Problem 53 - Elementary Algebra

Question

Suppose that we have a rectangular book cover. If the width is increased by 2 centimeters, and the length is decreased by 1 centimeter, then the area is increased by 28 square centimeters. However, if the width is decreased by 1 centimeter and the length is increased. by 2 centimeters, then the area is increased by 10 square centimeters. Find the dimensions of the book cover.

Step-by-Step Solution

Answer

The width is 18 cm and the length is 24 cm.

Step 1: Define Variables

Let the width of the book cover be denoted as \( w \) and the length as \( l \). This sets up our variables which represent the dimensions of the actual book cover.

Step 2: Create First Equation for Modified Dimensions

The width increases by 2 centimeters and the length decreases by 1 centimeter. The new dimensions become \( w+2 \) and \( l-1 \), respectively. The area now is \((w+2)(l-1)\), and it is given that this new area is 28 square centimeters more than the original area, \( wl \). Therefore, the equation is: \[(w+2)(l-1) = wl + 28\].

Step 3: Create Second Equation for Another Modification

The width is decreased by 1 centimeter and the length is increased by 2 centimeters. The new dimensions become \( w-1 \) and \( l+2 \). The new area is \((w-1)(l+2)\), which is 10 square centimeters more than the original, \( wl \). Hence, we have: \[(w-1)(l+2) = wl + 10\].

Step 4: Simplify the Equations

Expand both equations to simplify them. Expand equation from Step 2: \[wl + 2l - w - 2 = wl + 28 \rightarrow 2l - w - 2 = 28 \]. Simplifying gives: \[2l - w = 30 \] (Equation 1). Similarly, expand equation from Step 3: \[wl - l + 2w - 2 = wl + 10 \rightarrow 2w - l = 12 \] (Equation 2).

Step 5: Solve System of Linear Equations

We now have two equations: 1. \[2l - w = 30\] 2. \[2w - l = 12\] Solve either by substitution or elimination method. Multiply the second equation by 2: \[4w - 2l = 24\]. Add to the first equation: \[4w - 2l + 2l - w = 24 + 30\], which simplifies to \[3w = 54\]. Thus, \( w = 18 \). Substitute \( w = 18 \) back into Equation 1: \[2l - 18 = 30\] \[2l = 48\] \[l = 24\].

Step 6: Verify the Solution

Check if these dimensions satisfy both conditions. For the first condition (modified dimensions \( w+2 \) and \( l-1 \)): \[(18+2)(24-1) = 20 \times 23 = 460\]. Original area: \(18 \times 24 = 432\). Difference is 28. For the second condition (modified dimensions \( w-1 \) and \( l+2 \)): \[(18-1)(24+2) = 17 \times 26 = 442\]. Original area: 432. Difference is 10. Both conditions hold true.

Key Concepts

System of Equations

When faced with a problem in algebra, especially those involving multiple conditions, a system of equations can be extremely useful. This approach is used to find values that satisfy all given equations simultaneously. In our exercise, to solve for the dimensions of the book cover, two scenarios were presented based on changes to the width and length.

Each scenario resulted in a unique increase in the area of the rectangular book cover. By representing these conditions mathematically, we derived two equations, which together form a system of equations:

• First Equation: \(2l - w = 30\)
• Second Equation: \(2w - l = 12\)

The solution to this system involves finding the values of the variables that satisfy both equations.
This can be done through methods such as substitution or elimination. In this problem, using the elimination method helped in simplifying these equations to solve for the variables."

Variables in Algebra

In algebra, variables are symbols that represent unknown values which we're trying to find. Throughout various problems, especially in rectilinear shapes like rectangles, they help us set the problem.
For the book cover problem, we defined two primary variables:

• \( w \) for the width
• \( l \) for the length

These variables represent dimensions and they change based on certain conditions presented in the problem.
By adjusting these variables through arithmetic operations, we are able to correspond each scenario to equations that describe how the area is affected by those changes."

Rectangular Area Problems

Calculating area is a fundamental concept in geometry, especially for rectangles which is given by the formula: \(\text{Area} = \text{Length} \times \text{Width}\). Rectangular area problems often involve determining unknown dimensions when certain changes to these dimensions alter the area.

In our problem, two modifications were made to the dimensions of a rectangular book cover:

• Width was increased by 2cm and length decreased by 1cm.
• Width decreased by 1cm, length increased by 2cm.

Each alteration led to a different increase in the area, forming two essential equations. By solving them, one determines the actual dimensions of the rectangle. Solving area problems often involves setting up equations based on these changes and calculating how they affect the original area."