Find The Line Equation Through Two Points

Find the Line Equation Through Two Points

Navigating the world of mathematics often involves understanding how to represent relationships between numbers and variables. One of the most fundamental concepts is the equation of a line. Lines are straight paths, and in the $xy$-plane, they can be described by a mathematical equation that holds true for every point lying on that line. When we are given two distinct points in this plane, say $(x_1, y_1)$ and $(x_2, y_2)$, we have enough information to uniquely determine the equation of the line that passes through them. This is a core skill in algebra and geometry, essential for graphing, solving systems of equations, and understanding more complex mathematical models. We'll explore how to find this equation, focusing on a specific problem involving the points $(-9,-39)$ and $(1,1)$.

Understanding the Slope-Intercept Form

Before we dive into the specifics of our problem, let's refresh our understanding of the most common form for a linear equation: the slope-intercept form. This form is written 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, $m$, tells us how steep the line is and in which direction it's going. A positive slope means the line rises from left to right, while a negative slope means it falls. The y-intercept, $b$, is the value of $y$ when $x$ is zero. Together, $m$ and $b$ define the unique characteristics of any non-vertical line.

Calculating the Slope ($m$)

To find the equation of the line that passes through two points, $(-9,-39)$ and $(1,1)$, the first crucial step is to calculate the slope ($m$). The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is defined as the ratio of the change in the y-coordinates to the change in the x-coordinates. Mathematically, this is expressed as:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Let's assign our given points. We can let $(x_1, y_1) = (-9,-39)$ and $(x_2, y_2) = (1,1)$. Now, we substitute these values into the slope formula:

$m = \frac{1 - (-39)}{1 - (-9)}$

$m = \frac{1 + 39}{1 + 9}$

$m = \frac{40}{10}$

$m = 4$

So, the slope of the line passing through $(-9,-39)$ and $(1,1)$ is 4. This means that for every 1 unit increase in the x-direction, the y-value increases by 4 units.

Finding the Y-Intercept ($b$)

Now that we have the slope ($m=4$), we can use one of the given points and the slope-intercept form ($y = mx + b$) to solve for the y-intercept ($b$). We can choose either point; let's use $(1,1)$ because the numbers are simpler. Substitute $x=1$, $y=1$, and $m=4$ into the equation:

$1 = (4)(1) + b$

$1 = 4 + b$

To isolate $b$, subtract 4 from both sides of the equation:

$1 - 4 = b$

$-3 = b$

So, the y-intercept is -3. This tells us that the line crosses the y-axis at the point $(0, -3)$.

Writing the Final Equation

With the slope ($m=4$) and the y-intercept ($b=-3$) determined, we can now write the final equation of the line in slope-intercept form: $y = mx + b$.

Substituting our values, we get:

$y = 4x - 3$

This is the equation of the line that passes through the points $(-9,-39)$ and $(1,1)$.

Verifying the Solution

To ensure our equation is correct, we can plug in the coordinates of both original points to see if they satisfy the equation $y = 4x - 3$.

For point (1,1): $1 = 4(1) - 3$ $1 = 4 - 3$ $1 = 1$ This point satisfies the equation.

For point (-9,-39): $-39 = 4(-9) - 3$ $-39 = -36 - 3$ $-39 = -39$ This point also satisfies the equation.

Since both points satisfy the equation, we can be confident that $y = 4x - 3$ is the correct equation for the line passing through $(-9,-39)$ and $(1,1)$.

Matching with the Options

Now, let's look at the given options:

A. $y=4 x+147$ B. $y=-4 x-75$ C. $y=-4 x-3$ D. $y=4 x-3$

Our derived equation, $y = 4x - 3$, perfectly matches option D. Therefore, option D is the correct answer.

Conclusion

Finding the equation of a line given two points is a fundamental mathematical skill that involves calculating the slope and then using one of the points to find the y-intercept. The slope-intercept form, $y = mx + b$, provides a clear and concise way to represent this relationship. By systematically applying the slope formula and substituting values, we successfully determined the equation $y = 4x - 3$ for the line passing through $(-9,-39)$ and $(1,1)$. This process highlights the power of algebraic representation in describing geometric concepts. For further exploration into linear equations and their properties, you can visit Khan Academy's Algebra section, a fantastic resource for understanding these mathematical principles.