Equation Of A Line: Points (1,-1) And (3,5)

Hey guys! Today, we're diving into a fundamental concept in mathematics: finding the equation of a line. Specifically, we'll tackle the problem of determining the equation of a line that passes through two given points: $(1, -1)$ and $(3, 5)$. This is a classic problem that pops up in algebra, geometry, and even calculus, so mastering it is super important. So, grab your pencils, and let's get started!

Understanding the Basics

Before we jump into the calculations, let's refresh some key concepts about lines and their equations. A line in a two-dimensional plane can be represented in several forms, but the most common ones are:

• Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
• Point-slope form: y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line.
• Standard form: Ax + By = C, where A, B, and C are constants.

The slope (m) of a line measures its steepness and direction. It tells us how much the y-value changes for every unit change in the x-value. A positive slope means the line goes upwards from left to right, a negative slope means it goes downwards, a zero slope indicates a horizontal line, and an undefined slope signifies a vertical line. The slope is calculated using the formula:

m = (y₂ - y₁) / (x₂ - x₁)

where (x₁, y₁) and (x₂, y₂) are any two distinct points on the line.

The y-intercept (b) is the point where the line crosses the y-axis. It's the value of y when x is equal to 0. Knowing the slope and y-intercept allows us to write the equation of the line in slope-intercept form directly.

The point-slope form is particularly useful when we know the slope of the line and a point that it passes through. This form allows us to quickly write an equation for the line, which can then be converted to other forms if needed. It emphasizes the relationship between the slope, a specific point, and any other point (x, y) on the line.

Now, let’s look more closely at why understanding these basic concepts is so crucial. When dealing with linear equations, the slope-intercept form serves as a cornerstone. It not only provides a clear visual representation of the line—highlighting its steepness and where it intersects the y-axis—but it also simplifies graphing and comparison tasks. For instance, if you need to quickly sketch the line, knowing the slope and y-intercept makes it straightforward to plot one point (the y-intercept) and then use the slope to find another. Moreover, comparing two lines is a breeze; just look at their slopes and y-intercepts to understand their relative positions and behaviors.

On the other hand, the point-slope form shines when you have a point and the slope but aren't immediately sure about the y-intercept. Imagine a scenario where you know the line passes through a specific point and has a certain inclination. This form allows you to directly translate that information into an equation. It’s especially handy in practical problems where you might have a rate of change (the slope) and a starting value (a point), but figuring out the y-intercept directly might be cumbersome. The point-slope form becomes a powerful tool in these contexts, bridging the gap between graphical intuition and algebraic expression.

Lastly, the standard form might seem less intuitive at first glance, but it has its own set of advantages. It's particularly useful when dealing with systems of linear equations, where the standard form streamlines the process of elimination or substitution. By aligning the variables and constants in a predictable manner, the standard form simplifies the algebraic manipulations required to solve the system. It also presents a symmetric way of representing the line, which can be useful in certain theoretical discussions and geometrical interpretations. Each form, therefore, offers a unique perspective and practical advantage, making the mastery of all forms essential for a well-rounded understanding of linear equations.

Step-by-Step Solution

Okay, let's get back to our problem. We have two points, $(1, -1)$ and $(3, 5)$, and we want to find the equation of the line that passes through them. Here's how we can do it:

1. Calculate the Slope

The first step is to find the slope (m) of the line using the formula:

m = (y₂ - y₁) / (x₂ - x₁)

Let's plug in our points: (x₁, y₁) = (1, -1) and (x₂, y₂) = (3, 5).

m = (5 - (-1)) / (3 - 1) = (5 + 1) / 2 = 6 / 2 = 3

So, the slope of the line is 3. This means that for every 1 unit we move to the right along the x-axis, the line goes up 3 units along the y-axis.

2. Use the Point-Slope Form

Now that we have the slope, we can use the point-slope form to write the equation of the line:

y - y₁ = m(x - x₁)

We can use either of the given points, (1, -1) or (3, 5). Let's use (1, -1) as our (x₁, y₁).

y - (-1) = 3(x - 1)

Simplify the equation:

y + 1 = 3x - 3

3. Convert to Slope-Intercept Form (Optional)

If we want to write the equation in slope-intercept form (y = mx + b), we need to isolate y.

Subtract 1 from both sides:

y = 3x - 3 - 1

y = 3x - 4

So, the equation of the line in slope-intercept form is y = 3x - 4. This tells us that the line has a slope of 3 and crosses the y-axis at -4.

4. Convert to Standard Form (Optional)

To convert to standard form (Ax + By = C), we need to rearrange the equation so that x and y terms are on the same side and the constant is on the other side.

Starting from the slope-intercept form:

y = 3x - 4

Subtract 3x from both sides:

-3x + y = -4

Multiply both sides by -1 to make the coefficient of x positive (optional, but often preferred):

3x - y = 4

So, the equation of the line in standard form is 3x - y = 4.

Now, let's dive deeper into the implications of each step and why they are significant in the broader context of linear equations. When we calculate the slope, we are essentially quantifying the rate of change of the line. It’s not just a number; it’s a measure of how much the y-value changes for each unit change in the x-value. This understanding is vital because it connects the algebraic representation (the equation) with the geometric visualization (the line on a graph). For instance, a steep slope indicates a rapid change in y relative to x, while a shallow slope suggests a more gradual change. In real-world applications, the slope might represent anything from the speed of a car (change in distance over time) to the growth rate of a population (change in population size over time). This makes the concept of slope incredibly versatile and applicable beyond the classroom.

