Line Equation: Point (1,1) And Slope -5

by Alex Johnson 40 views

Have you ever wondered how to define a straight line using mathematical language? Understanding the equation of a line is a fundamental concept in mathematics, with applications ranging from basic geometry to advanced calculus and beyond. In this comprehensive guide, we'll explore how to find the equation of a line, focusing on a specific example: a line that passes through the point (1,1) and has a slope of -5. Let's dive in!

Understanding the Basics: Slope-Intercept Form

At the heart of defining a line lies the slope-intercept form. This form, expressed as y = mx + b, is your key to unlocking the secrets of any straight line. Let's break down each component:

  • y: Represents the vertical coordinate on the Cartesian plane.
  • x: Represents the horizontal coordinate on the Cartesian plane.
  • m: This is where the magic happens! m denotes the slope of the line, indicating its steepness and direction. A positive slope means the line goes upwards from left to right, a negative slope means it goes downwards, a slope of zero means a horizontal line, and an undefined slope represents a vertical line.
  • b: The y-intercept. This is the point where the line intersects the y-axis (where x = 0). It tells us the line's vertical position on the plane.

Understanding these components is crucial because the slope-intercept form provides a clear and concise way to describe any non-vertical line. It's the foundation upon which we'll build our understanding of finding the equation of a line.

The Point-Slope Form: Your Secret Weapon

While the slope-intercept form is powerful, sometimes we don't immediately have the y-intercept. This is where the point-slope form comes to the rescue. The point-slope form is another way to represent a linear equation and is particularly useful when you know a point on the line and the slope. The formula is:

y - y1 = m(x - x1)

Let's dissect this equation:

  • (x1, y1): This represents a known point on the line. It could be any point that the line passes through.
  • m: As before, this is the slope of the line.
  • x and y: These are the variables representing any point on the line, just like in the slope-intercept form.

The point-slope form allows us to construct the equation of a line even without knowing the y-intercept directly. By plugging in the coordinates of a known point and the slope, we can easily derive the equation. This form is particularly handy in situations where we are given a point and a slope, as in our example problem.

Solving Our Specific Problem: Line Through (1,1) with Slope -5

Now, let's apply our knowledge to solve the specific problem: finding the equation of the line that passes through the point (1,1) and has a slope of -5. We have all the information we need! We'll use the point-slope form to begin, and then convert to slope-intercept form for clarity.

  1. Identify the Given Information:

    • Point (x1, y1) = (1, 1)
    • Slope (m) = -5

  2. Apply the Point-Slope Form:

    • Substitute the values into the point-slope formula: y - y1 = m(x - x1)
    • This gives us: y - 1 = -5(x - 1)

  3. Simplify and Convert to Slope-Intercept Form:

    • Distribute the -5 on the right side: y - 1 = -5x + 5
    • Add 1 to both sides to isolate y: y = -5x + 6

Therefore, the equation of the line that passes through the point (1,1) and has a slope of -5 is y = -5x + 6. We have successfully found the equation using the point-slope form and converting it to the familiar slope-intercept form.

Visualizing the Line

To solidify our understanding, it's helpful to visualize the line. The equation y = -5x + 6 tells us a few things:

  • The line slopes downwards steeply (slope of -5).
  • The line intersects the y-axis at the point (0, 6) (y-intercept of 6).

If you were to graph this line on a Cartesian plane, you would see a line descending from left to right, crossing the y-axis at 6. You could also verify that the point (1, 1) lies on this line by substituting x = 1 into the equation and checking if y = 1.

Alternative Methods: Direct Substitution into Slope-Intercept Form

While we used the point-slope form as our primary approach, there's an alternative method to find the equation of the line. We can directly substitute the given point and slope into the slope-intercept form (y = mx + b) and solve for the y-intercept (b).

  1. Start with the Slope-Intercept Form:

    • y = mx + b

  2. Substitute the Given Slope (m) and Point (x, y):

    • We have m = -5 and the point (1, 1), so we substitute x = 1 and y = 1:
    • 1 = -5(1) + b

  3. Solve for the Y-Intercept (b):

    • Simplify: 1 = -5 + b
    • Add 5 to both sides: b = 6

  4. Write the Equation:

    • Now we have m = -5 and b = 6, so we can write the equation in slope-intercept form:
    • y = -5x + 6

As you can see, this method leads us to the same equation as the point-slope method. It's a valuable alternative, especially when you're comfortable with the slope-intercept form.

Common Mistakes to Avoid

When finding the equation of a line, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them.

  • Incorrectly Applying the Point-Slope Formula: Double-check that you are substituting the x and y coordinates into the correct places in the formula. It's easy to mix them up!
  • Sign Errors: Pay close attention to the signs, especially when dealing with negative slopes or coordinates. A misplaced negative sign can drastically change the equation.
  • Algebraic Errors During Simplification: Be careful when distributing, combining like terms, and isolating variables. A small arithmetic error can lead to an incorrect equation.
  • Forgetting to Convert to Slope-Intercept Form (If Required): If the question specifically asks for the equation in slope-intercept form, make sure you complete the simplification to get the equation in the form y = mx + b.

By being mindful of these common errors and double-checking your work, you can increase your accuracy and confidence in finding the equation of a line.

Real-World Applications of Linear Equations

The ability to find the equation of a line isn't just a theoretical exercise; it has practical applications in various real-world scenarios. Linear equations are used to model relationships between two variables that change at a constant rate. Here are a few examples:

  • Distance and Time: If you're traveling at a constant speed, the relationship between the distance you've traveled and the time elapsed can be represented by a linear equation. The slope represents your speed.
  • Cost and Quantity: The total cost of purchasing a certain number of items at a fixed price per item can be modeled using a linear equation. The slope represents the price per item, and the y-intercept might represent a fixed cost or fee.
  • Temperature Conversion: The relationship between Celsius and Fahrenheit temperatures is linear. You can use a linear equation to convert between the two scales.
  • Depreciation: The value of an asset that decreases linearly over time (e.g., a car) can be modeled using a linear equation. The slope represents the rate of depreciation.

Understanding linear equations allows you to analyze and predict trends in these and many other real-world situations. It's a fundamental tool in problem-solving and decision-making.

Conclusion: Mastering the Equation of a Line

Finding the equation of a line is a crucial skill in mathematics, and mastering it opens doors to a deeper understanding of various mathematical concepts and real-world applications. We've explored the slope-intercept form and the point-slope form, learned how to use them effectively, and even discussed common mistakes to avoid.

By practicing these techniques and applying them to different scenarios, you'll build confidence and proficiency in finding the equation of a line. Whether you're solving a geometry problem, analyzing data, or modeling a real-world situation, the ability to define a line mathematically is a valuable asset.

To further enhance your understanding of linear equations and related concepts, you can explore resources like Khan Academy's Linear Equations section. This is an excellent resource for learning and practicing mathematical skills.