Finding A Line's Equation From Intercepts: A Step-by-Step Guide

by Tim Redaksi 64 views
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Hey guys! Let's break down how to find the equation of a line when you're given its intercepts. We'll be using the handy-dandy intercept form: x/a + y/b = 1. This method is super useful, and I'll walk you through it step by step, so you can totally nail it. We're given two points: (0, 3) and (7, 0). These are our y-intercept and x-intercept respectively. Let's get started!

Understanding the Intercept Form: The Key to the Puzzle

First off, let's get friendly with the intercept form. This form of a linear equation is a real lifesaver when you're dealing with intercepts. In the equation x/a + y/b = 1:

  • a represents the x-intercept. This is the point where the line crosses the x-axis, and the y-coordinate is always zero.
  • b represents the y-intercept. This is the point where the line crosses the y-axis, and the x-coordinate is always zero.

Knowing this, we can directly plug in our given points into the intercept form. It’s like having a secret decoder ring! We know that the point (0, 3) is the y-intercept, and (7, 0) is the x-intercept. So, we've got our a and b values ready to go. Remember, the x-intercept is where the line meets the x-axis (y = 0), and the y-intercept is where it meets the y-axis (x = 0). Got it? Awesome!

Now, let's identify the values from the provided points. The x-intercept is 7, and the y-intercept is 3. It's that simple, my friends! Because the x-intercept is the point where y = 0, and the y-intercept is the point where x = 0. Got it?

Alright, let’s go a bit deeper, shall we? The intercept form provides a direct path to the equation. Once we grasp the meaning of 'a' and 'b,' the rest is a piece of cake. This formula is efficient and specifically tailored to situations where intercepts are given, making it a go-to method for these types of problems.

Plugging in the Values: Making it Real

Now comes the fun part: plugging in the values! We know that a (the x-intercept) is 7, and b (the y-intercept) is 3. Substitute these values into the intercept form equation:

x/7 + y/3 = 1

See how easy that was? We've successfully inserted our known values into the equation. It's like slotting puzzle pieces together, and bam, we're on our way to the solution!

So, why do we use this form? Because it simplifies things when we're given the intercepts directly. The intercept form is a direct route to understanding the line's characteristics. This approach is more streamlined than other methods, such as point-slope or slope-intercept forms, specifically when intercepts are given. It saves us a bunch of steps!

Also, let's not forget the strategic advantage of this form. Because it isolates the intercepts, it gives you a quick visual understanding of how the line interacts with the axes. This can be super helpful for sketching the line or understanding its position in the coordinate plane. Are you with me?

Clearing the Fractions and Simplifying

Next, we'll get rid of those pesky fractions. To do this, we need to find the least common multiple (LCM) of the denominators, which are 7 and 3. The LCM of 7 and 3 is 21. Multiply both sides of the equation by 21:

21 * (x/7 + y/3) = 21 * 1 21 * (x/7) + 21 * (y/3) = 21 3x + 7y = 21

This is where things start to look really nice, right? No more fractions, and we're getting closer to the standard form of a linear equation. We are converting the intercept form to a more common and easily recognizable format. The goal here is to make the equation user-friendly. By removing the fractions, we've made the equation easier to read and work with, setting the stage for the final step. So, what do you think? Are you enjoying this?

Why did we multiply by 21? Because it's the LCM of our denominators. Multiplying by the LCM clears the fractions, allowing us to deal with whole numbers. This is a common practice in algebra to simplify equations and make them easier to solve and manipulate.

Rearranging into Standard Form: The Final Touch

Almost there, guys! We need to rearrange the equation to match the standard form, which is Ax + By + C = 0. To do this, simply subtract 21 from both sides of the equation:

3x + 7y - 21 = 0

And there you have it! This is our final equation. It’s now in standard form, and we can easily compare it to the answer choices. We transformed our equation into a familiar and standard format, making it directly comparable to the answer choices. This is the format most commonly used and recognized in algebra. This step is about finalizing the equation so it's perfectly matched with the available options.

So, what does standard form give us? It gives us the ability to easily identify the coefficients (A, B, and C) of our linear equation. It also facilitates comparison with other equations or solutions. It's like the polished, final version of our equation. It is also a very useful form for various algebraic manipulations and analyses.

Choosing the Correct Answer: The Grand Finale

Now, let’s look at the multiple-choice options and see which one matches our final equation: 3x + 7y - 21 = 0.

The correct answer is B. 3x + 7y - 21 = 0. Congrats, we did it!

Let's recap what we've done:

  1. Understood the Intercept Form: We learned that x/a + y/b = 1, where 'a' is the x-intercept and 'b' is the y-intercept.
  2. Identified Intercepts: We found that (0, 3) gave us b = 3 and (7, 0) gave us a = 7.
  3. Plugged in Values: We put these values into the intercept form, which gave us x/7 + y/3 = 1.
  4. Cleared Fractions: We multiplied the whole equation by 21, resulting in 3x + 7y = 21.
  5. Rearranged to Standard Form: We subtracted 21 from both sides, arriving at 3x + 7y - 21 = 0.
  6. Selected the Correct Answer: We matched our final equation with the multiple-choice options and found our correct solution.

And that's the whole shebang, guys! You now know how to find the equation of a line when you're given its intercepts. Keep practicing, and you'll be a pro in no time! Remember, understanding the method and the different forms of linear equations is a key part of math. Great job, everyone!

I hope this explanation was helpful and easy to follow. If you have any more questions, feel free to ask. Keep up the awesome work!