When we move to the point-slope form, we’re leveraging a fundamental property of lines: a line is uniquely defined by its slope and any point it passes through. This form, y - y₁ = m(x - x₁), is not just a formula; it’s a statement of this property in algebraic terms. By substituting the slope and the coordinates of a known point, we’re essentially anchoring the line in a specific location while maintaining its orientation (as determined by the slope). This is particularly useful in scenarios where you have partial information about the line and need to construct its equation. The point-slope form allows you to build the equation incrementally, piece by piece, as you gather more information.

The subsequent steps of converting to slope-intercept form (y = mx + b) and standard form (Ax + By = C) serve different purposes. The slope-intercept form provides immediate insights into the line’s behavior: the slope (m) tells us the line’s steepness and direction, while the y-intercept (b) reveals where the line crosses the vertical axis. This form is exceptionally useful for graphing and comparing lines, as we’ve discussed. On the other hand, the standard form, although less intuitive at first, becomes invaluable when dealing with systems of linear equations. It organizes the equation in a way that facilitates algebraic manipulations, such as elimination and substitution, making it easier to solve for the unknowns. Furthermore, the standard form’s symmetry offers certain theoretical advantages in more advanced mathematical contexts.

Alternative Method: Using System of Equations

There's another cool way to solve this problem using a system of equations. Since the points (1, -1) and (3, 5) lie on the line, they must satisfy the equation of the line, which we can write in slope-intercept form: y = mx + b.

We can plug in the coordinates of each point to get two equations:

1. For (1, -1): -1 = m(1) + b => -1 = m + b
2. For (3, 5): 5 = m(3) + b => 5 = 3m + b

Now we have a system of two linear equations with two variables, m and b:

• m + b = -1
• 3m + b = 5

We can solve this system using substitution or elimination. Let's use elimination. Subtract the first equation from the second equation:

(3m + b) - (m + b) = 5 - (-1)

2m = 6

m = 3

Now that we have m, we can substitute it back into either equation to find b. Let's use the first equation:

3 + b = -1

b = -1 - 3

b = -4

So, we found m = 3 and b = -4. Plugging these values into the slope-intercept form, we get:

y = 3x - 4

Which is the same equation we found earlier! This alternative method demonstrates the versatility of algebra and how different approaches can lead to the same solution.

Let's further explore the advantages of using a system of equations. This method not only provides a different perspective on solving the problem but also reinforces key concepts in algebra. By setting up and solving a system of equations, we're essentially applying a more general problem-solving technique that extends beyond just finding the equation of a line. This skill is crucial in various mathematical and real-world contexts, where relationships between variables are often expressed as multiple equations that need to be solved simultaneously.

When we substitute the coordinates of the points into the slope-intercept form, y = mx + b, we're creating a concrete link between the abstract equation and the specific points on the line. Each equation in the system represents a constraint that the line must satisfy. The solution to the system, the values of m and b, then represents the unique line that satisfies both constraints, i.e., passes through both points. This process highlights the power of algebraic representation in capturing geometric relationships.

The elimination method, which we used to solve the system, is a classic technique in linear algebra. It involves manipulating the equations in a way that cancels out one of the variables, allowing us to solve for the other. This method is particularly effective when the coefficients of one of the variables are the same or simple multiples of each other, as was the case here with the b terms. The substitution method, on the other hand, involves solving one equation for one variable and substituting that expression into the other equation. Both methods are valuable tools in an algebraist's toolkit, and choosing the most efficient method often depends on the specific structure of the equations.

By solving for m and b, we're essentially deconstructing the line into its fundamental components: its slope and y-intercept. This provides a deeper understanding of the line's properties and allows us to express it in a standard form that is easy to interpret and compare with other lines. The fact that we arrived at the same equation using both the point-slope method and the system of equations method reinforces the consistency and reliability of these algebraic techniques. It also demonstrates that there's often more than one way to approach a mathematical problem, and choosing the right method can depend on the specific context and the solver's preferences.

Conclusion

So, there you have it! We've successfully found the equation of the line passing through the points (1, -1) and (3, 5) using both the slope-intercept and point-slope forms, as well as the system of equations method. The equation is y = 3x - 4. Remember, practice makes perfect, so try solving similar problems to solidify your understanding. Keep exploring the fascinating world of mathematics, guys! You've got this!

In summary, finding the equation of a line through two points is a foundational skill in mathematics with broad applications. Whether you opt for the direct approach using slope and point-slope form or the more systematic method of solving a system of equations, the key is to grasp the underlying concepts and choose the method that best suits the problem at hand. The ability to move fluidly between different representations of a line—be it the slope-intercept, point-slope, or standard form—is a hallmark of mathematical proficiency. It empowers you not only to solve specific problems but also to reason abstractly and apply mathematical thinking to diverse challenges.

As you continue your mathematical journey, remember that each problem solved is a step forward in building your problem-solving toolkit. The more techniques you master, the better equipped you'll be to tackle complex challenges. Linear equations, in particular, serve as a cornerstone for more advanced topics in calculus, linear algebra, and beyond. So, embrace the challenge, stay curious, and keep exploring the exciting world of mathematics!