Verifying If The Line Through C(-3, 2) And D(-2, 4) Satisfies 2y - 4x = 389
In this detailed exploration, we will delve into the realm of linear equations and coordinate geometry to determine whether a line defined by two points satisfies a given equation. Specifically, we will investigate if the line passing through points C(-3, 2) and D(-2, 4) adheres to the equation 2y - 4x = 389. This involves several key steps, including calculating the slope of the line, determining the equation of the line, and finally, verifying if the equation aligns with the given condition. Our journey will not only solidify your understanding of linear equations but also enhance your problem-solving skills in analytical geometry.
1. Determining the Slope of the Line
The slope of a line is a fundamental concept in coordinate geometry, representing the steepness and direction of the line. It quantifies the change in the y-coordinate for a unit change in the x-coordinate. Given two points on a line, the slope (m) can be 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, the points are C(-3, 2) and D(-2, 4). Let's designate C as (x₁, y₁) and D as (x₂, y₂). Plugging these values into the slope formula, we get:
m = (4 - 2) / (-2 - (-3)) m = 2 / 1 m = 2
Thus, the slope of the line passing through points C and D is 2. This positive slope indicates that the line rises as we move from left to right on the coordinate plane. The magnitude of the slope, 2, tells us that for every unit increase in x, the y-value increases by two units. This information is crucial for understanding the line's orientation and will be vital in determining the line's equation.
Knowing the slope is just the first step. We now need to find the equation of the line to see if it aligns with the equation 2y - 4x = 389. The slope provides the rate of change, which is a key component in defining the line's behavior across the coordinate plane. The next step will involve using this slope and one of the given points to derive the line's equation in slope-intercept or point-slope form. This will allow us to express the relationship between x and y for all points on the line, enabling us to compare it to the given equation.
2. Finding the Equation of the Line
Having determined the slope of the line, the next crucial step is to derive the equation of the line. Several methods exist for this purpose, but the point-slope form is particularly convenient when we have a slope (m) and a point (x₁, y₁) on the line. The point-slope form is given by:
y - y₁ = m(x - x₁)
We already know the slope, m = 2, and we have two points to choose from: C(-3, 2) and D(-2, 4). Let's use point C(-3, 2) as our (x₁, y₁). Substituting these values into the point-slope form, we get:
y - 2 = 2(x - (-3)) y - 2 = 2(x + 3)
Now, we can simplify this equation to obtain the slope-intercept form (y = mx + b), which is a more familiar and readily interpretable form. Distributing the 2 on the right side, we have:
y - 2 = 2x + 6
Adding 2 to both sides to isolate y, we get:
y = 2x + 8
This equation, y = 2x + 8, is the equation of the line passing through points C(-3, 2) and D(-2, 4). It tells us that the line has a slope of 2 and a y-intercept of 8. The y-intercept is the point where the line crosses the y-axis, which occurs when x = 0. In this case, the line intersects the y-axis at the point (0, 8).
Now that we have the equation of the line in slope-intercept form, we can compare it to the given equation, 2y - 4x = 389, to determine if they represent the same line or if the line passing through points C and D satisfies the given equation. This comparison will involve manipulating the equations to a common form and then examining their coefficients and constants.
3. Verifying the Equation
With the equation of the line passing through C(-3, 2) and D(-2, 4) determined as y = 2x + 8, our final task is to verify if this line satisfies the given equation 2y - 4x = 389. To do this, we need to see if the equation we derived is consistent with the given equation. A direct comparison isn't immediately possible due to the different forms of the equations. Therefore, we need to manipulate one or both equations to bring them into a comparable form.
Let's start by substituting the expression for y from our derived equation (y = 2x + 8) into the given equation:
2(2x + 8) - 4x = 389
Now, we simplify the equation by distributing the 2:
4x + 16 - 4x = 389
Combining like terms, we get:
16 = 389
This statement is clearly false. 16 is not equal to 389. This discrepancy indicates that the line represented by the equation y = 2x + 8 does not satisfy the equation 2y - 4x = 389. In other words, the line passing through points C(-3, 2) and D(-2, 4) is not a solution to the equation 2y - 4x = 389.
The fact that we arrived at a false statement means that there is no point on the line y = 2x + 8 that will also lie on the line 2y - 4x = 389. These two lines are parallel and do not intersect. This verification process highlights the importance of algebraic manipulation and logical deduction in analytical geometry. We were able to definitively conclude that the line defined by the two points does not satisfy the given equation.
Conclusion
In conclusion, we meticulously examined whether the line passing through points C(-3, 2) and D(-2, 4) satisfies the equation 2y - 4x = 389. Through a step-by-step process, we first determined the slope of the line, then derived its equation, and finally, verified if this equation aligns with the given condition. Our analysis revealed that the line passing through the points C and D has the equation y = 2x + 8. However, upon substituting this into the given equation, we arrived at a contradiction (16 = 389), conclusively demonstrating that the line does not satisfy the equation 2y - 4x = 389. This exercise underscores the importance of understanding the relationships between points, slopes, and equations in coordinate geometry. It also reinforces the power of algebraic manipulation in solving mathematical problems and verifying geometric conditions.
