Linear Equations: Finding The Right Fit

Hey guys! Ever stared at a math problem and felt like you needed a secret decoder ring just to understand it? Yeah, me too. Today, we're diving deep into the world of linear equations, specifically how to make sure you've got the right equation to represent a given line. Chin was given a sweet deal: a graph of a line that he knows passes through the point (1,7). He then boldly declared that the equation $f(x)=4x+3$ perfectly represents this line. Now, the real kicker is figuring out which of the provided options also describes this very same line. This isn't just about plugging in numbers; it's about understanding the underlying principles of linear equations and how different forms can represent the same relationship. We'll break down Chin's initial equation, analyze the point he was given, and then dissect each of the answer choices to see which one truly aligns. Get ready to flex those math muscles, because by the end of this, you'll be a pro at spotting the correct linear representation!

Unpacking Chin's Equation and the Given Point

First things first, let's talk about Chin's equation: $f(x)=4x+3$. When we're dealing with lines, $f(x)$ is just another way of saying 'y'. So, Chin's equation is essentially $y = 4x + 3$. This is in the slope-intercept form, which is super handy because it tells us two key pieces of information right away: the slope and the y-intercept. In this equation, the slope ($m$) is 4, and the y-intercept ($b$) is 3. The slope tells us how steep the line is – for every one unit we move to the right on the x-axis, the line goes up by 4 units on the y-axis. The y-intercept is the point where the line crosses the y-axis, which is at (0,3).

Now, Chin was also told that the line contains the point (1,7). This means that if we plug in $x=1$ into the correct equation for the line, we should get $y=7$. Let's check if Chin's equation, $y=4x+3$, actually holds true for this point. If we substitute $x=1$, we get $y = 4(1) + 3 = 4 + 3 = 7$. Bingo! The point (1,7) does lie on the line represented by $y=4x+3$. This confirms that Chin's initial equation is indeed correct for the given information. So, our mission now is to find another equation among the choices that describes this exact same line, meaning it must have the same slope (4) and also pass through the point (1,7).

Exploring Different Forms of Linear Equations

Before we jump into the answer choices, let's quickly recap the common forms of linear equations we might encounter. We've already seen the slope-intercept form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. Another super useful form is the point-slope form: $y - y_1 = m(x - x_1)$. This form is awesome because it directly uses a known point $(x_1, y_1)$ on the line and its slope $m$. If we have the slope and any point on the line, we can immediately write its equation in point-slope form. Sometimes, you might also see the standard form: $Ax + By = C$. While less intuitive for understanding slope and intercepts directly, it's a common way to express linear relationships.

Our goal is to find an equation that represents a line with a slope of 4 and passes through the point (1,7). Since Chin's equation $y=4x+3$ is in slope-intercept form, we know the slope is $m=4$. We also know the line passes through $(x_1, y_1) = (1,7)$. The point-slope form is tailor-made for this scenario. If we plug our slope $m=4$ and our point $(x_1, y_1) = (1,7)$ into the point-slope formula, we get: $y - 7 = 4(x - 1)$. This equation must represent the same line as $y=4x+3$ because it incorporates the same fundamental properties: the slope and a point on the line.

Now, let's consider how the other answer choices might relate. The point-slope form is quite versatile. We can rearrange it to get the slope-intercept form. If we take $y - 7 = 4(x - 1)$ and solve for $y$:

$y - 7 = 4x - 4$

Add 7 to both sides:

$y = 4x - 4 + 7$

$y = 4x + 3$

And voilà! We've arrived back at Chin's original equation. This confirms that any equation that correctly uses the slope $m=4$ and the point $(x_1, y_1)=(1,7)$ in the point-slope form will represent the same line.

Analyzing the Options: Spotting the Correct Equation

Alright, let's get down to business and dissect each of the answer choices provided. Remember, we're looking for an equation that represents a line with a slope of 4 and passes through the point (1,7). Chin's equation, $y = 4x + 3$, has a slope of 4. The point (1,7) satisfies this equation.

Option A: $y-7=3(x-1)$

Let's look at this one. This equation is in point-slope form, $y - y_1 = m(x - x_1)$. By comparing it to the general form, we can identify the slope and the point it represents. Here, $y_1 = 7$, $x_1 = 1$, and the slope $m = 3$. So, this equation represents a line that passes through the point (1,7), but its slope is 3, not 4. Since the slope is different from Chin's line, this is not the correct equation.

Option B: $y-1=3(x-7)$

Similar to option A, this is also in point-slope form. Here, $y_1 = 1$, $x_1 = 7$, and the slope $m = 3$. This equation represents a line with a slope of 3 that passes through the point (7,1). Both the slope and the point are different from what we need. So, this is definitely not the correct equation.

Option C: $y-7=4(x-1)$

Now, let's examine this option closely. It's in point-slope form. We can see that $y_1 = 7$, $x_1 = 1$, and the slope $m = 4$. This means the equation represents a line that passes through the point (1,7) and has a slope of 4. This perfectly matches the characteristics of the line we are looking for! We already showed by rearranging this equation that it simplifies to $y=4x+3$. Therefore, this is a strong contender for the correct answer.

Option D: $y-1=4(x-7)$

Let's check out this last option. It's also in point-slope form. Here, $y_1 = 1$, $x_1 = 7$, and the slope $m = 4$. This equation represents a line with a slope of 4, which is the correct slope. However, it represents a line passing through the point (7,1), not (1,7). Since the point is incorrect, this equation does not represent the same line as Chin's.

The Final Verdict: Confirming the Match

After carefully analyzing each option, we can confidently conclude that Option C: $y-7=4(x-1)$ is the equation that could represent the same line as Chin's $f(x)=4x+3$.

Why? Because both the slope and a point on the line are consistent. Chin's equation $y=4x+3$ has a slope of $m=4$ and passes through the point (1,7) (we verified this by plugging in $x=1$ and getting $y=7$). Option C, $y-7=4(x-1)$, is in point-slope form. It explicitly states that the slope is $m=4$ and that the line passes through the point $(x_1, y_1) = (1,7)$. These are identical characteristics.

We demonstrated earlier that by converting the point-slope form of Option C ($y-7=4(x-1)$) into slope-intercept form, we get $y=4x+3$, which is exactly Chin's original equation. This mathematical transformation proves they are indeed the same line.

It's crucial to remember that different forms of linear equations exist, but they all describe the same underlying relationship between $x$ and $y$ if they share the same slope and pass through the same points. The point-slope form is particularly powerful for constructing equations when you have a point and the slope, as it directly incorporates these pieces of information. Always double-check the slope and ensure the given point satisfies the equation. That's the key to navigating these types of problems like a math whiz!

So, next time you see a line on a graph or are given a point and a slope, you'll know exactly how to find or verify its equation. Keep practicing, guys, and you'll master these concepts in no time. Happy problem-solving!