Equation Of A Line Passing Through (2, -1/2) With Slope 3 Explained
Let's dive into this math problem together, guys! We've got a point (2, -1/2) and a slope of 3, and we need to figure out which equation represents the line that goes through them. No sweat, we can totally crack this.
Understanding the Point-Slope Form
To tackle this, we're going to use the point-slope form of a linear equation. You might remember it from algebra class – it's super handy for situations like this. The point-slope form looks like this:
y - y₁ = m(x - x₁)
Where:
- m is the slope of the line
- (x₁, y₁) is a point that the line passes through
This form is a lifesaver because it lets us plug in the information we already have (the slope and a point) and directly get an equation for the line. Let's break down why this form works so well and how it connects to the more familiar slope-intercept form (y = mx + b). The point-slope form is essentially a rearranged version of the slope formula. Remember that the slope (m) is calculated as the change in y divided by the change in x: m = (y₂ - y₁) / (x₂ - x₁). If we think of a general point on the line as (x, y) and the given point as (x₁, y₁), we can rewrite the slope formula as m = (y - y₁) / (x - x₁). Now, if we multiply both sides of this equation by (x - x₁), we get y - y₁ = m(x - x₁), which is exactly the point-slope form! This shows that the point-slope form is not just a random formula; it's a direct consequence of the definition of slope.
So, by using the point-slope form, we are essentially ensuring that the equation we construct will satisfy the condition of having the given slope and passing through the given point. This makes it a powerful tool for solving problems like this one, where we are given a point and a slope and asked to find the equation of the line. Furthermore, the point-slope form provides a clear and intuitive way to see how the slope and a specific point on the line determine the entire line. It emphasizes the idea that a line is uniquely defined by its steepness (slope) and a single location it passes through (a point). This understanding is crucial for grasping the fundamental concepts of linear equations and their graphical representations. The flexibility of the point-slope form also allows us to easily convert it into other forms, such as the slope-intercept form or the standard form, depending on the specific requirements of the problem or the desired format of the answer. In our case, we'll use the point-slope form to directly write the equation and then compare it to the given options. This approach will help us efficiently identify the correct equation without having to go through unnecessary algebraic manipulations.
Plugging in the Values
Okay, we know our point is (2, -1/2), so x₁ = 2 and y₁ = -1/2. We also know our slope, m, is 3. Let's plug these values into the point-slope form:
y - (-1/2) = 3(x - 2)
Notice how we carefully substituted the values, especially the negative sign for the y-coordinate of the point. Getting the signs right is super important in these kinds of problems!
Now, let's simplify this a bit. Subtracting a negative is the same as adding, so we have:
y + 1/2 = 3(x - 2)
And there you have it! This is the equation of the line in point-slope form that passes through the point (2, -1/2) and has a slope of 3. But don't stop here! It's essential to understand what this equation actually represents and how it visually translates into a line on a graph. The equation y + 1/2 = 3(x - 2) tells us a lot about the line. It tells us that for every increase of 1 in x, the value of y increases by 3 (that's the slope of 3). It also tells us that the line passes through the point (2, -1/2). If you were to graph this equation, you would start by plotting the point (2, -1/2). Then, using the slope of 3, you would move 1 unit to the right and 3 units up to find another point on the line. Connecting these points would give you the visual representation of the line described by the equation.
Understanding the connection between the equation and its graph is fundamental to mastering linear equations. It allows you to not only solve problems algebraically but also to visualize the solutions and interpret them in a real-world context. For example, if this equation represented a relationship between time (x) and distance (y), the slope would tell us the speed of an object, and the point (2, -1/2) would tell us the object's position at a specific time. The ability to move between the algebraic representation (the equation) and the graphical representation (the line) is a key skill in mathematics and its applications. In addition to the graphical interpretation, it's also worth noting that this equation can be manipulated into other forms, such as the slope-intercept form (y = mx + b) or the standard form (Ax + By = C). Converting between these forms can be useful in different situations, depending on what information you want to highlight or what calculations you need to perform. For instance, the slope-intercept form makes it easy to identify the y-intercept of the line, while the standard form is often used when dealing with systems of linear equations.
