Henry's Line Equation: A Step-by-Step Solution

Hey guys! Let's dive into a fun little math problem today that involves linear equations and graphs. We've got Adriana, who's written the equation y = 2x + 4 to represent a line. Now, Henry wants to write his own equation. His line needs to have the same slope as Adriana's, but its y-intercept should be 1 unit lower. Let's figure out which equation represents Henry's line!

Understanding the Basics: Slope and Y-intercept

Before we jump into solving the problem, let's quickly review what slope and y-intercept actually mean in a linear equation. Think of the equation y = mx + b. This is the slope-intercept form, a super handy way to represent a straight line. In this form:

• m represents the slope: The slope tells us how steep the line is and in what direction it's going. A positive slope 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 the slope, the steeper the line.
• b represents the y-intercept: The y-intercept is the point where the line crosses the y-axis. It's the value of y when x is equal to 0. So, if our equation is y = mx + b, the line crosses the y-axis at the point (0, b).

In Adriana's equation, y = 2x + 4, we can clearly see that the slope (m) is 2 and the y-intercept (b) is 4. This means her line goes upwards and is relatively steep, crossing the y-axis at the point (0, 4). Now, let's keep these concepts in mind as we tackle Henry's equation.

Finding Henry's Equation: A Step-by-Step Approach

Okay, so we know Henry's line has the same slope as Adriana's, which is 2. This means the m value in Henry's equation will also be 2. The crucial difference is the y-intercept. Henry's line has a y-intercept that's 1 unit lower than Adriana's. Adriana's y-intercept is 4, so Henry's y-intercept will be 4 - 1 = 3.

Now we have all the pieces we need to write Henry's equation. We know the slope (m) is 2 and the y-intercept (b) is 3. Plugging these values into the slope-intercept form (y = mx + b), we get:

y = 2x + 3

That's it! This is the equation that represents Henry's line. It has the same slope as Adriana's line (2) but a y-intercept that is 1 unit lower (3 instead of 4). See how breaking down the problem into smaller steps makes it much easier to solve? First, we reviewed the basics of slope and y-intercept. Then, we carefully considered the changes in the y-intercept while keeping the slope constant. Finally, we plugged the new values into the y = mx + b formula to arrive at the answer. You can use this same methodical approach to solve all sorts of linear equation problems!

Analyzing the Options: Which One Matches?

Let's say we had some multiple-choice options to choose from. For example, maybe the options were:

A. y = x + 4 B. y = 3x + 4 C. y = 2x + 3 D. y = 2x + 5

We've already figured out that Henry's equation is y = 2x + 3. Looking at the options, we can see that option C perfectly matches our solution. Options A and B have different slopes than Adriana's line, so we can rule them out immediately. Option D has the correct slope, but its y-intercept is higher than Adriana's, not lower. So, option C is definitely the correct answer.

This process of elimination can be a very powerful tool when you're solving math problems, especially on multiple-choice tests. If you're not sure of the answer right away, try ruling out the options that you know are incorrect. This can significantly increase your chances of choosing the right answer.

Graphing the Lines: A Visual Representation

Sometimes, visualizing the problem can help solidify your understanding. If we were to graph both Adriana's line (y = 2x + 4) and Henry's line (y = 2x + 3), we'd see two parallel lines. Parallel lines, remember, have the same slope but different y-intercepts. Adriana's line would cross the y-axis at the point (0, 4), while Henry's line would cross the y-axis at the point (0, 3). Notice how Henry's line is exactly one unit lower than Adriana's line across the entire graph. This visual representation confirms our algebraic solution and gives us a deeper understanding of the relationship between the two equations.

Graphing can be particularly helpful when you're dealing with more complex linear equation problems, such as finding the point of intersection between two lines or determining if lines are perpendicular. So, don't hesitate to sketch a quick graph if it helps you visualize the problem!

Practice Makes Perfect: Tackling Similar Problems

Now that we've successfully solved this problem, let's think about how we can apply these concepts to other similar situations. Linear equations are everywhere in math and science, so understanding them is super important. Let's consider a few variations of this problem to get some practice.

What if Henry's line had a y-intercept that was 2 units higher than Adriana's? How would that change the equation? In this case, we would add 2 to Adriana's y-intercept (4 + 2 = 6). So, Henry's equation would be y = 2x + 6. The slope remains the same, but the line shifts upwards on the graph.

Or, what if Henry's line had a different slope but the same y-intercept as Adriana's? Let's say Henry's line had a slope of 3 and the same y-intercept of 4. Then, Henry's equation would be y = 3x + 4. The line would cross the y-axis at the same point as Adriana's line, but it would be steeper because the slope is larger.

By playing around with these variations, you can develop a much stronger intuition for how changing the slope and y-intercept affects the graph of a line. Try creating your own variations and solving them! You can also graph the equations to check your answers visually.

Real-World Applications: Linear Equations in Action

You might be wondering,