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5.1. Word Problem Example
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Today, we have a real-world application of linear equations. Let's consider a scenario where a taxi charges a base fare plus a rate per kilometer. Does anyone want to suggest how we might start solving this?
We could start by identifying the fixed and variable costs.
Great! That's right. The fixed cost, or the base fare, is $5. Now, what about the cost per kilometer?
It's $2 for each kilometer driven.
Exactly! So if we represent the number of kilometers with 'x', what equation can we form?
The equation would be C = 2x + 5.
You got it! Now, if we want to calculate the cost for a ride of 10 kilometers, what do we do next?
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Now that we have our equation, let's calculate the cost for a 10 km ride. What substitution should we make?
We should substitute x with 10 in the equation C = 2x + 5.
Correct! Let’s perform the substitution. What do we get?
C = 2(10) + 5 = 20 + 5 = 25.
So the total cost for the ride is $25. This is a perfect example of how linear equations can be used in real-life situations. Any questions about this?
Overview
Short Summary
This section illustrates how to form and solve a linear equation using a real-world scenario.
Medium Summary
The section focuses on applying linear equations to real-life situations, specifically highlighting a taxi fare problem. It provides the formulation of the equation and a detailed calculation to find the total cost of a ride.
Detailed Summary
Word Problem Example
In this section, we explore the application of linear equations to a practical scenario involving taxi fares. Here, we need to develop a linear equation based on the problem statement, which states that a taxi charges a fixed rate plus a variable rate depending on distance traveled.
Key Details:
- Fixed Rate: $5 (base charge)
- Variable Rate: $2 per kilometer
- Objective: Formulate the equation and calculate the total cost for a ride of 10 kilometers.
Equation Formation
The cost, denoted as C, can be represented as:
C = 2x + 5
Where x is the number of kilometers traveled.
Calculation
For a ride of 10 km:
- C = 2(10) + 5 = 20 + 5 = $25
Through this example, students learn not only to write a linear equation from a word problem but also to perform calculations using the equation. This skill is essential in solving real-world problems using algebra.
Audio Book
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Create a free accountA taxi charges a fixed rate of 2 per km. Write the equation and calculate the cost for a 10 km ride.
Detailed Explanation
In this word problem, we are given the details of how a taxi charges its fare. The fixed base charge is 2. To express this mathematically, we can create an equation. We denote the total cost as C, and the distance driven in kilometers as x. The cost equation can be written as C = 2x + 5, where 2x corresponds to the additional charge for distance traveled while 5 is the fixed starting fare. To find the total cost for a ride of 10 kilometers, we substitute x with 10 in the equation: C = 2(10) + 5, which simplifies to C = 20 + 5, resulting in a total cost of $25.
Examples & Analogies
Think about it like going to a restaurant where you have to pay a base fee to just sit down at the table (2 per km driven). If you decide to order just a few drinks or stay longer, the overall cost increases based on how far you want to travel after you sit down.
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Create a free accountSolution: Equation: 𝐶 = 2𝑥 + 5 For 𝑥 = 10: 𝐶 = 2(10) + 5 = 25
Detailed Explanation
Once we have established the equation C = 2x + 5, we can calculate the total cost for a specific distance. Here, we replace x with 10, which is the total kilometers we plan to travel. The calculation proceeds as follows: first, we multiply 2 by 10, which equals 20. Then, we add the fixed cost of 5 to this amount. Thus, we have a total of C = 20 + 5 = 25, meaning the total fare for traveling 10 kilometers is $25.
Examples & Analogies
Imagine this calculation like doing a math problem in sports – say you're keeping track of points scored and bonus points for each goal. If your team scores 10 goals and you also get a bonus for being in the game, you simply multiply the goals by the points for a goal and then add in the bonus points at the end to get your final score. Just like calculating the taxi fare!
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Key Concepts
Examples
Step-by-step examples to apply the section's ideas and test your understanding.
A taxi charges a fixed rate of 2 per km. If you travel 10 km, the total cost can be calculated as C = 2x + 5, resulting in $25.
If a gym charges a monthly fee of 5 for each class attended, the cost for attending 4 classes would be C = 4(5) + 30, totaling $50.
Memory Aids
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Stories
Flash Cards
Glossary
Linear Equation
An equation that makes a straight line when graphed, with each term being a constant or a product of a constant and a single variable.
Variable Cost
A cost that varies depending on the quantity of usage, such as distance traveled in a taxi ride.
Fixed Cost
A cost that does not change with the level of output; for example, a base fare on a taxi ride.