Line Equation Through Two Points: Find The Solution

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Hey guys! Today, we're diving into a super common problem in algebra: finding the equation of a line when you're given two points it passes through. This might seem tricky at first, but I promise, with a few simple steps, you'll be a pro in no time. We're going to break down a specific example to really nail down the process. Let's get started!

Understanding the Problem

Okay, so the question we're tackling is: What is the equation of the line that passes through the points (-8, 11) and (4, 7/2)? We've also got some multiple-choice options to guide us:

A. y = -5/8 x + 16 B. y = -15/2 x + 71 C. y = -5/8 x + 6 D. y = -15/2 x - 49

To solve this, we need to remember the fundamental forms of a linear equation and how to use them. The most important one here is the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. Our mission is to find m and b using the given points.

Step 1: Calculate the Slope (m)

The slope is the measure of how steep a line is, and it's calculated using the formula:

m = (y2 - y1) / (x2 - x1)

Where (x1, y1) and (x2, y2) are our two points. In this case, let's say:

  • (x1, y1) = (-8, 11)
  • (x2, y2) = (4, 7/2)

Now, plug these values into the slope formula:

m = (7/2 - 11) / (4 - (-8))

First, let's simplify the numerator. We need to convert 11 to a fraction with a denominator of 2:

11 = 22/2

So, the numerator becomes:

7/2 - 22/2 = -15/2

Next, simplify the denominator:

4 - (-8) = 4 + 8 = 12

Now we have:

m = (-15/2) / 12

To divide a fraction by a whole number, we can rewrite the whole number as a fraction (12/1) and then multiply by the reciprocal:

m = (-15/2) * (1/12)

Multiply the numerators and the denominators:

m = -15 / 24

Finally, simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 3:

m = -5/8

So, the slope of our line is -5/8. Awesome! We've got the first piece of the puzzle.

Step 2: Find the y-intercept (b)

Now that we have the slope (m = -5/8), we can use the slope-intercept form (y = mx + b) and one of our points to solve for the y-intercept (b). Let's use the point (-8, 11). Plug the values into the equation:

11 = (-5/8) * (-8) + b

Simplify the equation:

11 = 5 + b

Subtract 5 from both sides to isolate b:

11 - 5 = b

b = 6

So, the y-intercept is 6. We're on a roll!

Step 3: Write the Equation

We've found the slope (m = -5/8) and the y-intercept (b = 6). Now we can plug these values back into the slope-intercept form (y = mx + b) to get the equation of the line:

y = (-5/8)x + 6

Step 4: Check the Answer Choices

Looking back at our multiple-choice options, we see that option C matches our equation:

C. y = -5/8 x + 6

Therefore, the correct answer is C.

Alternative Method: Point-Slope Form

Just to show you another way, let's briefly talk about the point-slope form of a linear equation. It's super useful, especially when you have a point and a slope. The point-slope form is:

y - y1 = m(x - x1)

Where (x1, y1) is a point on the line and m is the slope. We already found the slope (m = -5/8). Let's use the point (-8, 11) again. Plug these values into the point-slope form:

y - 11 = (-5/8)(x - (-8))

Simplify:

y - 11 = (-5/8)(x + 8)

Now, let's distribute the -5/8:

y - 11 = (-5/8)x - 5

Add 11 to both sides:

y = (-5/8)x + 6

See? We get the same equation as before! The point-slope form is just another tool in your arsenal for tackling these types of problems.

Key Takeaways

Okay, let's recap the main things we learned today:

  1. Slope Formula: Remember the slope formula: m = (y2 - y1) / (x2 - x1). It's the foundation for finding the steepness of a line.
  2. Slope-Intercept Form: Master the slope-intercept form: y = mx + b. Knowing this form allows you to easily write the equation of a line once you have the slope and y-intercept.
  3. Point-Slope Form: The point-slope form (y - y1 = m(x - x1)) is another powerful tool. It's especially handy when you have a point and the slope.
  4. Step-by-Step Approach: Break down the problem into smaller, manageable steps. First, find the slope. Then, find the y-intercept. Finally, write the equation.

By following these steps, you can confidently find the equation of a line given two points. Practice makes perfect, so try out a few more examples to really solidify your understanding.

Practice Problems

Want to test your skills? Here are a couple of practice problems you can try:

  1. Find the equation of the line that passes through the points (2, -3) and (5, 1).
  2. What is the equation of the line that passes through the points (-1, 4) and (3, -2)?

Work through these problems using the methods we discussed, and you'll be well on your way to mastering linear equations!

Conclusion

So, that's how you find the equation of a line that passes through two given points! It might seem like a lot of steps at first, but once you get the hang of it, it becomes second nature. Remember to focus on understanding the slope formula and the slope-intercept form, and you'll be able to solve these problems with ease. Keep practicing, and you'll become a math whiz in no time! If you have any questions, feel free to ask. Happy solving, guys!