In this post, we will explore real-life applications of systems of linear equations. Solving these types of problems allows us to find solutions to complex situations involving multiple variables. Let's dive in!

1. Example: Cost and Revenue

Imagine you run a small toy manufacturing company. You produce two types of toys: stuffed animals and building blocks. Each stuffed animal requires 2 hours of labor and generates a profit of $5, while each building block requires 1 hour of labor and generates a profit of $3. Suppose you want to know the number of stuffed animals and building blocks you need to produce in order to earn a profit of $100. We can represent this situation using a system of linear equations as follows:

2x + y = 100
5x + 3y = 100

To find the solution, we can use substitution or elimination methods we've previously discussed.

2. Example: Mixing Solutions

Consider a chemistry lab where two chemicals are mixed to create a new solution. Let's say that Chemical A is 40% pure and Chemical B is 60% pure. We want to obtain 100 liters of a mixture that is 50% pure. We can set up a system of linear equations to represent this situation:

0.4x + 0.6y = 0.5(100)
x + y = 100

Solving this system will help us determine the amount of each chemical needed to achieve the desired concentration.

3. Example: Distance and Time

Suppose two friends start driving towards each other from different cities. The first friend drives at a speed of 60 miles per hour, while the second friend drives at 70 miles per hour. If they are initially 500 miles apart and meet after 5 hours, we can use a system of linear equations to find the distance each friend has traveled:

x + y = 500
60x + 70y = 5

Once again, solving this system will give us the distance traveled by each friend.

Remember, systems of linear equations have numerous real-world applications, from solving cost and profit problems to determining mixtures and distances. By mastering the techniques we've discussed, you'll be equipped to tackle a wide range of practical scenarios!

Keep up the excellent work in your mathematical journey!