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Chapter 2: Problem 92

State one application of using the point-slope formula.

Short Answer

Expert verified
The point-slope formula is used to find the equation of a line given a point and the slope.

Step by step solution

01

Understand the Point-Slope Formula

The point-slope formula is given by \(y - y_1 = m(x - x_1)\,\text{where} (x_1, y_1)\text{is a point on the line and }m\) is the slope.
02

Determine an Application

An application of the point-slope formula is to find the equation of a line when you know a point on the line and the slope. This can be useful in various fields such as engineering, physics, and economics to model relationships between different variables.
03

Example: Road Gradient

Consider a road where the elevation changes from 200 meters to 400 meters over a horizontal distance of 2 kilometers. The point-slope formula helps in creating an equation representing the incline of the road, which can be used in construction planning.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Point-Slope Form
The point-slope form of an equation is a powerful tool in algebra. This form is written as: \( y - y_1 = m(x - x_1) \). In this formula, \( (x_1, y_1) \) represents a specific point on the line, and \( m \) represents the slope of the line. The slope \( m \) indicates the steepness of the line: how much the y-value changes for every unit change in the x-value. This form is particularly useful when you have one point on the line and the slope, which allows you to quickly create the equation of the line.
Equation of a Line
The equation of a line is a mathematical representation of a straight line. Within the context of the point-slope form, it's essential to grasp that any line can be represented if you know its slope and at least one point on the line. Building on the point-slope form \( y - y_1 = m(x - x_1) \), you can rearrange it to the slope-intercept form, another common form of a line equation. For example, if you have a point (2, 3) and a slope of 4, plug in the values:
\( y - 3 = 4(x - 2) \),
then solve for y to switch to slope-intercept form.
Slope-Intercept Form
The slope-intercept form of a line equation is written as \( y = mx + b \). Here, \( m \) is the slope, and \( b \) is the y-intercept, where the line crosses the y-axis. If you start from the point-slope form, it's straightforward to convert:
Consider \( y - 3 = 4(x - 2) \) from the previous section. Expanding and solving for \( y \), you get:
\( y - 3 = 4x - 8 \)
\( y = 4x - 8 + 3 \)
\( y = 4x - 5 \).
This last form \( y = 4x - 5 \) fits the slope-intercept structure, showing how the point-slope form connects directly to another widely-used form.
Real-World Applications
Understanding the point-slope form applies to many real-world situations. For example, road gradient calculations, as mentioned in the exercise solution, are vital in civil engineering. Knowing the elevation change, engineers can model and plan roadways. Another practical application is in economics, where the relationship between cost and production can be analyzed. Likewise, in physics, tracking the motion or speed of an object along a path often uses similar linear equations. These applications underline how crucial it is to master the point-slope form for problem-solving across various disciplines.

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Most popular questions from this chapter

Explain how the average rate of change of a function \(f\) on the interval \(\left[x_{1}, x_{2}\right]\) is related to slope.

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