Find A Point On A Line Given Slope And A Point
Hey math enthusiasts! Today, we're diving into a fun problem that combines our knowledge of points, slopes, and lines. We're going to explore how to find additional points on a line when we know one point and the slope. So, grab your thinking caps, and let's get started!
The Challenge: Decoding the Line
Our mission, should we choose to accept it, is to decipher the path of a line. We know this line bravely strides through the point (-7, 5) and boasts a slope of 1/2. The burning question is: Which of these points also lie on this very line?
- A. (-13, 9)
- B. (-9, 13)
- C. (9, 13)
- D. (13, 9)
This isn't just about picking an answer; it's about understanding the elegant dance between points and slopes. So, let's roll up our sleeves and explore the concepts that will guide us to the solution.
Unveiling the Slope-Intercept Form: The Line's Secret Identity
To crack this code, we need to tap into one of the most powerful tools in our mathematical arsenal: the slope-intercept form of a line. This form is like the line's secret identity, revealing its essence in a neat equation:
y = mx + b
Where:
- y represents the vertical coordinate of any point on the line.
- x represents the horizontal coordinate of any point on the line.
- m is the slope, indicating the line's steepness and direction.
- b is the y-intercept, the point where the line crosses the y-axis.
This equation is our key to unlocking the line's secrets. We already know the slope (m = 1/2), and we have a point (-7, 5) that the line passes through. Our next step is to use this information to find the y-intercept (b).
The Art of Substitution: Finding the Y-Intercept
Now comes the fun part: detective work! We'll use the point (-7, 5) and the slope (1/2) to solve for 'b' in our slope-intercept equation. Think of it like plugging in the clues we have to reveal the missing piece.
We substitute x = -7, y = 5, and m = 1/2 into the equation:
5 = (1/2)(-7) + b
Let's simplify:
5 = -7/2 + b
To isolate 'b', we add 7/2 to both sides of the equation:
5 + 7/2 = b
Converting 5 to a fraction with a denominator of 2, we get:
10/2 + 7/2 = b
17/2 = b
So, we've discovered that the y-intercept, 'b', is 17/2. This is a crucial piece of the puzzle!
The Line's Full Portrait: The Complete Equation
With the slope (m = 1/2) and the y-intercept (b = 17/2) in hand, we can now paint the full portrait of our line. We simply plug these values back into the slope-intercept form:
y = (1/2)x + 17/2
This equation is the line's complete identity! It tells us the relationship between the x and y coordinates of every single point on this line. Now, we're ready to put this equation to the test and see which of the given points fit the profile.
Point Verification: Testing the Suspects
We have four potential points, and it's our job to determine which ones lie on the line defined by our equation y = (1/2)x + 17/2. To do this, we'll substitute the x and y coordinates of each point into the equation and see if it holds true. If the equation is satisfied, then the point lies on the line; otherwise, it's an impostor!
Point A: (-13, 9)
Let's plug in x = -13 and y = 9:
9 = (1/2)(-13) + 17/2
9 = -13/2 + 17/2
9 = 4/2
9 = 2
This is a false statement! Point A does not lie on the line.
Point B: (-9, 13)
Now, let's test x = -9 and y = 13:
13 = (1/2)(-9) + 17/2
13 = -9/2 + 17/2
13 = 8/2
13 = 4
Again, this is incorrect. Point B is not on the line.
Point C: (9, 13)
Let's see if x = 9 and y = 13 fit the equation:
13 = (1/2)(9) + 17/2
13 = 9/2 + 17/2
13 = 26/2
13 = 13
Success! This statement is true. Point C lies on the line.
Point D: (13, 9)
Finally, let's check x = 13 and y = 9:
9 = (1/2)(13) + 17/2
9 = 13/2 + 17/2
9 = 30/2
9 = 15
This is false, so Point D is not on the line.
The Verdict: Point C is the Key!
After carefully examining each point, we've discovered that only Point C, (9, 13), satisfies the equation of the line. Therefore, the correct answer is:
C. (9, 13)
We've successfully navigated the world of slopes and points to find our solution. Great job, guys! This problem highlights the power of the slope-intercept form and the importance of careful substitution and verification.
Mastering Lines: Practice Makes Perfect
This adventure into the realm of lines and slopes is just the beginning. To truly master these concepts, practice is key. Try tackling similar problems with different points and slopes. You can even create your own line equations and challenge yourself to find points that lie on them.
Remember, the more you practice, the more comfortable you'll become with these mathematical tools. Soon, you'll be effortlessly navigating the world of lines and slopes, solving problems with confidence and flair.
Keep exploring, keep questioning, and most importantly, keep having fun with math! You've got this!
