Equation Of A Line Through Two Points: A Detailed Solution

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Equation of a Line Through Two Points: A Detailed Solution

Hey guys! Let's dive into a super important concept in math: finding the equation of a line when you're given two points. This is something that comes up a lot, whether you're in algebra, calculus, or even in real-world applications. So, let's break it down step by step. We'll tackle the question: What is the equation of the line that passes through the points A(2, 3) and B(4, 7)? Explain your answer.

Understanding the Basics of Linear Equations

Before we jump into solving this specific problem, it’s essential to understand the fundamental form of a linear equation. A linear equation represents a straight line on a coordinate plane, and the most common form we use is the slope-intercept form:

Slope-Intercept Form: y = mx + b

In this equation:

  • y represents the vertical coordinate.
  • x represents the horizontal coordinate.
  • m is the slope of the line, indicating how steep the line is and its direction (positive or negative).
  • b is the y-intercept, the point where the line crosses the y-axis.

Knowing this form is crucial because our goal is to find the values of m (slope) and b (y-intercept) for the line that passes through the given points. Once we have these values, we can write the equation of the line.

What is Slope?

The slope, often denoted as m, is a measure of the steepness and direction of a line. It tells us how much the y-value changes for each unit change in the x-value. In simpler terms, it’s the “rise over run” – the vertical change divided by the horizontal change between any two points on the line.

The formula to calculate the slope (m) between two points (x₁, y₁) and (x₂, y₂) is:

m = (y₂ - y₁) / (x₂ - x₁)

This formula is super important, so make sure you’ve got it down! The slope can be positive, negative, zero, or undefined:

  • Positive Slope: The line goes upwards from left to right.
  • Negative Slope: The line goes downwards from left to right.
  • Zero Slope: The line is horizontal.
  • Undefined Slope: The line is vertical.

Understanding the slope is key to visualizing and defining the line we’re trying to find.

What is the Y-Intercept?

The y-intercept, denoted as b, is the point where the line intersects the y-axis. At this point, the x-coordinate is always 0. So, the y-intercept is the y-value when x = 0. It’s the point (0, b).

The y-intercept is essential because it tells us where the line starts on the vertical axis. When we write the equation of a line in slope-intercept form (y = mx + b), the y-intercept is the constant term, b. This value helps us to fully define the line’s position on the coordinate plane.

Step-by-Step Solution to Find the Equation

Now that we've covered the basics, let's get to the heart of the problem. We're given two points, A(2, 3) and B(4, 7), and our mission is to find the equation of the line that passes through these points. Here’s how we’ll do it:

Step 1: Calculate the Slope (m)

First, we need to find the slope of the line. We use the slope formula:

m = (y₂ - y₁) / (x₂ - x₁)

Let’s plug in the coordinates of our points A(2, 3) and B(4, 7):

  • x₁ = 2
  • y₁ = 3
  • x₂ = 4
  • y₂ = 7

So, the slope m is:

m = (7 - 3) / (4 - 2) = 4 / 2 = 2

Awesome! We've found that the slope of our line is 2. This means that for every 1 unit we move to the right on the x-axis, the line goes up 2 units on the y-axis.

Step 2: Use the Slope-Intercept Form (y = mx + b)

Now that we have the slope, we can use the slope-intercept form of the equation, y = mx + b, and plug in the value of m:

y = 2x + b

We’re halfway there! Now we need to find the y-intercept, b.

Step 3: Find the Y-Intercept (b)

To find the y-intercept, we can use one of the given points (A or B) and substitute its coordinates into the equation y = 2x + b. Let’s use point A(2, 3):

  • x = 2
  • y = 3

Plug these values into our equation:

3 = 2(2) + b

Now, solve for b:

3 = 4 + b b = 3 - 4 b = -1

Great! We've found the y-intercept, b, which is -1. This means the line crosses the y-axis at the point (0, -1).

Step 4: Write the Equation of the Line

Now we have both the slope (m = 2) and the y-intercept (b = -1). We can plug these values into the slope-intercept form y = mx + b:

y = 2x + (-1)

Simplify it:

y = 2x - 1

And there you have it! The equation of the line that passes through the points A(2, 3) and B(4, 7) is y = 2x - 1.

Verifying Our Solution

To make sure our equation is correct, we can plug in the coordinates of both points A and B into the equation and see if they satisfy it. This is a great way to double-check your work and ensure you haven’t made any mistakes.

Verification with Point A(2, 3)

Let’s plug in x = 2 and y = 3 into our equation y = 2x - 1:

3 = 2(2) - 1 3 = 4 - 1 3 = 3

It checks out! Point A satisfies the equation.

Verification with Point B(4, 7)

Now, let’s do the same with point B(4, 7):

7 = 2(4) - 1 7 = 8 - 1 7 = 7

Perfect! Point B also satisfies the equation. Since both points fit the equation, we can be confident that our solution is correct.

Conclusion: Mastering Linear Equations

So, to recap, we found the equation of the line that passes through the points A(2, 3) and B(4, 7) by following these steps:

  1. Calculated the slope (m) using the slope formula.
  2. Used the slope-intercept form (y = mx + b) and plugged in the slope.
  3. Found the y-intercept (b) by substituting one of the points into the equation.
  4. Wrote the final equation using the values of m and b.
  5. Verified the solution by plugging in both points into the equation.

Understanding how to find the equation of a line given two points is a fundamental skill in math. It’s not just about memorizing formulas; it’s about understanding the relationship between points, slopes, and intercepts. Keep practicing, and you’ll become a pro in no time!

By understanding these concepts and practicing, you’ll be well-equipped to tackle any linear equation problem that comes your way. Keep up the great work, and happy math-solving, guys!