Find The Equation Of The Line That Passes Through The Points (-5, 1) And (-11, -3).

Finding linear equations is a fundamental concept in mathematics, with applications ranging from basic algebra to advanced calculus and data analysis. This article will provide a comprehensive guide on how to determine the equation of a line, focusing on the specific problem of finding the equation of the line passing through the points (-5, 1) and (-11, -3). We will cover the essential concepts, step-by-step methods, and provide detailed explanations to ensure a thorough understanding.

Understanding Linear Equations

Before diving into the problem, let's establish a solid understanding of what linear equations are and their significance. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. In simpler terms, when graphed on a coordinate plane, a linear equation represents a straight line. The general form of a linear equation is:

The Slope-Intercept Form

One of the most common and useful forms for representing linear equations is the slope-intercept form:

y = mx + b

Where:

• y is the dependent variable (typically plotted on the vertical axis).
• x is the independent variable (typically plotted on the horizontal axis).
• m represents the slope of the line, indicating its steepness and direction.
• b is the y-intercept, the point where the line crosses the y-axis (where x = 0).

The slope (m) is a crucial concept, defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. It quantifies how much the y-value changes for every unit change in the x-value. A positive slope indicates an increasing line (going upwards from left to right), while a negative slope indicates a decreasing line.

The y-intercept (b) is the point where the line intersects the y-axis. It provides a fixed reference point for the line's position on the coordinate plane. Knowing the slope and y-intercept allows us to easily visualize and graph the line.

The Point-Slope Form

Another important form is the point-slope form, which is particularly useful when we know a point on the line and its slope:

y - y1 = m(x - x1)

Where:

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

This form is derived directly from the definition of the slope and provides a straightforward way to construct the equation of a line when a point and the slope are known. It's especially helpful when dealing with problems where the y-intercept isn't immediately apparent.

Why Linear Equations Matter

Linear equations are fundamental in mathematics and have a wide array of applications in various fields. Understanding them is crucial for several reasons:

• Modeling Real-World Phenomena: Many real-world situations can be approximated using linear models. For example, the relationship between time and distance traveled at a constant speed, or the relationship between the number of items produced and the total cost.
• Solving Problems: Linear equations provide a framework for solving various problems, from simple algebraic puzzles to more complex optimization challenges.
• Foundation for Advanced Concepts: Linear equations serve as a building block for more advanced mathematical concepts, such as linear algebra, calculus, and differential equations.
• Data Analysis and Statistics: Linear regression, a powerful statistical technique, relies heavily on the principles of linear equations to model relationships between variables.

Step-by-Step Solution: Finding the Equation

Now, let's tackle the problem of finding the equation of the line passing through the points (-5, 1) and (-11, -3). We'll break down the solution into clear, manageable steps.

Step 1: Calculate the Slope (m)

The first step in finding the linear equation is to determine the slope (m) of the line. The slope represents the rate of change of y with respect to x. Given two points (x1, y1) and (x2, y2), the slope can be calculated using the formula:

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

In our case, the given points are (-5, 1) and (-11, -3). Let's assign the coordinates:

• (x1, y1) = (-5, 1)
• (x2, y2) = (-11, -3)

Now, substitute these values into the slope formula:

m = (-3 - 1) / (-11 - (-5))

Simplify the expression:

m = (-4) / (-11 + 5)

m = -4 / -6

m = 2/3

Therefore, the slope of the line is 2/3. This positive slope indicates that the line is increasing, meaning it goes upwards from left to right. For every 3 units we move to the right on the x-axis, the line rises 2 units on the y-axis.

Step 2: Use the Point-Slope Form

Now that we have calculated the slope (m = 2/3), we can use the point-slope form of a linear equation to find the equation of the line. The point-slope form is:

y - y1 = m(x - x1)

We have the slope (m = 2/3) and two points to choose from: (-5, 1) and (-11, -3). It doesn't matter which point we choose; the final equation will be the same. Let's use the point (-5, 1), so:

• x1 = -5
• y1 = 1

Substitute these values and the slope into the point-slope form:

y - 1 = (2/3)(x - (-5))

Simplify the equation:

y - 1 = (2/3)(x + 5)

This is the equation of the line in point-slope form. While this form is perfectly valid, it's often desirable to convert it to slope-intercept form for easier interpretation and graphing.

Step 3: Convert to Slope-Intercept Form (y = mx + b)

To convert the equation from point-slope form to slope-intercept form (y = mx + b), we need to isolate y. Start by distributing the 2/3 on the right side of the equation:

y - 1 = (2/3)x + (2/3)(5)

y - 1 = (2/3)x + 10/3

Now, add 1 to both sides of the equation to isolate y:

y = (2/3)x + 10/3 + 1

To add the fractions, we need a common denominator. Convert 1 to 3/3:

y = (2/3)x + 10/3 + 3/3

Combine the fractions:

y = (2/3)x + 13/3

This is the equation of the line in slope-intercept form. We can now clearly see the slope (m = 2/3) and the y-intercept (b = 13/3).

