Framing A Painting: Find The Length With Math!

Hey there, math enthusiasts! Ever found yourself staring at a beautiful painting and wondering about the math behind framing it? Well, today, let's dive into a classic problem involving perimeters and equations. We're going to help Lisa figure out the dimensions of her rectangular painting using a little algebra magic. So, grab your pencils, and let's get started!

Understanding the Problem: Lisa's Framing Challenge

Framing a painting can be a fun and rewarding project, but it also involves some careful measurements. In Lisa's case, she's dealing with a rectangular painting, and she knows she has 30 inches of framing material to work with. The tricky part? The length of the painting isn't just a simple number; it's related to the width in a specific way: it's three more than twice the width. This is where our algebraic skills come into play. To solve this problem effectively, we need to break it down into manageable steps. First, we need to define our variables. Let's use 'w' to represent the width of the painting and 'l' to represent the length. The problem tells us that the length is three more than twice the width, which we can translate into an equation: l = 2w + 3. This equation is crucial because it establishes the relationship between the length and the width. Next, we need to consider the perimeter of the rectangle. The perimeter is the total distance around the painting, which is the amount of framing material Lisa has. The formula for the perimeter of a rectangle is P = 2l + 2w. Since Lisa has 30 inches of framing material, we know that P = 30. Now we have two equations: l = 2w + 3 and 30 = 2l + 2w. This is a system of equations, and we can use substitution to solve it. We'll substitute the expression for 'l' from the first equation into the second equation. This will give us an equation with only one variable, 'w', which we can solve. Once we find the width, we can plug it back into the first equation to find the length. This step-by-step approach ensures that we address all aspects of the problem and arrive at the correct solution. Remember, the key to solving word problems is to carefully read and understand the information, define the variables, and translate the given relationships into equations. So, let's move on to the next section and see how we can put these equations to work!

Setting Up the Equation: Translating Words into Math

To set up the equation, we need to translate the word problem into mathematical language. This is a crucial step in solving any algebra problem, and it requires careful attention to detail. Let's revisit the information we have: Lisa has a rectangular painting, the length is three more than twice the width, and she uses 30 inches of framing material. The first thing we need to do is define our variables. Let's use 'w' to represent the width of the painting and 'l' to represent the length. This is a standard practice in algebra, and it helps us keep track of what we're trying to find. Now, let's translate the statement