Matching the Answer Choices
Now, let's look at the answer choices provided and see which one matches our equation:
A. y - 2 = 3(x + 1/2) B. y - 3 = 2(x + 1/2) C. y + 1/2 = 3(x - 2)
It's clear that option C is the winner! It's exactly the equation we derived using the point-slope form. Options A and B have different forms and don't match our calculated equation. So, we've successfully identified the equation that represents the line with the given slope and point. But wait, there's more to learn here! It's a great practice to not only find the correct answer but also to understand why the other options are incorrect. This helps solidify your understanding of the concepts and prevents you from making similar mistakes in the future. Let's analyze why options A and B are wrong. Option A, y - 2 = 3(x + 1/2), has the correct slope of 3, but the point it represents is ( -1/2, 2), which is not the point we were given (2, -1/2). So, while the slope is correct, the line doesn't pass through the required point.
Option B, y - 3 = 2(x + 1/2), has both an incorrect slope (2 instead of 3) and represents a different point (-1/2, 3). This option is completely different from the line we're looking for. By analyzing these incorrect options, we reinforce our understanding of how the point-slope form works and how changing the slope or the point affects the equation of the line. This kind of analysis is crucial for developing a deeper understanding of mathematics and for becoming a more confident problem solver. Furthermore, understanding why the other options are wrong can help you develop strategies for quickly eliminating incorrect answers on multiple-choice tests. By recognizing common errors or misconceptions, you can avoid falling into traps and increase your chances of selecting the correct answer. In this case, knowing that the point-slope form directly uses the given point and slope allows you to quickly discard options that don't fit the pattern.
Conclusion
Therefore, the equation that represents a line passing through (2, -1/2) with a slope of 3 is:
C. y + 1/2 = 3(x - 2)
We nailed it! Remember, guys, the point-slope form is your friend when you have a point and a slope. Keep practicing, and you'll be a pro at these types of problems in no time.
To understand the equation of a line, especially when given a specific point and slope, we need to use the point-slope form. This form is a powerful tool in algebra and is essential for solving problems like the one we discussed earlier. So, what exactly is the point-slope form, and why is it so useful? Let's break it down in a way that makes it super clear and easy to remember. The point-slope form is expressed as:
y - y₁ = m(x - x₁)
Where:
- m is the slope of the line, which represents its steepness and direction.
- (x₁, y₁) is a known point that the line passes through. This point anchors the line in the coordinate plane.
- (x, y) represents any other point on the line, and these are the variables that will appear in the equation.
This equation might look a bit abstract at first, but it's actually quite intuitive once you understand its components. The slope m tells you how much the y-value changes for every one unit change in the x-value. For instance, a slope of 2 means that y increases by 2 for every increase of 1 in x. A negative slope indicates that y decreases as x increases. The point (x₁, y₁) acts as a reference point. It's a specific location on the line that we know for sure. The equation ensures that the line passes through this point and has the desired slope. Think of it as setting the line's direction and then pinning it down at a particular spot.
Now, let's dig deeper into why this form is so valuable. One of the main reasons is its directness. When you're given a point and a slope, the point-slope form allows you to write the equation of the line in a single step. You simply plug in the values of m, x₁, and y₁, and you're done! There's no need to calculate intercepts or solve for other variables. This makes it a very efficient tool for problem-solving. But the benefits don't stop there. The point-slope form also provides a clear understanding of how the slope and a point together define a line. It highlights the fact that a line is uniquely determined by its steepness (slope) and a single location it passes through (a point). This is a fundamental concept in linear algebra and is crucial for grasping the nature of linear relationships.
Furthermore, the point-slope form is flexible. It can be easily converted into other forms of linear equations, such as the slope-intercept form (y = mx + b) or the standard form (Ax + By = C). This flexibility allows you to work with lines in the form that's most convenient for a particular situation. For example, you might use the point-slope form to write the equation initially, then convert it to slope-intercept form to easily identify the y-intercept, or to standard form if you're working with a system of equations. Let's illustrate this with an example. Suppose we have a line that passes through the point (3, -2) and has a slope of 1/2. Using the point-slope form, we can immediately write the equation as:
y - (-2) = (1/2)(x - 3)
Simplifying, we get:
y + 2 = (1/2)(x - 3)
This is the equation of the line in point-slope form. Now, let's convert it to slope-intercept form. To do this, we need to isolate y on one side of the equation. First, distribute the 1/2 on the right side:
y + 2 = (1/2)x - 3/2
Next, subtract 2 from both sides:
y = (1/2)x - 3/2 - 2
y = (1/2)x - 3/2 - 4/2
y = (1/2)x - 7/2
Now we have the equation in slope-intercept form, where the slope is 1/2 and the y-intercept is -7/2. This example demonstrates how easily you can move between different forms of linear equations using the point-slope form as a starting point. In summary, the point-slope form is a powerful and versatile tool for working with linear equations. It provides a direct way to write the equation of a line when you know a point and the slope, it offers a clear understanding of how lines are defined, and it can be easily converted into other forms. Mastering the point-slope form is essential for anyone studying algebra and beyond.
