Persamaan Garis: Titik (3, -8) Dan Gradien 4

by SLV Team 45 views

Okay guys, let's dive into the fascinating world of straight lines! Specifically, we're going to figure out how to find the equation of a line when we know a point it passes through and its slope (or gradient). In this case, we're given the point (3, -8) and a gradient of 4. So, grab your pencils, and let's get started!

Memahami Konsep Dasar Persamaan Garis

First, let's quickly recap the fundamental concepts. A straight line can be represented by the equation:

y = mx + c

Where:

  • y is the dependent variable (usually plotted on the vertical axis)
  • x is the independent variable (usually plotted on the horizontal axis)
  • m is the gradient (or slope) of the line, indicating its steepness and direction
  • c is the y-intercept, which is the point where the line crosses the y-axis

The gradient (m) tells us how much the y-value changes for every unit change in the x-value. A positive gradient means the line slopes upwards from left to right, while a negative gradient means it slopes downwards. The steeper the line, the larger the absolute value of the gradient.

The y-intercept (c) is simply the value of y when x is 0. It's the point where the line intersects the vertical axis.

Understanding this basic equation is key to solving our problem. We already know the gradient (m), but we need to find the y-intercept (c) using the given point.

Menggunakan Bentuk Titik-Gradien

Now, there's a super handy formula called the point-slope form of a linear equation. This form is perfect for situations like ours, where we have a point and a gradient. The formula looks like this:

y - y₁ = m(x - x₁)

Where:

  • (x₁, y₁) is the given point that the line passes through
  • m is the gradient

This formula is derived directly from the definition of slope, and it provides a straightforward way to determine the equation of a line.

In our case, we have the point (3, -8), so x₁ = 3 and y₁ = -8. We also know the gradient is 4, so m = 4. Let's plug these values into the point-slope form:

y - (-8) = 4(x - 3)

See how we've simply substituted the given values into the formula? Now, it's just a matter of simplifying the equation.

Menyederhanakan Persamaan

Let's simplify the equation step-by-step:

  1. First, handle the double negative: y - (-8) becomes y + 8.

    y + 8 = 4(x - 3)

  2. Next, distribute the 4 on the right side of the equation:

    y + 8 = 4x - 12

  3. Finally, isolate y by subtracting 8 from both sides:

    y = 4x - 12 - 8

    y = 4x - 20

And there you have it! We've simplified the equation into the slope-intercept form (y = mx + c).

Persamaan Garis Akhir

The equation of the line that passes through the point (3, -8) with a gradient of 4 is:

y = 4x - 20

This equation tells us everything we need to know about the line. The gradient is 4, meaning the line slopes upwards steeply. The y-intercept is -20, meaning the line crosses the y-axis at the point (0, -20).

Verifikasi Jawaban

It's always a good idea to double-check our work. One way to do this is to plug the original point (3, -8) into the equation and see if it holds true:

-8 = 4(3) - 20

-8 = 12 - 20

-8 = -8

Since the equation holds true, we can be confident that our answer is correct. Another way to verify is to plot the line using the equation we found and visually confirm that it passes through the given point and has the correct slope. You can use graphing software or even draw it by hand to do this. Visual confirmation can be a powerful tool for understanding and verifying your results.

Contoh Soal Serupa

To really nail this concept, let's look at a few similar examples. This will help solidify your understanding and build your problem-solving skills.

Contoh 1:

Find the equation of the line that passes through the point (-2, 5) with a gradient of -2.

  • Solution:

    Using the point-slope form:

    • y - 5 = -2(x - (-2))
    • y - 5 = -2(x + 2)
    • y - 5 = -2x - 4
    • y = -2x + 1

Contoh 2:

Find the equation of the line that passes through the point (1, -1) with a gradient of 1/2.

  • Solution:

    Using the point-slope form:

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

By working through these examples, you can see how the same process applies regardless of the specific point and gradient given. The key is to remember the point-slope form and apply it systematically.

Kiat Sukses Tambahan

Here are a few extra tips to help you master finding the equation of a line:

  • Always write down the point-slope form: This will help you remember the formula and avoid mistakes.
  • Be careful with signs: Pay close attention to negative signs, especially when substituting values into the formula.
  • Simplify carefully: Take your time when simplifying the equation to avoid arithmetic errors.
  • Verify your answer: Plug the original point into your equation to make sure it holds true.
  • Practice, practice, practice: The more problems you solve, the more comfortable you'll become with the process.

Kesimpulan

So, there you have it! Finding the equation of a line when given a point and a gradient is a straightforward process using the point-slope form. Remember the formula, substitute the values carefully, simplify the equation, and always verify your answer. With a little practice, you'll be a pro at this in no time! This skill is fundamental in many areas of mathematics and physics, so mastering it now will definitely pay off in the long run. Keep practicing, keep learning, and most importantly, have fun with it! Remember, math can be challenging, but it's also incredibly rewarding when you finally grasp a concept. You got this!

If you have any questions or want to explore more about linear equations, feel free to ask. There's always more to learn, and the journey of mathematical discovery is a fascinating one.