5. Applications of Linear Equations

Linear equations model a variety of real-life problems:

Linear equations are not just abstract concepts in mathematics; they have practical applications in various fields. They help model real-life situations where there is a consistent relationship between different quantities.

Think of linear equations as a way to keep track of your spending. For instance, if you always spend $20 on groceries plus additional costs, you can use a linear equation to predict your total spending based on how many items you buy.

• Budgeting: If a phone plan charges $10 base +$0.50 per minute: 𝐶 = 0.5𝑚+10

This example demonstrates how to model costs with a linear equation. The equation 𝐶 = 0.5𝑚 + 10 shows that the total cost (C) is made up of two components: a fixed base fee of $10 and a variable cost of$0.50 for each minute used (m). This means that as the number of minutes increases, the cost goes up linearly.

Imagine you have a subscription service that charges a flat rate every month plus extra fees based on how much you use the service. If you know how long you will be using it, you can easily predict how much you will spend.

• Motion: Distance = Speed × Time 𝑑 = 𝑠𝑡

This equation is fundamental in physics and demonstrates a linear relationship between distance (d), speed (s), and time (t). It states that the total distance traveled is equal to the speed multiplied by the time traveled. This creates a straight line when graphed, with time on one axis and distance on the other.

Think of a car traveling on a highway. If you drive at a constant speed of 60 km/h, in one hour you'll cover 60 km, in two hours you'll cover 120 km, and so on. This conversion helps you plan your trips.

• Business: Profit = Revenue – Cost 𝑃 = 𝑅−𝐶

In business, profits can be calculated using this linear equation. Here, profit (P) is determined by subtracting costs (C) from revenue (R). This relationship illustrates how businesses can predict their profit based on their sales and expenses.

Imagine you own a small bakery. If you sell cakes for $20 each, and your costs (ingredients, labor) add up to$10 per cake, this equation helps you see that you earn $10 profit for each cake sold. Knowing this helps you set sales goals!

🔹 Word Problem Example A taxi charges a fixed rate of $5 plus$2 per km. Write the equation and calculate the cost for a 10 km ride. Solution: Equation: 𝐶 = 2𝑥+5 For 𝑥 = 10: 𝐶 = 2(10)+5 = 25

This word problem demonstrates how to formulate and solve a real-world situation using a linear equation. The fixed charge is $5, and it costs$2 per kilometer. Thus, for x kilometers traveled, the cost (C) can be modeled with the equation C = 2x + 5. When we substitute x = 10 into the equation, we find the total cost for a 10 km taxi ride.

Consider this a practical scenario: If you've taken a taxi before, you'd notice that there is always an initial fee plus a charge per distance traveled. By writing this in equation form, you can easily determine your fare ahead of time.

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