Alright, guys, let's break down the process of finding the equation of a line when we're given a point and a slope. We'll go through it step by step, so it's crystal clear. We've already established that the point-slope form is our go-to tool for this, but let's really nail down how to use it effectively. So, as we know the point-slope form is:
y - y₁ = m(x - x₁)
Where m is the slope, and (x₁, y₁) is the given point. The process involves a few key steps:
Step 1: Identify the Slope and the Point
The very first thing you need to do is carefully identify the slope (m) and the coordinates of the given point (x₁, y₁). Read the problem statement closely, and make sure you've correctly extracted this information. It's easy to make a mistake if you rush through this step, so take your time and double-check. For example, if the problem states that the line has a slope of 3 and passes through the point (2, -1/2), then we have m = 3, x₁ = 2, and y₁ = -1/2. This is the foundation of our solution, so accuracy here is crucial. Misidentifying the slope or the point will lead to an incorrect equation.
Step 2: Substitute the Values into the Point-Slope Form
Once you have the slope and the point, the next step is to plug these values into the point-slope form. This is a straightforward substitution, but it's important to pay attention to the signs. Replace m with the slope, x₁ with the x-coordinate of the point, and y₁ with the y-coordinate of the point. For our example, substituting m = 3, x₁ = 2, and y₁ = -1/2 into the point-slope form gives us:
y - (-1/2) = 3(x - 2)
Notice how we carefully substituted the negative value for y₁. This is a common area for errors, so be extra cautious with signs. The resulting equation is the equation of the line in point-slope form. It satisfies the condition of having the given slope and passing through the given point.
Step 3: Simplify the Equation (if necessary)
The equation you get after substitution might need a bit of cleaning up. This usually involves simplifying any double negatives or distributing the slope. In our example, we have y - (-1/2) = 3(x - 2). The double negative can be simplified to a positive, giving us:
y + 1/2 = 3(x - 2)
This is the simplified point-slope form of the equation. Sometimes, you might need to distribute the slope on the right side, depending on the instructions or the desired form of the equation. For example, if we distribute the 3 in our equation, we get:
y + 1/2 = 3x - 6
This is still a valid form of the equation, but it's less common to leave it in this form when using the point-slope form. The main goal of simplification is to make the equation as clear and concise as possible.
Step 4: Convert to Other Forms (if required)
Depending on the question or the context, you might need to convert the equation from point-slope form to another form, such as slope-intercept form (y = mx + b) or standard form (Ax + By = C). We discussed earlier how to convert from point-slope form to slope-intercept form. Let's complete that conversion for our example. We have:
y + 1/2 = 3x - 6
To isolate y, subtract 1/2 from both sides:
y = 3x - 6 - 1/2
y = 3x - 12/2 - 1/2
y = 3x - 13/2
Now we have the equation in slope-intercept form, where the slope is 3 and the y-intercept is -13/2. If we wanted to convert to standard form, we would need to eliminate the fraction and rearrange the terms. Multiplying both sides of y = 3x - 13/2 by 2 gives:
2y = 6x - 13
Subtracting 2y from both sides and adding 13 to both sides gives:
6x - 2y = 13
This is the equation in standard form. The ability to convert between these forms is a valuable skill in algebra. Each form has its own advantages and is useful in different situations. By mastering these steps, you can confidently find the equation of a line given a point and a slope, and you can express that equation in various forms as needed. Remember, practice makes perfect, so work through plenty of examples to solidify your understanding.
