This week’s focus is on systems of equations. Describe a profession that might use this on a regular basis, and give an example with a solution (not covered in the text) of a math problem with a real-life application.

What profession do you want to consider?

I want to be a carpenter

A profession that might use systems of equations on a regular basis is civil engineering. Civil engineers often encounter situations where they need to solve a system of equations to design and analyze structures such as bridges, buildings, and road networks.

Let's consider an example of traffic flow analysis to demonstrate the use of systems of equations in civil engineering. Suppose a civil engineer is tasked with optimizing traffic flow at a busy intersection with two roads, let's call them Road A and Road B. The engineer wants to determine the optimal signal timing for the traffic lights at the intersection.

To solve this problem, the engineer can set up a system of equations based on the traffic flow rates, cycle lengths, and signal timings. Let's denote the number of vehicles passing through Road A and Road B during a signal cycle as A and B, respectively. We want to find the optimal values of A and B that minimize congestion and maximize efficiency.

Here's an example system of equations that represents traffic flow at the intersection:

Equation 1: A + B = 100 (Total number of vehicles passing through the intersection in one cycle is 100)

Equation 2: 0.5A + 0.8B = 60 (Based on traffic flow rates and percentages, part of the vehicles pass through each road)

In Equation 1, we assume that the total number of vehicles passing through the intersection in one cycle is 100. In Equation 2, we assume that 50% of the vehicles from Road A and 80% of the vehicles from Road B make it through the intersection.

To solve this system of equations, we can use various methods like substitution, elimination, or matrix operations. Solving this specific example using the substitution method, we isolate one variable in Equation 1 and substitute it into Equation 2:

A + B = 100
A = 100 - B (Isolating A in terms of B)

0.5(100 - B) + 0.8B = 60 (Substituting into Equation 2)
50 - 0.5B + 0.8B = 60
0.3B = 10
B = 33.33

Substituting B back into Equation 1:
A + 33.33 = 100
A = 66.67

Therefore, the optimal traffic flow for Road A and Road B at the intersection is approximately 66.67 and 33.33 vehicles per cycle, respectively.

By solving this system of equations, civil engineers can determine the optimal signal timings that balance traffic flow at intersections, reduce congestion, and improve overall transportation efficiency.