Finding The Equation Of A Line Passing Through (-6, 7) And (-3, 6)

Hey guys! Today, we're diving into a common problem in algebra: finding the equation of a line when you're given two points it passes through. Specifically, we're going to tackle the question: Which equation represents the line that passes through the points $(-6, 7)$ and $(-3, 6)$? Let's break it down step by step.

Understanding the Problem

Before we jump into calculations, let's make sure we understand what the question is asking. We're given two points, $(-6, 7)$ and $(-3, 6)$. These points exist on a coordinate plane, and we're trying to find the equation of the straight line that connects them. Remember, the equation of a line is typically written in slope-intercept form, which looks like this: y = mx + b, where:

• m is the slope of the line, which tells us how steep the line is and whether it's going uphill or downhill.
• b is the y-intercept, which is the point where the line crosses the y-axis.

Our goal is to find the values of m and b that fit the line passing through our given points. Once we have those, we can write the equation of the line.

Step 1: Calculate the Slope (m)

The slope is the first piece of the puzzle. The formula for calculating the slope (m) between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

m = (y₂ - y₁) / (x₂ - x₁)

In our case, we have:

• $(x_1, y_1) = (-6, 7)$
• $(x_2, y_2) = (-3, 6)$

Let's plug these values into the formula:

m = (6 - 7) / (-3 - (-6))

Simplify the equation:

m = (-1) / (-3 + 6)

m = -1 / 3

So, the slope of our line is -1/3. This means that for every 3 units we move to the right on the graph, the line goes down 1 unit. A negative slope indicates that the line is decreasing (going downhill) as we move from left to right.

Step 2: Use the Point-Slope Form

Now that we have the slope, we can use the point-slope form of a linear equation to find the equation of the line. The point-slope form is:

**y - y₁ = m(x - x₁) **

Where:

• m is the slope (which we just calculated).
• $(x_1, y_1)$ is one of the points on the line. We can use either $(-6, 7)$ or $(-3, 6)$. Let's use $(-6, 7)$ for this example.

Plug in the values:

y - 7 = (-1/3)(x - (-6))

Simplify:

y - 7 = (-1/3)(x + 6)

Step 3: Convert to Slope-Intercept Form (y = mx + b)

To match the answer choices, we need to convert our equation from point-slope form to slope-intercept form (y = mx + b). Let's distribute the -1/3 on the right side:

y - 7 = (-1/3)x - 2

Now, add 7 to both sides to isolate y:

y = (-1/3)x - 2 + 7

y = (-1/3)x + 5

Step 4: Verify the Answer

We've found our equation: y = (-1/3)x + 5. But it's always a good idea to double-check our work. We can do this by plugging in the coordinates of our other point, $(-3, 6)$, into the equation to see if it holds true:

6 = (-1/3)(-3) + 5

6 = 1 + 5

6 = 6

The equation holds true for both points, so we can be confident that our equation is correct.

The Answer

Looking at the answer choices, we can see that option B. y = (-1/3)x + 5 matches our calculated equation. So, that's the correct answer!

Why This Matters

Understanding how to find the equation of a line is a fundamental skill in algebra and has lots of real-world applications. For example, it can help you model relationships between two variables, like distance and time, or cost and quantity. Mastering this skill will set you up for success in more advanced math courses and in various problem-solving situations.

Alternative Method: Using Both Points to Create a System of Equations

There's another cool way to solve this problem, and it involves using both points to create a system of equations. Let's explore this method too!

We know the equation of a line is y = mx + b. We have two points, and each point gives us an x and a y value. So, we can plug these values into the equation to create two separate equations:

For point $(-6, 7)$:

7 = m(-6) + b which simplifies to 7 = -6m + b

For point $(-3, 6)$:

6 = m(-3) + b which simplifies to 6 = -3m + b

Now we have a system of two equations with two variables (m and b):

1. 7 = -6m + b
2. 6 = -3m + b

We can solve this system using several methods, such as substitution or elimination. Let's use elimination. Notice that both equations have a b term. If we subtract the second equation from the first, the b terms will cancel out:

(7 = -6m + b) - (6 = -3m + b)

This gives us:

1 = -3m

Now, solve for m:

m = -1/3

Awesome! We got the same slope as before. Now we can plug this value of m into either of our original equations to solve for b. Let's use the second equation:

6 = -3(-1/3) + b

6 = 1 + b

Subtract 1 from both sides:

b = 5

So, we found that the slope m is -1/3 and the y-intercept b is 5. Plugging these values into the slope-intercept form y = mx + b, we get:

y = (-1/3)x + 5

Again, this matches answer choice B and confirms our solution using a different method!

Key Takeaways

• Finding the equation of a line through two points involves calculating the slope and then using either the point-slope form or a system of equations to find the y-intercept.
• There are often multiple ways to solve a math problem. Knowing different methods can help you check your work and deepen your understanding.
• Practice makes perfect! The more you work through these types of problems, the more comfortable you'll become with the concepts and the faster you'll be able to solve them.

Let's Practice!

Okay, guys, you've got the tools! Now it's your turn to try some practice problems. Finding the equation of a line is like riding a bike – it might feel wobbly at first, but with a little practice, you'll be cruising in no time. Here are a couple of questions to get you started:

1. Find the equation of the line that passes through the points (2, 3) and (4, 7).
2. What is the equation of the line that goes through (-1, 5) and (1, 1)?

Work through these problems, and don't be afraid to revisit the steps we've covered in this article. Remember, the key is to understand the process, not just memorize the formula. And if you get stuck, don't worry! Reach out to a friend, a teacher, or even search online for help. There are tons of resources available to support your learning journey.

Wrapping Up

So, there you have it! We've explored how to find the equation of a line that passes through two given points using both the slope-intercept form and the point-slope form. Remember to calculate the slope first, then use either point-slope form or a system of equations to find the y-intercept. With a little practice, you'll be a pro at solving these problems. Keep up the great work, and I'll catch you in the next math adventure! Keep practicing and you'll be acing those math problems in no time!