Finding Linear Equations Passing Through Specific Points A Mathematical Exploration

In the realm of mathematics, linear equations form the backbone of numerous concepts and applications. When given two points, the task of finding the equation of the line that passes through them becomes a fundamental skill. This article delves into how to determine the correct linear equation from a set of options, using the points (2, 15) and (0, 5) as our guide. We will explore different forms of linear equations and methods to verify if a given equation satisfies the provided points.

Understanding the Basics of Linear Equations

Before diving into the specific problem, it’s essential to grasp the core concepts of linear equations. A linear equation represents a straight line on a coordinate plane. The most common form is the slope-intercept form, expressed as y = mx + b, where 'm' represents the slope of the line and 'b' represents the y-intercept (the point where the line crosses the y-axis). The slope indicates the steepness and direction of the line, while the y-intercept provides a fixed point on the line.

Another significant form is the point-slope form, given by y - y1 = m(x - x1), where (x1, y1) is a known point on the line and 'm' is the slope. This form is particularly useful when you have a point and the slope or when you have two points and need to find the equation. The standard form of a linear equation is Ax + By = C, where A, B, and C are constants. Each form has its advantages, but they all describe the same straight line.

Calculating the Slope and Y-Intercept

The first step in finding the equation of a line passing through two points is determining the slope. 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. Using the given points (2, 15) and (0, 5), we can calculate the slope as follows:

m = (15 - 5) / (2 - 0) = 10 / 2 = 5

Thus, the slope of the line passing through the points (2, 15) and (0, 5) is 5. This means that for every one unit increase in x, y increases by five units. The slope is a crucial parameter that dictates the line's inclination. Next, we need to find the y-intercept. The y-intercept is the value of y when x is 0. Fortunately, we are given the point (0, 5), which directly tells us that the y-intercept is 5. Therefore, b = 5.

With the slope (m = 5) and the y-intercept (b = 5), we can now write the equation of the line in slope-intercept form: y = 5x + 5. This equation is a crucial benchmark for assessing the given options. It explicitly defines the line's behavior, allowing us to compare it with other forms of the equation. Understanding the slope and y-intercept not only helps in writing the equation but also in visualizing the line's position and direction on the coordinate plane.

Analyzing the Given Equations

Now that we have established the fundamental equation y = 5x + 5, we can analyze the provided options to determine which one matches our line. Let's examine each equation in detail:

Equation I: y + 10 = 5(x + 2)

To determine if Equation I, y + 10 = 5(x + 2), represents the same line, we need to manipulate it into a more recognizable form, such as the slope-intercept form (y = mx + b). Let’s expand and simplify the equation:

y + 10 = 5x + 10

Subtracting 10 from both sides, we get:

y = 5x

This equation, y = 5x, has a slope of 5, which matches the slope we calculated earlier. However, the y-intercept is 0, not 5. This means the line represented by Equation I passes through the origin (0, 0) but not through the point (0, 5), which we know is on our line. To further confirm, we can substitute the given points (2, 15) and (0, 5) into Equation I:

For (2, 15): 15 + 10 = 5(2 + 2) → 25 = 20, which is false. For (0, 5): 5 + 10 = 5(0 + 2) → 15 = 10, which is also false.

Since neither point satisfies Equation I, it does not represent the line we are looking for. The discrepancy in the y-intercept and the failure of the points to satisfy the equation clearly indicate that Equation I is not a valid solution. Therefore, we can confidently exclude this option.

Equation II: 5x - y = -6

Next, we analyze Equation II: 5x - y = -6. To determine if this equation represents the line passing through (2, 15) and (0, 5), we need to convert it to slope-intercept form (y = mx + b). Let’s rearrange the equation:

Subtract 5x from both sides:

-y = -5x - 6

Multiply both sides by -1:

y = 5x + 6

Now, we can see that the slope of this line is 5, which matches the slope we calculated earlier. However, the y-intercept is 6, not 5. This indicates that the line represented by Equation II intersects the y-axis at a different point than our target line. To verify this further, let's substitute the given points (2, 15) and (0, 5) into the original Equation II:

For (2, 15): 5(2) - 15 = -6 → 10 - 15 = -6 → -5 = -6, which is false. For (0, 5): 5(0) - 5 = -6 → -5 = -6, which is also false.

Since neither point satisfies Equation II, it does not represent the line we are looking for. The different y-intercept and the failure of the points to satisfy the equation confirm that Equation II is not a valid solution. This rigorous verification process helps to ensure accuracy in determining the correct equation.

Point-Slope Form and Verification

Another effective method to verify the equations is by using the point-slope form, which is y - y1 = m(x - x1). Using the slope m = 5 and one of the points, say (0, 5), we can write the equation as:

y - 5 = 5(x - 0)

Simplifying this, we get:

y - 5 = 5x

y = 5x + 5

This matches our initial slope-intercept form equation, y = 5x + 5. This confirms our calculated slope and y-intercept. To further solidify our understanding, let’s use the point (2, 15) in the point-slope form:

y - 15 = 5(x - 2)

Expanding and simplifying:

y - 15 = 5x - 10

y = 5x + 5

Again, we arrive at the same equation, y = 5x + 5. This consistent result from different methods reinforces the accuracy of our equation. By using the point-slope form, we can quickly generate and verify equations using different points, ensuring that the final equation correctly represents the line passing through the given points. This method is particularly useful in situations where the slope and a point are known, providing a direct pathway to the linear equation.

Conclusion: Finding the Right Fit

In conclusion, to determine which equation represents a line that passes through the points (2, 15) and (0, 5), it is crucial to calculate the slope and y-intercept accurately. By calculating the slope as 5 and identifying the y-intercept as 5, we derived the equation y = 5x + 5. Through a detailed analysis of the given equations, we found that neither Equation I (y + 10 = 5(x + 2)) nor Equation II (5x - y = -6) matches the correct equation.

The step-by-step approach, including converting equations to slope-intercept form and substituting the given points, ensures a thorough and accurate solution. This method not only provides the answer but also reinforces the understanding of linear equations and their properties. Linear equations are a cornerstone of mathematics, and mastering the techniques to find and verify them is essential for further studies in algebra and calculus. This exploration highlights the importance of a systematic approach and the versatility of different forms of linear equations in problem-solving.