The Equation Behind the Fare

Oct 13, 2025 by windailys.com 134 views

The Equation Behind the Fare

Jeremiah lives in New York City and takes a taxi almost everywhere he goes. To calculate the price of his taxi ride, Jeremiah came up with the equation $P = 1.80x + 2.50$ , where $x$ is the number of miles or partial miles traveled in the taxi. This equation is a linear function that represents the cost of the taxi ride based on the distance traveled.

Breaking Down the Equation

The equation $P = 1.80x + 2.50$ consists of two main components: the slope-intercept form and the constant term. The slope-intercept form is represented by the coefficient of $x$ , which is $1.80$ . This coefficient indicates that for every mile or partial mile traveled, the cost of the taxi ride increases by $1.80. The constant term, $2.50, represents the base cost of the taxi ride, which is the minimum amount that Jeremiah has to pay regardless of the distance traveled.

Interpreting the Equation

To understand the equation better, let's consider a few examples. Suppose Jeremiah takes a taxi ride of 5 miles. Using the equation, we can calculate the cost of the ride as follows:

$P = 1.80(5) + 2.50$ $P = 9.00 + 2.50$ $P = 11.50$

Therefore, the cost of the taxi ride for 5 miles is $11.50. Similarly, if Jeremiah takes a taxi ride of 10 miles, the cost of the ride would be:

$P = 1.80(10) + 2.50$ $P = 18.00 + 2.50$ $P = 20.50$

Graphing the Equation

To visualize the equation, we can graph it on a coordinate plane. The graph of the equation $P = 1.80x + 2.50$ is a straight line with a slope of $1.80 and a y-intercept of $2.50. The graph shows that as the distance traveled increases, the cost of the taxi ride also increases at a constant rate of $1.80 per mile.

Real-World Applications

The equation $P = 1.80x + 2.50 has several real-world applications. For instance, taxi companies can use this equation to calculate the cost of a taxi ride based on the distance traveled. Additionally, passengers can use this equation to estimate the cost of a taxi ride before they even get in the taxi.

Conclusion

In conclusion, the equation $P = 1.80x + 2.50 is a linear function that represents the cost of a taxi ride in New York City. The equation consists of a slope-intercept form and a constant term, which can be used to calculate the cost of a taxi ride based on the distance traveled. By understanding this equation, taxi companies and passengers can make informed decisions about taxi rides and estimate the cost of a ride before it even begins.

Understanding the Graph of the Equation

The Graph of a Linear Function

The graph of a linear function is a straight line that can be represented by the equation $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. In the case of the equation $P = 1.80x + 2.50, the slope is $1.80 and the y-intercept is $2.50.

Interpreting the Graph

To interpret the graph of the equation, let's consider a few points. Suppose we want to find the cost of a taxi ride for 5 miles. We can use the equation to calculate the cost as follows:

$P = 1.80(5) + 2.50$ $P = 9.00 + 2.50$ $P = 11.50$

Therefore, the cost of the taxi ride for 5 miles is $11.50. Similarly, if we want to find the cost of a taxi ride for 10 miles, we can use the equation to calculate the cost as follows:

$P = 1.80(10) + 2.50$ $P = 18.00 + 2.50$ $P = 20.50$

Real-World Applications

The graph of the equation $P = 1.80x + 2.50 has several real-world applications. For instance, taxi companies can use this graph to calculate the cost of a taxi ride based on the distance traveled. Additionally, passengers can use this graph to estimate the cost of a taxi ride before they even get in the taxi.

Conclusion

In conclusion, the graph of the equation $P = 1.80x + 2.50 is a straight line that represents the cost of a taxi ride in New York City. The graph can be used to calculate the cost of a taxi ride based on the distance traveled and estimate the cost of a ride before it even begins.

Solving Systems of Equations

Solving Systems of Linear Equations

A system of linear equations is a set of two or more linear equations that have the same variables. To solve a system of linear equations, we can use the method of substitution or the method of elimination.

Method of Substitution

The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. For example, suppose we have the following system of linear equations:

$2x + 3y = 7$ $x - 2y = -3$

We can solve the first equation for $x$ as follows:

$x = 7 - 3y$

Substituting this expression into the second equation, we get:

$(7 - 3y) - 2y = -3$

Simplifying the equation, we get:

$7 - 5y = -3$

Solving for $y$ , we get:

$-5y = -10$ $y = 2$

Substituting this value of $y$ into one of the original equations, we can solve for $x$ . For example, substituting $y = 2$ into the first equation, we get:

$2x + 3(2) = 7$ $2x + 6 = 7$ $2x = 1$ $x = 0.5$

Therefore, the solution to the system of linear equations is $x = 0.5$ and $y = 2$ .