The Final Equation

The equation of the line passing through the points (-5, 1) and (-11, -3) is:

y = (2/3)x + 13/3

This equation tells us that the line has a slope of 2/3 and intersects the y-axis at the point (0, 13/3). We can use this equation to find any point on the line by substituting a value for x and solving for y, or vice versa.

Verification and Alternative Methods

To ensure our solution is correct, we can verify it by substituting the coordinates of the given points into the equation. If both points satisfy the equation, then our solution is correct.

Verification with Point (-5, 1)

Substitute x = -5 and y = 1 into the equation:

1 = (2/3)(-5) + 13/3

1 = -10/3 + 13/3

1 = 3/3

1 = 1

The equation holds true for the point (-5, 1).

Verification with Point (-11, -3)

Substitute x = -11 and y = -3 into the equation:

-3 = (2/3)(-11) + 13/3

-3 = -22/3 + 13/3

-3 = -9/3

-3 = -3

The equation also holds true for the point (-11, -3). Since both points satisfy the equation, we can be confident that our solution is correct.

Alternative Method: Using Two-Point Form

Another method to find the equation of a line given two points is to use the two-point form directly. The two-point form is:

(y - y1) / (x - x1) = (y2 - y1) / (x2 - x1)

Using the points (-5, 1) and (-11, -3), we have:

• (x1, y1) = (-5, 1)
• (x2, y2) = (-11, -3)

Substitute these values into the two-point form:

(y - 1) / (x - (-5)) = (-3 - 1) / (-11 - (-5))

(y - 1) / (x + 5) = -4 / -6

(y - 1) / (x + 5) = 2/3

Cross-multiply:

3(y - 1) = 2(x + 5)

3y - 3 = 2x + 10

Isolate y:

3y = 2x + 13

y = (2/3)x + 13/3

This method directly yields the slope-intercept form of the equation, confirming our previous result.

Common Mistakes to Avoid

When working with linear equations, several common mistakes can lead to incorrect results. Being aware of these pitfalls can help you avoid them.

• Incorrectly Calculating the Slope: The slope formula involves subtraction, so ensure you subtract the coordinates in the correct order. A mix-up in the order can lead to a sign error, resulting in an incorrect slope.
• Sign Errors: Pay close attention to signs, especially when dealing with negative numbers. A small sign error can significantly change the equation and the resulting line.
• Incorrectly Substituting Values: Ensure you substitute the values into the correct places in the formula. Mixing up x and y coordinates or using the wrong point can lead to errors.
• Algebraic Errors: When simplifying and rearranging equations, be careful with algebraic manipulations. Distribute terms correctly, combine like terms accurately, and avoid basic arithmetic errors.
• Forgetting the Y-Intercept: When converting from point-slope form to slope-intercept form, remember to isolate y completely. Forgetting to add or subtract a constant can result in an incorrect y-intercept.

Applications of Linear Equations

Linear equations are not just abstract mathematical concepts; they have numerous practical applications in various fields. Understanding these applications can provide a deeper appreciation for the importance of linear equations.

• Physics: Linear equations are used to describe motion with constant velocity, relationships between force and acceleration (Newton's second law), and Ohm's law in electrical circuits.
• Economics: Linear equations can model supply and demand curves, cost functions, and revenue functions. They are used to analyze market equilibrium and make economic predictions.
• Engineering: Linear equations are used in structural analysis, circuit design, and control systems. They help engineers model and analyze systems with linear behavior.
• Computer Graphics: Linear equations are fundamental in computer graphics for transformations, projections, and rendering. They are used to manipulate objects in 2D and 3D space.
• Data Analysis: Linear regression, a statistical technique based on linear equations, is widely used to model relationships between variables in datasets. It is used in forecasting, prediction, and understanding trends.

Conclusion

In this article, we have explored the process of finding linear equations, with a specific focus on determining the equation of the line passing through the points (-5, 1) and (-11, -3). We covered the essential concepts of slope, y-intercept, point-slope form, and slope-intercept form. By following the step-by-step method, we found that the equation of the line is:

y = (2/3)x + 13/3

We also verified the solution using both given points and explored an alternative method using the two-point form. By understanding the common mistakes and the wide range of applications of linear equations, you can develop a solid foundation in this fundamental mathematical concept. Mastering linear equations is crucial for success in various fields, and the knowledge gained here will serve as a valuable asset in your mathematical journey. Whether you're a student learning the basics or a professional applying these concepts in your work, a strong understanding of linear equations is essential for problem-solving and analytical thinking.

This comprehensive guide aims to provide you with the tools and understanding necessary to confidently tackle problems involving linear equations. Remember to practice regularly, review the concepts, and apply your knowledge to real-world scenarios to solidify your understanding. With dedication and effort, you can master linear equations and unlock their potential in various fields of study and application.