Understanding the Equation of a Line: Your Guide to Slope and Y-Intercept

What is the standard form of the equation of a line given the slope of 2 and y-intercept of 3?

• A. y = 2x - 3
• B. y = 2x + 3
• C. y = -2x + 3
• D. y = -2x - 3

Answer: (B)

To determine the correct form of the equation of a line given a slope and y-intercept, we use the slope-intercept form of a linear equation, which is expressed as \(y = mx + b\). In this formula, \(m\) represents the slope of the line, and \(b\) represents the y-intercept. In this case, the slope \(m\) is given as 2, and the y-intercept \(b\) is given as 3. By substituting these values into the slope-intercept form, we get: \[ y = 2x + 3 \] This equation indicates that for every unit increase in \(x\), \(y\) increases by 2, and the line crosses the y-axis at the point (0, 3). The other options do not match this description. They either have incorrect signs for the slope or intercept, or they do not follow the substitution of the given slope and y-intercept into the correct equation format. Thus, the standard form of the equation with the specified slope and y-intercept is correctly expressed as \(y = 2x + 3\).

When it comes to mastering math concepts, understanding the equation of a line is fundamental. Think of it like a roadmap for understanding how values change with respect to one another—it's all about connections. So, let’s tackle the question: What is the standard form of the equation of a line given a slope of 2 and a y-intercept of 3?

First things first, you've got to know about the slope-intercept form of a linear equation. This is where math gets a little zen: the equation is written as (y = mx + b). Here, (m) represents the slope, while (b) is that important point where the line meets the y-axis—your y-intercept.

Okay, so back to our numbers. In this case, the slope (m) is 2, and the y-intercept (b) is 3. If we substitute these into our trusty equation format, we’ve got:

[

y = 2x + 3

]

And there it is! This tells us a couple of things: first, for every single unit increase in (x), (y) increases by 2. Now, who doesn’t love a little predictability in their math? Additionally, we can see that the line crosses the y-axis at the point (0, 3)—this is a crucial intersection that defines our line’s relationship with the rest of the graph.

Now, if we skim through those other options presented— the ones that proposed alternative equations—it's apparent they don’t quite fit the bill. They either mess up the sign for the slope or completely misrepresent the intercept. Therefore, only (y = 2x + 3) holds true here.

But maybe you’re wondering, why does the equation format matter so much? Well, understanding these concepts is about more than just memorizing rules. It's a stepping stone that prepares you for everything from graphing to solving real-world problems, like budgeting or even planning your trip routes!

As you continue to study, remember that each mathematical journey has its own rhythm. Sometimes it’s a slow build where you reinforce foundational skills. Other times, it’s a sprint to tackle complex problems head-on. And that’s what studying for the California Assessment of Student Performance and Progress (CAASPP) Math Exam is all about—building confidence in your math abilities, one equation at a time. Remember, practice is key, and before you know it, you’ll recognize various forms of linear equations like you recognize your favorite songs!

So, whether you’re graphing lines or tackling equations, keep your chin up and dive into those math challenges. You’ve got this!