Which Equation Represents The Line That Passes Through And

Decoding the Equation of a Line: Passing Through (2, 3) and (4, 1)

Finding the equation of a line that passes through two given points is a fundamental concept in algebra and coordinate geometry. This article will thoroughly explore how to determine this equation, focusing on the line passing through the points (2, 3) and (4, 1). We’ll cover various methods, explain the underlying principles, and provide practical examples to solidify your understanding. This comprehensive guide will equip you with the skills to tackle similar problems effectively and confidently.

Understanding the Slope-Intercept Form (y = mx + c)

The most common way to represent a linear equation is using the slope-intercept form: y = mx + c, where:

• m represents the slope of the line (the steepness or incline). It indicates how much y changes for every unit change in x. A positive slope means the line rises from left to right, while a negative slope means it falls.
• c represents the y-intercept, which is the point where the line intersects the y-axis (where x = 0).

To find the equation of the line passing through (2, 3) and (4, 1), we first need to calculate the slope (m).

Calculating the Slope (m)

The slope (m) is calculated using the formula:

m = (y2 - y1) / (x2 - x1)

Where (x1, y1) and (x2, y2) are the coordinates of the two points. In our case:

• (x1, y1) = (2, 3)
• (x2, y2) = (4, 1)

Substituting these values into the slope formula:

m = (1 - 3) / (4 - 2) = -2 / 2 = -1

Therefore, the slope of the line passing through (2, 3) and (4, 1) is -1.

Finding the y-intercept (c)

Now that we have the slope (m = -1), we can use either of the given points and the slope-intercept form (y = mx + c) to find the y-intercept (c). Let's use the point (2, 3):

3 = (-1) * 2 + c

Solving for c:

c = 3 + 2 = 5

So, the y-intercept is 5.

The Equation of the Line

Now that we have both the slope (m = -1) and the y-intercept (c = 5), we can write the equation of the line in slope-intercept form:

y = -x + 5

This is the equation of the line that passes through the points (2, 3) and (4, 1).

Verifying the Equation

Let's verify this equation by substituting the coordinates of both points:

Point (2, 3):

3 = -(2) + 5 3 = 3 (This is correct)

Point (4, 1):

1 = -(4) + 5 1 = 1 (This is also correct)

Both points satisfy the equation, confirming its accuracy.

Alternative Method: Point-Slope Form

Another method to find the equation of a line is using the point-slope form:

y - y1 = m(x - x1)

Where (x1, y1) is one of the points and m is the slope. Using the point (2, 3) and the slope m = -1:

y - 3 = -1(x - 2)

Simplifying this equation:

y - 3 = -x + 2 y = -x + 5

This yields the same equation as before, demonstrating the equivalence of the methods.

Understanding the Significance of the Slope

The slope, -1, tells us that for every one unit increase in x, y decreases by one unit. This negative slope indicates a downward trend of the line. Understanding the slope provides valuable insights into the relationship between the variables x and y. A steeper slope (larger magnitude) indicates a stronger relationship, while a slope close to zero suggests a weak relationship.

Visualizing the Line

Graphing the line y = -x + 5 helps visualize its position and relationship to the given points. The line will intersect the y-axis at 5 and have a negative slope, passing through (2, 3) and (4, 1). Using graphing tools or plotting the points manually will provide a clear picture of the line's characteristics.

Applications of Linear Equations

Finding the equation of a line has numerous applications across various fields:

• Physics: Representing motion, relationships between variables like distance and time.
• Economics: Modeling supply and demand, cost functions, and other economic relationships.
• Engineering: Designing structures, calculating slopes and gradients.
• Computer Science: Creating algorithms for graphics and simulations.
• Data Analysis: Linear regression uses lines to represent trends in data sets.

Advanced Concepts and Extensions

This fundamental concept can be extended to more complex situations:

• Parallel Lines: Parallel lines have the same slope. Any line parallel to y = -x + 5 will have a slope of -1.
• Perpendicular Lines: The product of the slopes of perpendicular lines is -1. A line perpendicular to y = -x + 5 would have a slope of 1.
• Lines with Undefined Slopes: Vertical lines have undefined slopes (division by zero).
• Lines with Zero Slopes: Horizontal lines have a slope of zero.
• Systems of Linear Equations: Multiple linear equations can be solved simultaneously to find intersection points or determine if lines are parallel or coincident.

Solving Related Problems

To solidify your understanding, try solving these related problems:

1. Find the equation of the line passing through (1, 5) and (3, 1).
2. Find the equation of the line passing through (0, 2) and (4, 6).
3. Determine if the lines y = 2x + 1 and y = -1/2x + 3 are parallel, perpendicular, or neither.
4. Find the equation of the line parallel to y = 3x - 2 and passing through the point (1, 4).
5. Find the equation of the line perpendicular to y = -x + 5 and passing through the origin (0,0).

By practicing these problems, you will further enhance your understanding of linear equations and their applications. Remember to clearly identify the slope and y-intercept (or use the point-slope form) to construct the equation accurately. This article provided a comprehensive guide to finding the equation of a line passing through two points, along with practical applications and extensions to more complex scenarios. Mastering this fundamental skill is essential for success in various mathematical and scientific pursuits.