Method of Elimination

The method of elimination involves adding or subtracting the equations to eliminate one of the variables. For example, suppose we have the following system of linear equations:

$2x + 3y = 7$ $x - 2y = -3$

We can add the two equations to eliminate the variable $x$ . Adding the two equations, we get:

$(2x + 3y) + (x - 2y) = 7 + (-3)$ $3x + y = 4$

Solving for $x$ , we get:

$3x = 4 - y$ $x = (4 - y)/3$

Substituting this expression into one of the original equations, we can solve for $y$ . For example, substituting $x = (4 - y)/3$ into the first equation, we get:

$2((4 - y)/3) + 3y = 7$ $(8 - 2y)/3 + 3y = 7$ $8 - 2y + 9y = 21$ $7y = 13$ $y = 13/7$

Substituting this value of $y$ into one of the original equations, we can solve for $x$ . For example, substituting $y = 13/7$ into the first equation, we get:

$2x + 3(13/7) = 7$ $2x + 39/7 = 7$ $14x + 39 = 49$ $14x = 10$ $x = 10/14$ $x = 5/7$

Therefore, the solution to the system of linear equations is $x = 5/7$ and $y = 13/7$ .

Conclusion

In conclusion, the method of substitution and the method of elimination are two methods that can be used to solve systems of linear equations. By using these methods, we can solve systems of linear equations and find the values of the variables.

Real-World Applications

Real-World Applications of Systems of Linear Equations

Systems of linear equations have several real-world applications. For instance, in economics, systems of linear equations can be used to model the behavior of supply and demand. In business, systems of linear equations can be used to optimize production and minimize costs. In engineering, systems of linear equations can be used to design and optimize systems.

Example 1: Supply and Demand

Suppose we have the following system of linear equations:

$P = 2Q + 100$ $Q = 50 - P$

We can solve this system of linear equations using the method of substitution. Substituting $Q = 50 - P$ into the first equation, we get:

$P = 2(50 - P) + 100$ $P = 100 - 2P + 100$ $3P = 200$ $P = 200/3$ $P = 66.67$

Substituting this value of $P$ into the second equation, we get:

Q: What is the equation P = 1.80x + 2.50?

A: The equation P = 1.80x + 2.50 is a linear function that represents the cost of a taxi ride in New York City. The equation consists of a slope-intercept form and a constant term, which can be used to calculate the cost of a taxi ride based on the distance traveled.

Q: What is the slope of the equation P = 1.80x + 2.50?

A: The slope of the equation P = 1.80x + 2.50 is 1.80. This means that for every mile or partial mile traveled, the cost of the taxi ride increases by $1.80.

Q: What is the y-intercept of the equation P = 1.80x + 2.50?

A: The y-intercept of the equation P = 1.80x + 2.50 is $2.50. This means that the minimum cost of a taxi ride is $2.50, regardless of the distance traveled.

Q: How can I use the equation P = 1.80x + 2.50 to calculate the cost of a taxi ride?

A: To use the equation P = 1.80x + 2.50 to calculate the cost of a taxi ride, simply plug in the distance traveled (x) and solve for P. For example, if you take a taxi ride of 5 miles, the cost of the ride would be:

P = 1.80(5) + 2.50 P = 9.00 + 2.50 P = 11.50

Q: Can I use the equation P = 1.80x + 2.50 to estimate the cost of a taxi ride before I even get in the taxi?

A: Yes, you can use the equation P = 1.80x + 2.50 to estimate the cost of a taxi ride before you even get in the taxi. Simply plug in the distance you expect to travel and solve for P. This will give you an estimate of the cost of the taxi ride.

Q: What are some real-world applications of the equation P = 1.80x + 2.50?

A: Some real-world applications of the equation P = 1.80x + 2.50 include:

Calculating the cost of a taxi ride based on the distance traveled
Estimating the cost of a taxi ride before you even get in the taxi
Modeling the behavior of supply and demand in economics
Optimizing production and minimizing costs in business
Designing and optimizing systems in engineering

Q: Can I use the equation P = 1.80x + 2.50 to solve systems of linear equations?

A: Yes, you can use the equation P = 1.80x + 2.50 to solve systems of linear equations. However, the equation P = 1.80x + 2.50 is a single linear equation, not a system of linear equations. To solve a system of linear equations, you would need to use a different method, such as the method of substitution or the method of elimination.

Q: What are some common mistakes to avoid when using the equation P = 1.80x + 2.50?

A: Some common mistakes to avoid when using the equation P = 1.80x + 2.50 include:

Not plugging in the correct value for x (the distance traveled)
Not solving for P correctly
Not using the correct units (e.g. miles vs. kilometers)
Not considering other factors that may affect the cost of a taxi ride (e.g. traffic, time of day, etc.)

Q: Where can I find more information about the equation P = 1.80x + 2.50?

A: You can find more information about the equation P = 1.80x + 2.50 in various online resources, including:

Online math textbooks and tutorials
Math websites and forums
Online calculators and tools
Math apps and software

Q: Can I use the equation P = 1.80x + 2.50 to solve problems in other areas of mathematics?

A: Yes, you can use the equation P = 1.80x + 2.50 to solve problems in other areas of mathematics, such as algebra, geometry, and calculus. However, the equation P = 1.80x + 2.50 is a specific equation that represents the cost of a taxi ride in New York City, and may not be applicable to other areas of mathematics.