Line Equation Through Two Points: Solved

Jan 13, 2026 by ADMIN 41 views

Hey everyone, let's dive into a super common math problem: finding the equation of a line when you've got two points. This is a foundational skill, guys, and once you get the hang of it, you'll see these types of problems everywhere. Today, we're tackling a specific one: Which equation represents the line that passes through $(-8,11)$ and $(4, rac{7}{2})$ ? We'll break down each step, explain the reasoning, and even look at why the other options are a bit off. So, grab your notebooks, and let's get this done!

Understanding the Goal: The Equation of a Line

So, what are we actually trying to find here? We're looking for the equation of a line that connects two specific points in the coordinate plane. Remember, a line has a constant slope, and its equation typically takes the form $y = mx + b$ , where 'm' is the slope and 'b' is the y-intercept. Our mission is to figure out what 'm' and 'b' are for the line that goes through our given points, $(-8,11)$ and $(4, rac{7}{2})$ . We'll be using a few key formulas and concepts to get there. First off, we need to calculate the slope. The slope tells us how steep our line is and in which direction it's going. A positive slope means the line goes up from left to right, while a negative slope means it goes down. The formula for slope is pretty straightforward: $m = rac{y_2 - y_1}{x_2 - x_1}$ . This basically means you take the difference in the y-coordinates and divide it by the difference in the x-coordinates. It doesn't matter which point you designate as $(x_1, y_1)$ and which as $(x_2, y_2)$ , as long as you're consistent. Once we have the slope, we can use one of the points and the slope to find the y-intercept ('b'). This is where the point-slope form of a linear equation comes in handy: $y - y_1 = m(x - x_1)$ . We'll plug in our slope and one of the points, then rearrange the equation to solve for 'b'. It sounds like a lot, but we'll go through it step-by-step. The goal is to arrive at an equation in the $y = mx + b$ format that satisfies both points. Let's make sure we have our points clearly identified: Point 1 is $(-8, 11)$ , so $x_1 = -8$ and $y_1 = 11$ . Point 2 is $(4, rac{7}{2})$ , so $x_2 = 4$ and $y_2 = rac{7}{2}$ . Got it? Great! Now, let's get calculating.

Step 1: Calculate the Slope (m)

Alright guys, the very first thing we need to do is calculate the slope (m) of the line. This is crucial because the slope defines the direction and steepness of our line. If you remember, the formula for slope given two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = rac{y_2 - y_1}{x_2 - x_1}$ . Let's plug in our coordinates. We have our first point as $(-8, 11)$ and our second point as $(4, rac{7}{2})$ . So, we can set $x_1 = -8$ , $y_1 = 11$ , $x_2 = 4$ , and $y_2 = rac{7}{2}$ .

Now, substitute these values into the slope formula:

$m = rac{ rac{7}{2} - 11}{4 - (-8)}$

First, let's handle the numerator: $rac{7}{2} - 11$ . To subtract these, we need a common denominator, which is 2. So, $11$ becomes $rac{22}{2}$ .

$rac{7}{2} - rac{22}{2} = rac{7 - 22}{2} = rac{-15}{2}$

Next, let's handle the denominator: $4 - (-8)$ . Subtracting a negative is the same as adding a positive.

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

Now, we can put the numerator and denominator back together to find the slope:

$m = rac{ rac{-15}{2}}{12}$

To divide a fraction by a whole number, you can multiply the fraction by the reciprocal of the whole number. The reciprocal of 12 is $rac{1}{12}$ .

$m = rac{-15}{2} imes rac{1}{12}$

$m = rac{-15 imes 1}{2 imes 12}$

$m = rac{-15}{24}$

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3.

$m = rac{-15 ilde{1} 3}{24 ilde{1} 3}$

$m = rac{-5}{8}$

So, the slope of the line is $- rac{5}{8}$ . This is a super important piece of information! It tells us that for every 8 units we move to the right on the x-axis, the line goes down by 5 units on the y-axis. Seeing a negative slope in our answer options will immediately narrow down our choices. Let's keep this value handy as we move on to the next step, which is finding the y-intercept.

Step 2: Find the Y-Intercept (b)

Now that we've got our slope, $m = - rac{5}{8}$ , the next step is to find the y-intercept (b). The y-intercept is the point where the line crosses the y-axis, and it's represented by 'b' in our standard line equation format, $y = mx + b$ . We can use the point-slope form of a linear equation, which is $y - y_1 = m(x - x_1)$ , or we can directly plug our slope and one of the points into the $y = mx + b$ equation and solve for 'b'. Let's go with the direct substitution method, as it's often quicker.

