that these two equations have infinite solutions, we can see that Equation 2 is a multiple of Equation 1. Therefore, they represent the same line and have infinite solutions.
2. No solution:
Equation 1: 3x + 2y = 10
Equation 2: 3x + 2y = 5
To show that these two equations have no solution, we can see that they represent parallel lines with the same slope but different y-intercepts. Therefore, they will never intersect and have no solution.
Real-world scenario: A company produces two types of products, Product A and Product B. The first equation represents the cost of producing both products together and the second equation represents the revenue generated from selling them. However, due to market conditions, the company realizes that they cannot sell the products for as much as anticipated, causing the revenue equation to change. This results in the system having no solution as the costs remain constant but the revenue decreases.
3. One solution:
Equation 1: 2x + 5y = 13
Equation 2: 3x + 4y = 14
To find the solution for this system, we can solve it using the substitution or elimination method:
From Equation 1: 2x + 5y = 13
Rearranging, we get: x = (13 - 5y) / 2
Substitute x into Equation 2:
3((13 - 5y) / 2) + 4y = 14
(39 - 15y) / 2 + 4y = 14
39 - 15y + 8y = 28
-7y = -11
y = 11 / 7
Substitute y back into x:
x = (13 - 5(11/7)) / 2
x = (13 - 55/7) / 2
x = (91/7 - 55/7) / 2
x = 36 / 7
Therefore, the solution to this system of equations is x = 36/7, y = 11/7.
Real-world scenario: This system could represent a situation where a company needs to produce a certain number of two types of products to meet a specific production goal. The company has calculated the cost of production and revenue for each product, and by solving the system, they can determine the exact quantities of each product needed to reach their goal.