Finding The Equation Of Line CD - A Step-by-Step Guide

by THE IDEN 55 views

In mathematics, one of the fundamental concepts in coordinate geometry is determining the equation of a line. Given two points on a line, we can uniquely define the line and express its relationship between the x and y coordinates using various forms of linear equations. In this article, we will delve into the process of finding the equation of a line, specifically focusing on the line CD which passes through the points (0, 2) and (4, 6). Understanding this process is crucial for various applications in mathematics, physics, engineering, and computer graphics. This guide aims to provide a comprehensive explanation, making it accessible for students and enthusiasts alike.

Understanding Linear Equations

Before we dive into the specifics of line CD, let's first establish a solid understanding of linear equations. A linear equation represents a straight line on a coordinate plane. The most common form of a linear equation is the slope-intercept form, expressed as:

y = mx + b

where:

  • y represents the dependent variable (typically plotted on the vertical axis).
  • x represents the independent variable (typically plotted on the horizontal axis).
  • m represents the slope of the line, indicating its steepness and direction.
  • b represents the y-intercept, the point where the line crosses the y-axis.

Another useful form is the point-slope form, given by:

y - y1 = m(x - x1)

where:

  • (x1, y1) is a known point on the line.
  • m is the slope of the line.

The point-slope form is particularly useful when we have a point and the slope, or when we need to find the equation of a line given two points.

Step-by-Step Solution to Finding the Equation of Line CD

To find the equation of line CD, we will follow a systematic approach, utilizing the concepts and formulas discussed above. Let's break down the process into manageable steps:

Step 1: Calculate the Slope (m)

The slope of a line represents its steepness and direction. It is defined as the change in the y-coordinate divided by the change in the x-coordinate between two points on the line. Given two points (x1, y1) and (x2, y2), the slope m can be calculated using the formula:

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

For line CD, the given points are (0, 2) and (4, 6). Let's assign these points as follows:

  • (x1, y1) = (0, 2)
  • (x2, y2) = (4, 6)

Now, substitute these values into the slope formula:

m = (6 - 2) / (4 - 0) = 4 / 4 = 1

Therefore, the slope of line CD is 1. This indicates that for every unit increase in the x-coordinate, the y-coordinate also increases by one unit.

Step 2: Determine the y-intercept (b)

The y-intercept is the point where the line intersects the y-axis. This occurs when x = 0. Fortunately, we are given the point (0, 2), which directly provides the y-intercept. At this point, x = 0 and y = 2. Therefore, the y-intercept b is 2.

Alternatively, if we didn't have the y-intercept readily available, we could use the slope-intercept form (y = mx + b) and substitute one of the given points and the calculated slope to solve for b. Let's demonstrate this using the point (4, 6) and the slope m = 1:

6 = 1 * 4 + b 6 = 4 + b b = 6 - 4 = 2

As expected, we obtain the same y-intercept, b = 2.

Step 3: Form the Equation in Slope-Intercept Form

Now that we have determined the slope (m = 1) and the y-intercept (b = 2), we can write the equation of line CD in slope-intercept form (y = mx + b). Substitute the values of m and b into the equation:

y = 1 * x + 2 y = x + 2

Thus, the equation of line CD is y = x + 2. This equation represents all the points that lie on line CD. It tells us that the y-coordinate of any point on the line is equal to the x-coordinate plus 2.

Step 4: Verify the Equation

To ensure the accuracy of our equation, it's good practice to verify it using the given points. Let's substitute the coordinates of the points (0, 2) and (4, 6) into the equation y = x + 2 and check if they satisfy the equation.

  • For point (0, 2):

    2 = 0 + 2 2 = 2 (The equation holds true)

  • For point (4, 6):

    6 = 4 + 2 6 = 6 (The equation holds true)

Since both points satisfy the equation, we can confidently conclude that y = x + 2 is the correct equation for line CD.

Alternative Method: Using Point-Slope Form

As mentioned earlier, the point-slope form of a linear equation can also be used to find the equation of a line. This method is particularly useful when we have the slope and a point on the line. Let's use the point-slope form to find the equation of line CD and compare the result with our previous solution.

The point-slope form is given by:

y - y1 = m(x - x1)

We already know the slope m is 1. We can choose either of the given points, (0, 2) or (4, 6), to use as (x1, y1). Let's use the point (0, 2):

  • m = 1
  • (x1, y1) = (0, 2)

Substitute these values into the point-slope form:

y - 2 = 1(x - 0) y - 2 = x y = x + 2

As we can see, the equation obtained using the point-slope form is the same as the one we found using the slope-intercept form, y = x + 2. This confirms the consistency of our solution.

Now, let's use the point (4, 6) to see if we obtain the same equation:

  • m = 1
  • (x1, y1) = (4, 6)

Substitute these values into the point-slope form:

y - 6 = 1(x - 4) y - 6 = x - 4 y = x - 4 + 6 y = x + 2

Again, we arrive at the same equation, y = x + 2, demonstrating that the choice of point does not affect the final equation.

Graphing the Line CD

To visualize line CD, we can plot the two given points (0, 2) and (4, 6) on a coordinate plane and draw a straight line through them. The line should intersect the y-axis at the point (0, 2), which is the y-intercept. The slope of the line, which is 1, indicates that for every unit we move to the right along the x-axis, we move one unit up along the y-axis. This graphical representation provides a visual confirmation of the equation y = x + 2. The line extends infinitely in both directions, representing all the points that satisfy the equation.

Applications of Linear Equations

Understanding linear equations is fundamental in mathematics and has wide-ranging applications in various fields. Here are a few examples:

  • Physics: Linear equations are used to describe motion with constant velocity, where the relationship between distance, time, and velocity is linear.
  • Engineering: In structural engineering, linear equations are used to analyze the forces and stresses in structures.
  • Economics: Linear equations can model supply and demand curves, cost functions, and revenue functions.
  • Computer Graphics: Linear equations are used to represent lines and planes in 3D graphics, which are essential for rendering images and animations.
  • Data Analysis: Linear regression, a statistical technique, uses linear equations to model the relationship between variables in a dataset.

Conclusion

In this comprehensive guide, we have explored the process of finding the equation of a line given two points, specifically focusing on line CD passing through the points (0, 2) and (4, 6). We started by understanding the fundamental concepts of linear equations, including the slope-intercept form and the point-slope form. We then followed a step-by-step approach to calculate the slope, determine the y-intercept, and form the equation in slope-intercept form. We also demonstrated an alternative method using the point-slope form and verified our equation using the given points. Furthermore, we discussed the graphical representation of the line and highlighted the wide-ranging applications of linear equations in various fields.

By understanding these concepts and techniques, you can confidently solve similar problems and apply them to real-world situations. The equation of line CD, y = x + 2, is a prime example of how we can use mathematical tools to describe and analyze the world around us. Remember, practice is key to mastering these concepts, so continue exploring different problems and applications to solidify your understanding.