We have our slope $m = - rac{5}{8}$ . We can choose either of our original points, $(-8, 11)$ or $(4, rac{7}{2})$ , to plug into the equation. Let's use the point $(-8, 11)$ first. So, $x = -8$ and $y = 11$ .

Substitute these values into $y = mx + b$ :

$11 = ig(- rac{5}{8}ig)(-8) + b$

Now, let's simplify the multiplication:

$11 = rac{-5 imes -8}{8} + b$

$11 = rac{40}{8} + b$

$11 = 5 + b$

To solve for 'b', we just need to subtract 5 from both sides of the equation:

$11 - 5 = b$

$6 = b$

So, the y-intercept is $b = 6$ . That was pretty clean! Now, just to be absolutely sure, let's try using the other point, $(4, rac{7}{2})$ , with our slope $m = - rac{5}{8}$ and see if we get the same 'b'. This is a great way to double-check our work, guys.

Substitute $x = 4$ and $y = rac{7}{2}$ into $y = mx + b$ :

$rac{7}{2} = ig(- rac{5}{8}ig)(4) + b$

Simplify the multiplication:

$rac{7}{2} = rac{-5 imes 4}{8} + b$

$rac{7}{2} = rac{-20}{8} + b$

We can simplify the fraction $rac{-20}{8}$ by dividing both numerator and denominator by 4:

$rac{-20 ilde{1} 4}{8 ilde{1} 4} = rac{-5}{2}$

So, the equation becomes:

$rac{7}{2} = - rac{5}{2} + b$

To solve for 'b', add $rac{5}{2}$ to both sides:

$rac{7}{2} + rac{5}{2} = b$

$rac{7 + 5}{2} = b$

$rac{12}{2} = b$

$6 = b$

Awesome! We got the same y-intercept, $b = 6$ , using both points. This confirms that our calculations for both the slope and the y-intercept are correct. We now have all the pieces needed to write the final equation of the line.

Step 3: Write the Equation of the Line

We've done the heavy lifting, guys! We've successfully calculated the slope, $m = - rac{5}{8}$ , and found the y-intercept, $b = 6$ . Now, all we need to do is put these values back into the standard slope-intercept form of a linear equation, which is $y = mx + b$ .

Simply substitute the values we found:

$y = ig(- rac{5}{8}ig)x + 6$

So, the equation of the line that passes through the points $(-8,11)$ and $(4, rac{7}{2})$ is $y = - rac{5}{8}x + 6$ . This is our final answer!

Analyzing the Options: Why Other Choices Are Incorrect

Let's take a quick look at the options provided to see why they aren't the correct representation of the line passing through our points. This is a great way to solidify our understanding and practice our critical thinking skills.

A. $y=- rac{11}{2} x+71$ : This equation has a slope of $- rac{11}{2}$ . Our calculated slope is $- rac{5}{8}$ . Since the slopes don't match, this option is incorrect. The y-intercept also doesn't match what we found.
B. $y=- rac{4}{8} x+6$ : Let's simplify this. $- rac{4}{8}$ simplifies to $- rac{1}{2}$ . So, this equation is $y = - rac{1}{2}x + 6$ . The y-intercept $b=6$ actually matches what we found! However, the slope $m = - rac{1}{2}$ does not match our calculated slope of $m = - rac{5}{8}$ . So, this option is incorrect.
C. $y=- rac{5}{8} x+16$ : This option has a slope of $m = - rac{5}{8}$ , which does match our calculated slope! That's a good sign. However, the y-intercept is $b = 16$ . Our calculated y-intercept was $b = 6$ . Since the y-intercept doesn't match, this equation does not pass through both points. So, this option is incorrect.
D. $y=- rac{15}{2} x-49$ : The slope here is $m = - rac{15}{2}$ , which does not match our calculated slope of $- rac{5}{8}$ . Therefore, this option is incorrect.

As you can see, only our derived equation, $y = - rac{5}{8}x + 6$ , has both the correct slope and the correct y-intercept that satisfy the given points.

Conclusion: Mastering Linear Equations

So there you have it, guys! We successfully found the equation of the line that passes through the points $(-8,11)$ and $(4, rac{7}{2})$ . The key steps involved calculating the slope (m) using the formula $m = rac{y_2 - y_1}{x_2 - x_1}$ , and then using that slope along with one of the points to find the y-intercept (b) in the $y = mx + b$ equation. Our final answer is $y = - rac{5}{8}x + 6$ .

Remember, understanding how to find the equation of a line from two points is a fundamental skill in algebra and is applicable in countless scenarios, from graphing to analyzing data trends. Keep practicing these types of problems, and don't be afraid to check your work by plugging your final equation back into the original points to ensure they satisfy it. You've got this!