Points Excluded A Line Passing Through (0,1) With Positive Slope

Hey guys! Let's dive into a fascinating geometric puzzle. Imagine a line gracefully gliding through the point (0, 1), its slope tilted upwards towards the heavens – a positive slope, as we mathematicians like to say. Now, we've got a lineup of potential points: (12, 3), (-2, -5), (-3, 1), (1, 15), and (5, -2). Our mission, should we choose to accept it, is to figure out which of these points that line cannot possibly pass through. Buckle up, because we're about to embark on a journey through slopes, intercepts, and the very essence of linear equations.

The Line's Tale: Slope and Intercept

To crack this code, we need to understand the fundamental nature of our line. It's not just any line; it's a line with a story. It has a definite starting point – the y-intercept – and a certain inclination – the slope. The y-intercept is a piece of cake; we know the line passes through (0, 1), so that's our y-intercept. It’s like the line's anchor, firmly planted at the point where the line crosses the y-axis. Now, the real star of the show is the slope. A positive slope means that as we move from left to right along the line, we're always climbing upwards. It's like a gentle incline, the steeper the slope, the faster we ascend. Think of it as the line's personality, defining its direction and steepness.

Diving Deep into the Slope

In the world of mathematics, we have a neat way to quantify this slope. We call it 'm', and it’s calculated as the change in the y-coordinate divided by the change in the x-coordinate. It’s essentially a measure of how much the line rises (or falls) for every unit we move horizontally. Since our line has a positive slope, this 'm' value will always be greater than zero. That is a crucial piece of information that will guide us in our quest to identify the points that cannot belong to this line. We know that our line starts at (0, 1). As we move towards the right (increasing x), the y-values must also increase to maintain a positive slope. This simple principle will be the key to unlocking the solution.

The Equation of the Line: Our Guiding Star

Now, let’s bring in the big guns: the equation of a line. The most common form is the slope-intercept form: y = mx + b. Here, 'y' and 'x' are the coordinates of any point on the line, 'm' is the slope (which we know is positive), and 'b' is the y-intercept (which we know is 1). So, our line's equation looks like this: y = mx + 1, where m > 0. This equation is our guiding star. It dictates the relationship between the x and y coordinates of any point that lies on our line. If a point's coordinates don't satisfy this equation for any positive 'm', then that point is an imposter – it simply cannot reside on our line.

The Point-by-Point Investigation

Alright, time to put on our detective hats and investigate each point. We'll plug the x and y coordinates of each point into our line equation (y = mx + 1) and see if we can find a positive value for 'm' that makes the equation true. If we can find such an 'm', the point is a possible resident of our line. But if we hit a dead end – if no positive 'm' can make the equation work – then we've found our culprit, a point that cannot belong.

Point (12, 3): A Possible Candidate?

Let's start with (12, 3). Plugging these coordinates into our equation, we get: 3 = 12m + 1. Now, let’s solve for 'm'. Subtracting 1 from both sides gives us 2 = 12m. Dividing both sides by 12, we find m = 1/6. Bingo! We've found a positive value for 'm'. This means that a line with a slope of 1/6 passing through (0, 1) would indeed also pass through (12, 3). So, this point is safe, it could be on our line.

Point (-2, -5): An Intriguing Case

Next up is (-2, -5). Plugging these into our equation, we have: -5 = -2m + 1. Subtracting 1 from both sides gives us -6 = -2m. Dividing by -2, we get m = 3. Another positive value for 'm'! This means a line with a slope of 3 could pass through both (0, 1) and (-2, -5). This point is also a potential resident.

Point (-3, 1): A Tricky Customer

Let's examine (-3, 1). Our equation becomes: 1 = -3m + 1. Subtracting 1 from both sides yields 0 = -3m. Dividing by -3, we get m = 0. Hmm, this is interesting. We found a value for 'm', but it's 0, not a positive value. Remember, our line has a positive slope. Therefore, no line with a positive slope passing through (0, 1) can also pass through (-3, 1). This is our first point that cannot be on the line!

Point (1, 15): Another Potential Resident

Now, let’s check (1, 15). Plugging in the coordinates: 15 = 1m + 1. Subtracting 1 gives us 14 = m. We have a positive slope, m = 14. So, a line with a slope of 14 would happily accommodate this point.

Point (5, -2): The Final Suspect

Finally, we have (5, -2). Plugging in: -2 = 5m + 1. Subtracting 1 gives us -3 = 5m. Dividing by 5, we get m = -3/5. Uh oh! We've got a negative slope. This point is another one that cannot lie on our line, as our line strictly has a positive slope.

The Verdict: Points That Cannot Be

After our meticulous investigation, the verdict is in! The points that the line cannot pass through are (-3, 1) and (5, -2). These points simply do not align with the positive-slope nature of our line, their coordinates defying the equation y = mx + 1 for any positive 'm'.

Visualizing the Solution

To truly grasp this concept, it's helpful to visualize it. Imagine a line pivoting around the point (0, 1). As the line rotates to create a positive slope, it can cover a certain region of the coordinate plane. Points within this region are potential candidates, while points outside this region are simply unreachable. The points (-3, 1) and (5, -2) lie outside this positive-slope zone.

The Takeaway: Slope is Key

The key takeaway here is the importance of the slope. It dictates the direction and steepness of the line, and it ultimately determines which points can and cannot lie on the line. By understanding the relationship between slope, y-intercept, and the equation of a line, we can solve a wide array of geometric puzzles.

So, there you have it, guys! We've successfully unraveled the mystery of the points that lie beyond a line's reach. Remember, in the world of math, every line has a story to tell, and the slope is its most captivating chapter.