Now that we've mastered the process of finding the equation of a line using the point-slope form, let's talk about how to apply this knowledge to multiple-choice questions. Often, these questions will present you with several answer choices, and your task is to identify the one that correctly represents the line with the given slope and point. This requires a slightly different approach than simply writing the equation from scratch. You need to be able to analyze the given options and compare them to what you know about the line. So, let's break down the strategy for tackling these types of problems. The key is to understand how the point-slope form translates into the answer choices and to look for telltale signs that an option is correct or incorrect. Remember, the point-slope form is:
y - y₁ = m(x - x₁)
Where m is the slope, and (x₁, y₁) is the point. Here's a step-by-step guide to analyzing answer choices:
Step 1: Mentally Formulate the Correct Equation
Before you even look at the answer choices, take a moment to mentally formulate the correct equation using the point-slope form. This is a crucial step because it gives you a clear target to compare the options against. It prevents you from being swayed by incorrect options that might look similar to the correct one. For instance, if the problem gives you a slope of 3 and a point of (2, -1/2), you should immediately think:
y - (-1/2) = 3(x - 2)
y + 1/2 = 3(x - 2)
Having this equation in mind will make it much easier to evaluate the answer choices.
Step 2: Identify the Slope First
The slope is often the easiest part to check, so start there. Look for the coefficient of the (x - x₁) term. This coefficient is the slope. Eliminate any answer choices that have the wrong slope immediately. This can significantly narrow down your options. In our example, we're looking for an equation with a slope of 3. If any of the answer choices have a different slope, you can cross them out without further consideration. This is a time-saving strategy that helps you focus on the most promising options.
Step 3: Check the Point
Once you've narrowed down the options based on the slope, the next step is to check the point. Remember that the point (x₁, y₁) appears in the equation as (x - x₁) and (y - y₁). This means that the x-coordinate of the point will have the opposite sign in the equation, and the y-coordinate will also have the opposite sign. For example, if the point is (2, -1/2), you should see (x - 2) and (y + 1/2) in the equation. Be careful with the signs here, as this is a common source of errors. An answer choice might have the correct slope but represent a different point if the signs are incorrect. To check a point, look for the values that are being subtracted from x and y in the equation. These values, with the signs reversed, should match the coordinates of the given point. If they don't, eliminate that answer choice.
Step 4: Compare and Select the Best Option
After checking the slope and the point, you should hopefully have only one answer choice remaining. This is the correct equation. If you have more than one option left, double-check your work and make sure you haven't made any mistakes in your analysis. It's also a good idea to plug the given point into the remaining equations to see which one is satisfied. This is a foolproof way to verify your answer. In our example, we're looking for an equation that has a slope of 3 and represents the point (2, -1/2). After applying the steps above, you should be able to confidently select the correct option.
Step 5: Analyze Incorrect Options (if time allows)
If you have extra time, it's beneficial to analyze why the incorrect options are wrong. This helps solidify your understanding of the concepts and prevents you from making similar mistakes in the future. Look for common errors, such as incorrect signs, wrong slopes, or points that don't satisfy the equation. By understanding why the other options are wrong, you'll be better prepared to tackle similar problems in the future. Let's illustrate this with an example. Suppose the answer choices are:
A. y - 2 = 3(x + 1/2) B. y - 3 = 2(x + 1/2) C. y + 1/2 = 3(x - 2) D. y - 1/2 = 3(x + 2)
We know the slope should be 3 and the point should be (2, -1/2). Analyzing the options:
- Option A has the correct slope but represents the point (-1/2, 2), which is incorrect.
- Option B has an incorrect slope of 2 and represents the point (-1/2, 3), which is also incorrect.
- Option C has the correct slope and represents the point (2, -1/2), so it's the correct answer.
- Option D has the correct slope but represents the point (-2, 1/2), which is incorrect.
By analyzing the incorrect options, we reinforce our understanding of how the point-slope form works and how changes in the equation affect the line it represents. This kind of analysis is crucial for developing a deeper understanding of mathematics and for becoming a more confident problem solver. By following these steps, you can effectively analyze answer choices in multiple-choice questions and select the correct equation with confidence. Remember, practice makes perfect, so work through plenty of examples to hone your skills.
The correct answer is C. y + 1/2 = 3(x - 2). This equation represents a line that passes through the point (2, -1/2) and has a slope of 3. We successfully used the point-slope form to solve this problem, guys! Keep up the awesome work!
