Framed Picture Length: An Algebraic Expression

Hey there, math enthusiasts! Today, let's dive into a fun problem involving framing a picture. It's a fantastic way to see how algebra pops up in everyday situations. We'll break down the problem step by step, making sure everyone's on board. So, let's get started!

The Picture Framing Problem

Imagine Alissa is adding a frame to her rectangular picture. The picture itself is 10 inches long and 8 inches wide. Now, let's say the frame has a width of x inches. The big question we want to answer is: how do we represent the length of the framed picture in inches? This might seem tricky at first, but don't worry, we'll tackle it together. It’s like figuring out how much extra wall space you’ll need once you hang that awesome piece of art!

Breaking Down the Dimensions

To really understand this, let’s visualize what's happening. The original picture has a length of 10 inches. When Alissa adds a frame, she's essentially adding to the length on both sides of the picture. Think of it like this: there's a frame on the left side and a frame on the right side. Each of these frame sections adds x inches to the overall length. So, we're adding x inches twice. This is a crucial point to grasp. We're not just adding the frame width once; it affects both ends of the picture. It's similar to putting a border around a garden; you have to account for the border on all sides, not just one!

Visualizing the Frame

Imagine the picture as a rectangle. Now, picture the frame surrounding it. The frame adds width x on the left and width x on the right. So, the total additional length due to the frame is x + x, which simplifies to 2x. This is because the frame extends the picture's length on both ends. The original length was 10 inches, and we're adding 2x inches for the frame. Therefore, the length of the framed picture will be the original length plus the added frame length, which is 10 + 2x. This is the algebraic expression that represents the total length of the framed picture.

Why Not Just x + 10?

You might be wondering, why isn't the answer simply x + 10? That's a great question! Remember, the frame isn't just adding width to one side; it's adding width to both sides of the picture. So, if we only added x to 10, we'd be neglecting the extra width added by the frame on the other side. It's crucial to account for both sides to get the correct total length. Thinking of it in practical terms, if you were building the frame, you'd need to consider the material required for both the left and right sides, not just one!

The Correct Expression

So, after carefully considering how the frame affects both sides of the picture, we arrive at the correct expression for the length of the framed picture: 10 + 2x. This expression tells us exactly how the total length changes based on the width of the frame. If the frame is 1 inch wide (x = 1), the framed picture's length would be 10 + 2(1) = 12 inches. If the frame is 2 inches wide (x = 2), the framed picture's length would be 10 + 2(2) = 14 inches. See how the expression neatly captures this relationship? This is the power of algebra in action!

Extending the Concept: The Width

Now that we've nailed the length, let's think about the width of the framed picture. The original picture has a width of 8 inches. Just like the length, the frame adds width on both the top and bottom of the picture. Each of these additions is x inches. So, the total additional width due to the frame is also 2x inches. This means the total width of the framed picture can be represented by the expression 8 + 2x. It follows the same logic as the length calculation, but now we're applying it to the width.

Putting it All Together

We've now figured out the expressions for both the length and the width of the framed picture. The length is 10 + 2x inches, and the width is 8 + 2x inches. These expressions are incredibly useful. If Alissa decides she wants a specific frame width, say 3 inches, she can simply plug x = 3 into these expressions to find the overall dimensions of the framed picture. The length would be 10 + 2(3) = 16 inches, and the width would be 8 + 2(3) = 14 inches. It's like having a magic formula to calculate the final size!

Why This Matters

You might be thinking,