Which Point Is On The Line? Solve Y = (3/4)x + 5

Alright, guys, ever stared at a math problem and thought, "Where do I even begin?" Well, if you're looking at an equation like $y = \frac{3}{4} x + 5$ and wondering which tiny little point actually belongs on its graph, you've landed in the perfect spot! This isn't just about getting the right answer for your homework; it's about understanding the magic behind how graphs and equations are connected. We're going to break down exactly how to figure out which point lies on the graph of a line, making it super clear and, dare I say, even a little fun. You'll walk away knowing not just the answer to our specific problem – which point among A. (5,8), B. (1,6), C. (-3,3), or D. (-4,2) fits – but also the why and how behind it all. We'll cover everything from the basics of linear equations to a super simple substitution method that'll make you a pro at this in no time. So, grab a snack, settle in, and let's get ready to decode this awesome math puzzle. Understanding linear graphs and identifying points on them is a fundamental skill, and mastering it will seriously boost your confidence in algebra. We're talking about taking an abstract mathematical statement, $y = \frac{3}{4} x + 5$, and turning it into something you can visualize and interact with, by finding specific coordinates that make that statement true. It's like finding the perfect puzzle piece that fits exactly where it's supposed to. So, let's dive into the core concepts, examine our equation closely, and then systematically test each option to pinpoint the correct answer. You'll see that once you grasp the underlying principles, these types of problems become much less intimidating and a lot more satisfying to solve. Trust me, by the end of this, you'll be able to tackle similar challenges with a cool, calm, and collected attitude. Get ready to level up your math game!

Understanding Linear Equations: The Basics, Guys!

Before we jump into finding our special point, let's make sure we're all on the same page about what a linear equation actually is. When we talk about a linear equation, we're literally talking about an equation that, when you graph it, forms a perfectly straight line. No curves, no wiggles, just a nice, clean line. The most common and super helpful form of a linear equation is called the slope-intercept form, which looks like this: y = mx + b. Sounds a bit like a secret code, right? But it's actually pretty straightforward once you break it down.

Here’s the lowdown:

• y: This represents the vertical coordinate on your graph. Think of it as how high or low a point is.
• x: This represents the horizontal coordinate. Think of it as how far left or right a point is.
• m: This, my friends, is the slope of the line. The slope tells us two really important things: how steep the line is and in which direction it's going (up or down from left to right). A positive slope (like our $\frac{3}{4}$) means the line goes upwards as you move from left to right, while a negative slope means it goes downwards. The larger the absolute value of m, the steeper the line. Our $m = \frac{3}{4}$ means for every 4 units you move to the right on the graph, the line goes up 3 units. It's like the rise over run! Understanding the slope is key because it dictates the entire slant of our linear graph, fundamentally influencing which points lie on the line.
• b: And finally, b is the y-intercept. This is where your straight line crosses or intercepts the y-axis. It's the point where $x = 0$. So, for any linear equation, when $x$ is 0, $y$ will be $b$. It's a really easy point to spot on the graph and can often be a great starting point for drawing your line.

Now, let's look at our specific equation: $y = \frac{3}{4} x + 5$. Can you spot the m and the b? That's right! Our m (the slope) is $\frac{3}{4}$, and our b (the y-intercept) is 5. So, this line is going to go upwards from left to right, and it's going to cross the y-axis at the point (0, 5). Pretty cool, huh? Knowing these two components gives us a fantastic mental picture of the line even before we start plotting points or doing any heavy calculations. It sets the stage for identifying points on the graph because it defines the very path the line takes. This foundational understanding of linear equations in their slope-intercept form is absolutely crucial for verifying point coordinates. It's not just about memorizing a formula; it's about grasping what each part of that formula means for the visual representation of the line. This knowledge empowers you to not only solve problems but also to intuitively understand the behavior of any given linear graph and confidently determine which specific points belong to it, rather than just guessing. This forms the bedrock of our strategy for checking each of the given options.

The Secret Sauce: How to Test a Point on a Line

Alright, now that we're pros at understanding what our equation $y = \frac{3}{4} x + 5$ actually means, let's get to the most important part: how do we actually check if a given point is on this line? This is where the magic of substitution comes into play, and trust me, it’s not nearly as complicated as it sounds. It's like a simple truth test for each point.

Here’s the core idea: If a point with coordinates $(x, y)$ truly lies on the graph of the line, then when you plug its x-value and y-value into the equation, the equation must hold true. It needs to balance perfectly, like a scale. If it balances, the point is on the line. If it doesn't, well, then that point is just chilling somewhere else in the coordinate plane, not on our specific line.

