Find The Equation Of A Line: Point-Slope To Standard Form

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Find the Equation of a Line: Point-Slope to Standard Form

Hey guys! Let's break down how to find the equation of a line when you're given a point it passes through and its slope. This is a classic problem in algebra, and mastering it will definitely help you out. We'll go through it step by step, making sure everything's crystal clear. Let's dive in!

Understanding the Problem

The problem we're tackling is: Which equation in standard form represents a line that goes through the point (-4, 2) and has a slope of 9/2? We have four options to choose from:

A. 9x - 2y = 36 B. 9x - 2y = 26 C. 9x - 2y = -40 D. 9x - 2y = -10

To solve this, we'll need to use the point-slope form of a line equation and then convert it to standard form. Don't worry; we'll take it one step at a time.

Point-Slope Form

The point-slope form is a super handy way to write the equation of a line when you know a point (x₁, y₁) on the line and the slope (m). The formula looks like this:

y - y₁ = m(x - x₁)

In our case, we have the point (-4, 2), so x₁ = -4 and y₁ = 2. We also have the slope m = 9/2. Let's plug these values into the point-slope form:

y - 2 = (9/2)(x - (-4))

Simplifying the Equation

First, let's simplify the equation by dealing with that double negative inside the parentheses:

y - 2 = (9/2)(x + 4)

Next, we'll distribute the 9/2 across the (x + 4) term:

y - 2 = (9/2)x + (9/2)(4) y - 2 = (9/2)x + 18

Now, let's get rid of the fraction by multiplying every term in the equation by 2:

2(y - 2) = 2((9/2)x + 18) 2y - 4 = 9x + 36

Converting to Standard Form

The standard form of a linear equation is Ax + By = C, where A, B, and C are integers, and A is usually positive. We need to rearrange our equation to match this form. Currently, we have:

2y - 4 = 9x + 36

Let's move the terms around to get the x and y terms on the same side and the constant on the other. We'll subtract 9x from both sides:

2y - 4 - 9x = 9x + 36 - 9x -9x + 2y - 4 = 36

Next, add 4 to both sides:

-9x + 2y - 4 + 4 = 36 + 4 -9x + 2y = 40

Remember, we want A to be positive, so let's multiply the entire equation by -1:

(-1)(-9x + 2y) = (-1)(40) 9x - 2y = -40

Matching the Answer

Alright, we've arrived at the standard form equation: 9x - 2y = -40. Now, let's compare this to our options:

A. 9x - 2y = 36 B. 9x - 2y = 26 C. 9x - 2y = -40 D. 9x - 2y = -10

It looks like option C, 9x - 2y = -40, matches our result perfectly! So, the correct answer is C.

Quick Recap

To recap, here's what we did:

  1. Used the point-slope form: y - y₁ = m(x - x₁)
  2. Plugged in the given point (-4, 2) and slope 9/2.
  3. Simplified the equation.
  4. Converted the equation to standard form: Ax + By = C
  5. Found the matching option.

Why This Works

The beauty of this method is that it breaks down a potentially tricky problem into manageable steps. The point-slope form is derived directly from the definition of slope, which is the change in y over the change in x. When you have a specific point and a slope, you have enough information to define the line uniquely.

Converting to standard form helps us compare the equation easily with given options and also fits a common convention in algebra. It's like speaking the same language! Standard form makes it straightforward to identify coefficients and constants, which can be useful for other types of analysis and graphing.

Common Mistakes to Avoid

It's easy to make little mistakes when you're working through these problems, so let's highlight a few common pitfalls:

  • Sign Errors: Be super careful with those negative signs! A wrong sign can throw off your entire calculation.
  • Distribution Errors: When you distribute, make sure you multiply every term inside the parentheses. Missing one term can lead to an incorrect equation.
  • Fraction Phobia: Don't let fractions scare you. Multiplying through by the denominator is a neat trick to clear them out and simplify the equation.
  • Standard Form Mix-up: Remember, standard form is Ax + By = C. Make sure your equation matches this format at the end.

Practice Makes Perfect

The best way to get comfortable with these types of problems is to practice. Try some similar problems with different points and slopes. You can even make up your own problems and solve them. The more you practice, the more natural the process will become.

Real-World Applications

You might be wondering, "When am I ever going to use this in real life?" Well, linear equations are everywhere! They can model relationships between variables in all sorts of situations, such as:

  • Physics: Describing motion with constant velocity.
  • Economics: Modeling cost and revenue functions.
  • Engineering: Designing structures and systems.
  • Everyday Life: Calculating the cost of a taxi ride or figuring out how much to save each month to reach a financial goal.

Understanding linear equations is a foundational skill that opens the door to many other areas of math and science.

Let's Try Another Example

Let’s try one more example to really nail this down. Suppose we want to find the equation of a line in standard form that passes through the point (1, -3) and has a slope of -2.

  1. Use the point-slope form: y - y₁ = m(x - x₁)
  2. Plug in the values: y - (-3) = -2(x - 1)
  3. Simplify: y + 3 = -2x + 2
  4. Convert to standard form: 2x + y = -1

See? It's the same process, just with different numbers. Keep practicing, and you'll become a pro in no time!

Final Thoughts

So, there you have it! We've walked through how to find the equation of a line in standard form when given a point and a slope. Remember, the key is to use the point-slope form, simplify, and then convert to standard form. Keep an eye out for those common mistakes, and don't be afraid to practice. You've got this!

If you have any questions or want to try more examples, just let me know. Happy problem-solving!