Equation In Slope-Intercept Form Line With Slope 0 Through (2 3)

Hey everyone! Let's dive into a fun math problem today. We're going to figure out which equation represents a line in slope-intercept form that has a slope of 0 and passes through the point (2,3). Sounds like a puzzle, right? Let's break it down step by step. Understanding slope-intercept form and how a slope of 0 affects the line's equation are key here. We'll explore each option, making sure we understand why some fit and others don't. So, grab your thinking caps, and let's get started!

Understanding Slope-Intercept Form

Okay, first things first, let's quickly recap what the slope-intercept form actually is. This form is a super handy way to write linear equations because it immediately tells us two crucial things about the line: its slope and its y-intercept. The general form looks like this:

y = mx + b

Where:

• y is the vertical coordinate.
• m is the slope of the line.
• x is the horizontal coordinate.
• b is the y-intercept (the point where the line crosses the y-axis).

So, when you see an equation in this form, the number sitting right in front of x is your slope (m), and the constant term (b) is where the line cuts the y-axis. Knowing this form inside and out is like having a secret decoder ring for linear equations! It's essential for quickly understanding and visualizing lines. Remember, the slope tells us how steep the line is and whether it's going uphill or downhill, while the y-intercept gives us a fixed point where the line intersects the y-axis. This simple equation, y = mx + b, unlocks a wealth of information about any straight line you can imagine, making it a cornerstone of linear algebra and coordinate geometry. Understanding this foundation is critical for tackling problems involving linear equations and graphs.

The Significance of a Slope of 0

Now, let's talk about slopes. The slope tells us how much a line is inclined. A positive slope means the line goes uphill as you move from left to right, a negative slope means it goes downhill, and a slope of zero? Well, that's a special case! A slope of 0 means the line is perfectly horizontal – it doesn't go uphill or downhill at all. It's flat, like the horizon on a calm sea. Think of it this way: for every step you take to the right (or left) on the x-axis, you don't move up or down on the y-axis. The y-value stays the same. This is super important because it drastically simplifies our equation. If the slope (m) is 0, then the mx part of our slope-intercept form (y = mx + b) becomes 0, leaving us with just y = b. This means a line with a slope of 0 is simply a horizontal line where the y-value is constant. Recognizing that a zero slope corresponds to a horizontal line is key to solving this type of problem quickly and accurately. It's a fundamental concept that bridges the abstract idea of a slope to the visual representation of a line on a graph. This understanding is crucial not just for this specific problem, but for any scenario where you're dealing with horizontal lines or trying to interpret the meaning of slope in real-world contexts.

Incorporating the Point (2,3)

Okay, so we know we're looking for a horizontal line (slope 0), but we also have another piece of information: the line must pass through the point (2,3). This point gives us a specific location on the coordinate plane that our line needs to include. Remember, the coordinates (2,3) tell us that when x is 2, y is 3. Since we're dealing with a horizontal line, the y-value will be the same for every x-value. This is the crucial piece of the puzzle. A horizontal line has the same y-value for every point on the line. Because our line passes through (2,3), the y-value must always be 3. This significantly narrows down our options. We're not just looking for any horizontal line; we're looking for the specific horizontal line that has a y-value of 3. This point acts like a constraint, ensuring that our line isn't just flat, but also positioned correctly on the coordinate plane. Visualizing this point and the horizontal line passing through it can help solidify your understanding. Imagine a flat line slicing through the point (2,3) – that's the line we're trying to describe with an equation. This understanding is vital for connecting the geometric representation of a line with its algebraic equation.

Analyzing the Answer Choices

Alright, let's get down to business and examine our answer choices. We need to find the equation that's in slope-intercept form (or easily convertible to it), has a slope of 0, and passes through the point (2,3).

• A. x = 2

  This equation represents a vertical line, not a horizontal one. Vertical lines have undefined slopes, not a slope of 0. So, this isn't our answer. Think about it – a vertical line goes straight up and down, meaning the x-value is constant while the y-value can be anything. This is the opposite of what we're looking for. Vertical lines are fundamentally different from horizontal lines in terms of their equations and slopes. Recognizing this difference is a key skill in coordinate geometry.

• B. y = x + 2

  This is in slope-intercept form, but the slope is 1 (the coefficient of x), not 0. This line is slanted upwards, not horizontal. Plus, if you plug in x=2, you get y=4, not 3, so it doesn't pass through the point (2,3) either. This option fails on two counts: incorrect slope and incorrect point. It highlights the importance of checking both the slope and the point when verifying an equation. Remember, the slope determines the line's inclination, and the point ensures it's in the correct location on the graph.

• C. y = 3

  Aha! This looks promising. This equation is in slope-intercept form (we can think of it as y = 0x + 3), has a slope of 0, and passes through the point (2,3). This is because no matter what x-value you plug in, y will always be 3. This equation perfectly fits our criteria. It represents a horizontal line at a y-value of 3, which includes the point (2,3). This is the correct answer! This option demonstrates the power of the simplified equation y = b for horizontal lines, where b is the y-coordinate the line passes through.

• D. y = x + 3

  This is also in slope-intercept form, but the slope is 1, not 0. This line is slanted upwards and while it has a y-intercept of 3, it doesn't represent a horizontal line through (2,3). If you plug in x=2, you get y=5, not 3. This option is similar to option B in that it has the wrong slope and doesn't pass through the specified point. It reinforces the need to carefully check both the slope and the point when evaluating linear equations.

Conclusion: The Correct Equation

So, after analyzing all the options, the winner is C. y = 3. This equation is in slope-intercept form (with a slope of 0 and a y-intercept of 3) and represents a horizontal line that passes through the point (2,3). Yay, we solved it! This problem was a great way to reinforce our understanding of slope-intercept form, slopes of 0, and how points relate to linear equations. Remember, practice makes perfect, so keep those math muscles flexed! Remember, the key takeaway is that a horizontal line has a slope of 0 and its equation is of the form y = b, where b is the y-coordinate of any point on the line. By understanding this fundamental concept, you can quickly and confidently solve similar problems. And that's the power of math – once you grasp the core principles, you can tackle a wide range of challenges!