Finding The Correct Equation Of A Line Passing Through A Point With A Given Slope

Jul 12, 2025 by ADMIN 82 views

Finding the Correct Equation of a Line: A Comprehensive Guide

Finding the equation of a line is a fundamental concept in algebra and is crucial for various applications in mathematics and other fields. When given a point and a slope, there are several ways to determine the correct equation. This article will explore the different methods and provide a step-by-step guide to solve the problem effectively. We will focus on the given point (-2, 1) and slope m = 1/2, and evaluate the provided options to identify the correct equations.

Understanding Slope-Intercept Form

One of the most common ways to represent a linear equation is the slope-intercept form, which is given by:

y = mx + b

Where:

y is the dependent variable (usually plotted on the vertical axis).
m is the slope of the line, representing the rate of change of y with respect to x.
x is the independent variable (usually plotted on the horizontal axis).
b is the y-intercept, the point where the line crosses the y-axis (i.e., when x = 0).

Applying Slope-Intercept Form to the Problem

To find the equation of a line passing through the point (-2, 1) with a slope of 1/2, we can start by plugging the slope m into the slope-intercept form:

y = (1/2)x + b

Now, we need to find the y-intercept (b). We can do this by substituting the coordinates of the given point (-2, 1) into the equation:

1 = (1/2)(-2) + b

Solving for b:

1 = -1 + b

b = 2

So, the equation in slope-intercept form is:

y = (1/2)x + 2

This makes option B a correct answer. Let's analyze the other options to see if they are equivalent or incorrect.

Point-Slope Form: An Alternative Approach

Another useful form for representing a linear equation is the point-slope form. This form is particularly helpful when you have a point (x₁, y₁) and a slope m. The point-slope form is given by:

y - y₁ = m(x - x₁)

Using Point-Slope Form to Solve the Problem

Given the point (-2, 1) and the slope m = 1/2, we can plug these values into the point-slope form:

y - 1 = (1/2)(x - (-2))

y - 1 = (1/2)(x + 2)

This matches option C, which is another correct equation for the line. The point-slope form is advantageous because it directly incorporates the given point and slope, making it easy to write the equation without needing to solve for the y-intercept initially.

Converting Between Forms

It's important to understand that linear equations can be expressed in different forms, and these forms can be converted into one another. For instance, we can convert the point-slope form equation we found (option C) into slope-intercept form to verify that it is equivalent to option B.

Starting with:

y - 1 = (1/2)(x + 2)

Distribute the 1/2 on the right side:

y - 1 = (1/2)x + 1

Add 1 to both sides:

y = (1/2)x + 2

As we can see, this matches the slope-intercept form equation we derived earlier, confirming that options B and C are indeed equivalent.

Standard Form of a Linear Equation

The standard form of a linear equation is represented as:

Ax + By = C

Where A, B, and C are constants, and A and B are not both zero. This form is useful for various algebraic manipulations and comparisons. Let's convert our slope-intercept form equation into standard form.

Starting with:

y = (1/2)x + 2

To eliminate the fraction, multiply the entire equation by 2:

2y = x + 4

Rearrange the equation to get the standard form:

x - 2y = -4

This matches option D, indicating that it is also a correct equation for the line. Understanding how to convert between different forms of linear equations is crucial for solving problems efficiently and verifying answers.

Analyzing Incorrect Options

Now that we've confirmed that options B, C, and D are correct, let's analyze option A to understand why it is incorrect.

Option A states:

y = -2x + (1/2)

This equation has a slope of -2, which is not the given slope of 1/2. Additionally, even if the slope were correct, the y-intercept is 1/2. Let's check if the point (-2, 1) satisfies this equation:

1 = -2(-2) + (1/2)

1 = 4 + (1/2)

1 = 4.5

This is clearly false, so option A is not a correct equation for the line passing through (-2, 1) with a slope of 1/2. By understanding why an option is incorrect, we reinforce our understanding of the concepts involved.

Step-by-Step Solution

To summarize, here is a step-by-step solution to find the correct equation of the line:

Identify Given Information: Point (-2, 1), Slope m = 1/2.
Use Slope-Intercept Form: y = mx + b. Substitute m:

y = (1/2)x + b
Substitute Point Coordinates: Plug in (-2, 1) to find b:

1 = (1/2)(-2) + b

1 = -1 + b

b = 2

The equation is y = (1/2)x + 2 (Option B).
Use Point-Slope Form: y - y₁ = m(x - x₁). Substitute the point and slope:

y - 1 = (1/2)(x + 2) (Option C).
Convert to Standard Form: Start with y = (1/2)x + 2:

2y = x + 4

x - 2y = -4 (Option D).
Analyze Incorrect Option: Option A has the wrong slope and does not satisfy the given point.

By following these steps, you can systematically find the correct equation of a line given a point and a slope. Understanding the different forms of linear equations and how to convert between them is key to solving these problems effectively.

Practical Applications of Linear Equations

Linear equations are not just abstract mathematical concepts; they have numerous practical applications in real-world scenarios. Understanding how to work with linear equations can be beneficial in various fields.

Physics

In physics, linear equations are used to describe motion with constant velocity. For example, the equation d = vt + d₀ (where d is the distance, v is the velocity, t is the time, and d₀ is the initial distance) is a linear equation. Determining the position of an object moving at a constant speed over time is a direct application of linear equations.

Economics

Linear equations are used to model supply and demand curves. The point where the supply and demand curves intersect represents the equilibrium price and quantity in a market. Economists use these equations to analyze market trends and make predictions.

Engineering

Engineers use linear equations to design structures and systems. For example, the relationship between stress and strain in a material can often be approximated by a linear equation within certain limits. This helps engineers determine how much load a structure can bear before it fails.

Computer Graphics

Linear equations are fundamental in computer graphics for tasks like scaling, rotation, and translation of objects. Linear transformations, which are based on linear equations, are used to manipulate the position and orientation of objects in 3D space.

Data Analysis

In data analysis, linear regression is a common technique for modeling the relationship between two variables. A linear equation is used to find the best-fit line through a set of data points, allowing analysts to make predictions and understand trends.

Conclusion

In conclusion, finding the equation of a line given a point and a slope is a fundamental skill in algebra with wide-ranging applications. By understanding the slope-intercept form, point-slope form, and standard form, and by practicing conversions between these forms, one can effectively solve such problems. Options B (y = (1/2)x + 2), C (y - 1 = (1/2)(x + 2)), and D (x - 2y = -4) are all correct equations for the line passing through the point (-2, 1) with a slope of 1/2. This comprehensive guide not only provides the solution but also explains the underlying concepts and the practical importance of linear equations in various fields.