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.
A. System with infinite solutions:
Let's consider the real-world scenario of finding the relationship between the amount spent on advertising and the overall revenue of a company.
Suppose we have two equations that express the relationship between advertising cost (x) and the revenue (y):
Equation 1: y = 2x + 40
Equation 2: y - 2x = 40
When we try to solve these equations, we see that they are essentially the same:
y - 2x = 40 => y = 2x + 40
Since the two equations are the same, they will have infinitely many solutions - this means that there is a linear relationship between the advertising cost and revenue for this company, and any point on this line will satisfy the given relationship.
B. System with no solution:
Consider the following system of linear equations:
Equation 1: 4x - 2y = 6
Equation 2: 2x - y = 4
Let's try to solve these equations:
Divide the first equation by 2:
2x - y = 3
Now, we have:
Equation 1: 2x - y = 3
Equation 2: 2x - y = 4
These two equations have the same slope, but different y-intercepts, which means that they are parallel lines that will never intersect. Therefore, there is no solution to this system of linear equations.
C. System with one solution:
Let's consider the real-world scenario of finding the intersection point of two bike paths in a park. Each path is represented by a linear equation:
Path A: y = 3x + 2
Path B: y = -x + 5
To find the intersection point, we can solve this system of linear equations:
Equation 1: y = 3x + 2
Equation 2: y = -x + 5
Let's equate the two equations to find the value of x:
3x + 2 = -x + 5
Solve for x:
4x = 3
x = 3 / 4
Now, plug in the value of x into either equation to find the value of y. Using equation 1:
y = 3(3 / 4) + 2
y = 9 / 4 + 2
y = 17 / 4
So, the intersection point of the two paths is (3 / 4, 17 / 4).
Therefore, we have created and solved a system with infinite solutions, no solution, and one solution while also providing a real-world scenario for one of the systems.