Finding Parallel Lines Equation Through A Point A Step-by-Step Guide

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Hey guys! Let's dive into a fun math problem where we're going to find the equation of a line. This line has to be parallel to a given line and pass through a specific point. Sounds like a mission? Let's get started!

Understanding Parallel Lines

Before we jump into the problem, let's refresh our memory about parallel lines. In the world of coordinate geometry, parallel lines are lines that never intersect. They run alongside each other forever without ever meeting. The most important characteristic of parallel lines is that they have the same slope. This means if we have a line with a certain steepness (slope), any line parallel to it will have the exact same steepness.

The Slope-Intercept Form

The equation of a line is often written in what we call slope-intercept form: y = mx + b. In this form:

  • m represents the slope of the line.
  • b represents the y-intercept, which is the point where the line crosses the y-axis.

Understanding this form is crucial because it helps us quickly identify the slope of a line, which is exactly what we need for our problem.

Point-Slope Form: The Key to Our Solution

Now, there's another form of a linear equation that's going to be super handy for us: the point-slope form. This form is written as:

y - y₁ = m(x - x₁)

Where:

  • m is the slope of the line.
  • (x₁, y₁) is a point that the line passes through.

The point-slope form is perfect for situations where we know a point on the line and the slope, which is exactly the kind of information we have in our problem.

Our Specific Problem: A Step-by-Step Solution

Okay, let's get down to the problem at hand. We need to find the equation of a line that is parallel to the line y = -9x - 1 and passes through the point (2, -3).

Step 1: Identify the Slope of the Given Line

The first thing we need to do is figure out the slope of the line y = -9x - 1. Remember our slope-intercept form (y = mx + b)? The slope (m) is the coefficient of the x term. In this case, the slope of the given line is -9. That's a pretty steep line going downwards from left to right!

Step 2: Parallel Lines Have the Same Slope

Since we want a line that's parallel to y = -9x - 1, our new line will also have a slope of -9. This is the golden rule for parallel lines – same slope, different y-intercept (otherwise, they'd be the same line!).

Step 3: Use the Point-Slope Form

Now, we're going to use the point-slope form to build the equation of our new line. We know the slope (m = -9) and a point it passes through ((2, -3)). Let's plug these values into the point-slope form:

y - y₁ = m(x - x₁) y - (-3) = -9(x - 2)

Notice how we substituted y₁ with -3 and x₁ with 2. This is where the point-slope form really shines!

Step 4: Simplify the Equation

Let's simplify the equation we got in the last step. First, we'll deal with the double negative:

y + 3 = -9(x - 2)

Next, we'll distribute the -9 on the right side of the equation:

y + 3 = -9x + 18

Finally, let's isolate y to get the equation in slope-intercept form. We'll subtract 3 from both sides:

y = -9x + 15

The Green Box Value

Now, let's revisit the original question. We were asked to find the value that belongs in the green box in the equation:

y - [?] = [ ](x - [ ])

This is the point-slope form we used earlier. We plugged in the point (2, -3) into the equation, which gave us:

y - (-3) = -9(x - 2)

So, the value that belongs in the green box next to y is -3 because we have y - (-3). This is the y-coordinate of the point (2, -3) that the line passes through. Remember that subtracting a negative number is the same as adding a positive number, so y - (-3) is the same as y + 3.

Rewriting the Equation

The question also has blanks for the slope and the x-coordinate of the point. Let's fill those in as well.

We already know the slope is -9 because the line is parallel to y = -9x - 1. The x-coordinate of the point (2, -3) is 2.

So, the complete equation in point-slope form is:

y - (-3) = -9(x - 2)

Or, simplified:

y + 3 = -9(x - 2)

Putting It All Together

So, guys, we've successfully found the equation of the line parallel to y = -9x - 1 that passes through the point (2, -3). We did it by:

  1. Identifying the slope of the given line.
  2. Understanding that parallel lines have the same slope.
  3. Using the point-slope form of a linear equation.
  4. Plugging in our values and simplifying.

We found that the value that belongs in the green box is -3. And we rewrote the complete equation in point-slope form as y + 3 = -9(x - 2).

Visualizing the Solution

It's always helpful to visualize what we've done. Imagine a coordinate plane with the line y = -9x - 1. It's a steep line sloping downwards. Now, picture another line parallel to it, with the same steepness, but passing through the point (2, -3). That's the line we just found! y = -9x + 15

If you were to graph these two lines, you'd see that they never intersect, confirming that they are indeed parallel.

Why This Matters

Finding equations of parallel lines isn't just a math exercise. It has real-world applications in fields like:

  • Architecture: Ensuring that structures have parallel lines for stability and aesthetics.
  • Engineering: Designing roads and bridges that run parallel to each other.
  • Computer Graphics: Creating parallel lines in 2D and 3D models.

Understanding the concepts behind parallel lines and how to find their equations is a valuable skill in many areas.

Let's Recap: Key Takeaways

Before we wrap up, let's quickly recap the key concepts we've learned:

  • Parallel lines have the same slope.
  • The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept.
  • The point-slope form of a line is y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line.
  • We can use the point-slope form to find the equation of a line when we know the slope and a point it passes through.

Practice Makes Perfect

Now that we've walked through this problem together, the best way to solidify your understanding is to practice! Try finding equations of parallel lines with different slopes and points. You can even create your own problems and challenge yourself.

Extra Challenge:

What if we wanted to find a line perpendicular to y = -9x - 1 that passes through (2, -3)? How would that change our approach? (Hint: Perpendicular lines have slopes that are negative reciprocals of each other.)

Final Thoughts

So there you have it, guys! We've successfully navigated the world of parallel lines and found the equation of a line that fits our criteria. Remember, math can be fun, especially when we break it down step by step. Keep practicing, keep exploring, and keep those math skills sharp!