Let’s break down the step-by-step process for testing each point:

1. Identify the x and y values of the point: Every point is given as an ordered pair $(x, y)$. The first number is always your x-coordinate, and the second number is always your y-coordinate. Super simple, right?
2. Substitute these values into the equation: Take your line's equation (in our case, $y = \frac{3}{4} x + 5$) and carefully replace the 'y' with the point's y-value and the 'x' with the point's x-value. Be extra careful with fractions and negative numbers here, guys! This is where most people make tiny, fixable mistakes.
3. Calculate both sides of the equation: After substituting, simplify both sides of the equation. Usually, the left side (where you substituted 'y') will just be a number. The right side will require some multiplication and addition.
4. Compare the results: If the number on the left side is exactly equal to the number on the right side, congratulations! That point lies on the line. If they are different, then the point does not lie on the line.

See? It's like solving a mini-puzzle for each point. The beauty of this substitution method is its reliability. It's a foolproof way to verify point membership on any linear graph. This methodical approach ensures we don't miss anything and provides a definitive answer for which points satisfy the equation. So, armed with this powerful technique, we are now ready to systematically examine each of the options provided in our original problem. This method ensures accuracy and eliminates guesswork when trying to find points on the line $y = \frac{3}{4} x + 5$. It's a critical skill in algebra, proving whether specific coordinates truly fit the geometric path defined by the equation. Don't underestimate the power of careful substitution and calculation, as it's the gateway to confidently identifying the correct answer among multiple choices, solidifying your understanding of how points interact with linear equations in the coordinate plane.

Let's Get Down to Business: Testing Our Options!

Alright, guys, this is where the rubber meets the road! We've got our linear equation, $y = \frac{3}{4} x + 5$, and we've got our trusty substitution method in hand. Now, let's meticulously go through each of the given options and see which one makes our equation sing true. We’re looking for the single point that perfectly fits this mathematical description. This process will definitively show us which point lies on the graph of the line.

Option A: The Curious Case of (5, 8)

Let's start with point A, which is $(5, 8)$. Here, our $x = 5$ and our $y = 8$. We'll plug these values into our equation:

$y = \frac{3}{4} x + 5$

$8 = \frac{3}{4} (5) + 5$

First, let's multiply $\frac{3}{4}$ by 5. Remember, you can think of 5 as $\frac{5}{1}$:

$8 = \frac{3 \times 5}{4 \times 1} + 5$

$8 = \frac{15}{4} + 5$

Now, we need to add $\frac{15}{4}$ and 5. To do this, we should give 5 a common denominator of 4. Since $5 = \frac{20}{4}$:

$8 = \frac{15}{4} + \frac{20}{4}$

$8 = \frac{35}{4}$

Is $8$ equal to $\frac{35}{4}$? Well, $\frac{35}{4}$ is $8.75$. Since $8 \neq 8.75$, point A is not on the line.

Option B: Diving into (1, 6)

Next up, we have point B: $(1, 6)$. So, $x = 1$ and $y = 6$. Let's substitute these into our equation:

$y = \frac{3}{4} x + 5$

$6 = \frac{3}{4} (1) + 5$

$6 = \frac{3}{4} + 5$

Again, we need a common denominator for adding $\frac{3}{4}$ and 5. We know $5 = \frac{20}{4}$:

$6 = \frac{3}{4} + \frac{20}{4}$

$6 = \frac{23}{4}$

Is $6$ equal to $\frac{23}{4}$? $\frac{23}{4}$ is $5.75$. Since $6 \neq 5.75$, point B is also not on the line.

Option C: What About (-3, 3)?

Moving on to point C: $(-3, 3)$. Here, $x = -3$ and $y = 3$. Let's plug 'em in:

$y = \frac{3}{4} x + 5$

$3 = \frac{3}{4} (-3) + 5$

Multiply $\frac{3}{4}$ by -3:

$3 = \frac{-9}{4} + 5$

To add, let's convert 5 to $\frac{20}{4}$:

$3 = \frac{-9}{4} + \frac{20}{4}$

$3 = \frac{11}{4}$

Is $3$ equal to $\frac{11}{4}$? $\frac{11}{4}$ is $2.75$. Since $3 \neq 2.75$, point C is not on the line.

Option D: The Moment of Truth with (-4, 2)

Finally, we're at point D: $(-4, 2)$. Our last hope! Let $x = -4$ and $y = 2$. Let's substitute and see:

$y = \frac{3}{4} x + 5$

$2 = \frac{3}{4} (-4) + 5$

Multiply $\frac{3}{4}$ by -4. This is cool because the 4 in the denominator and the -4 in the numerator will simplify nicely:

$2 = 3 \times (-1) + 5$

$2 = -3 + 5$

$2 = 2$

Bingo! Both sides of the equation are equal! This means that point $(-4, 2)$ does lie on the graph of the line $y = \frac{3}{4} x + 5$. This systematic approach, testing each option with careful substitution, ensures we find the correct coordinates that satisfy the linear equation. This demonstration of how to find points on the line by checking each option reinforces the reliability of our method.

Why This Matters (Beyond Just Homework!)

So, you've just mastered how to tell which point lies on the graph of a line, specifically for our equation $y = \frac{3}{4} x + 5$. You might be thinking,