Problem 47 - Elementary Algebra

Question

Solve \(A=L W\) for \(W\). Find the value of \(W\) when \(A=26\) and \(L=2\).

Step-by-Step Solution

Answer

#tag_title# Step 2: Apply the Isolated Expression #tag_content# Now that we have the isolated expression \(W = \frac{A}{L}\), we can plug in the given values of \(A\) and \(L\) to find the value of \(W\). Given: \(A = 75 \text{ square meters}\) and \(L = 15 \text{ meters}\) \(W = \frac{75}{15}\) \(W = 5\) So, the width \(W\) is 5 meters.

Step 1: Write the Equation

We need to solve: \(A=L W\).

Step 2: Isolate the Variable

We apply algebraic operations to isolate the variable. This may include adding, subtracting, multiplying, or dividing both sides by the same value.

Step 3: Solve for the Variable

After performing the algebraic manipulations, we find the solution.

Step 4: State the Solution

#tag_title# Step 2: Apply the Isolated Expression #tag_content# Now that we have the isolated expression \(W = \frac{A}{L}\), we can plug in the given values of \(A\) and \(L\) to find the value of \(W\). Given: \(A = 75 \text{ square meters}\) and \(L = 15 \text{ meters}\) \(W = \frac{75}{15}\) \(W = 5\) So, the width \(W\) is 5 meters.

Key Concepts

Equation Solving

Equation solving is a foundational concept in algebra where we determine the value of unknown variables. To solve an equation means to find a value for the variable that makes the equation true. It usually involves basic operations like addition, subtraction, multiplication, and division. For example, in the equation \(A = LW\), our goal is to find a way to express \(W\) when the values of \(A\) and \(L\) are known.
This involves understanding the balance of an equation, akin to balancing scales.

• Maintain equality by performing the same operation on both sides.
• Identify operations that reverse each other's effects such as multiplication and division.

Learning these fundamentals helps tackle even the most complex algebraic problems.

Isolating Variables

Isolating variables is a key skill that allows you to express a specific variable in terms of others. This involves manipulating an equation so that only the desired variable appears on one side. Think of it as "unpacking" the variable from other terms. When isolating \(W\) in the equation \(A = LW\), you need to "undo" the multiplication by \(L\) using division.
Steps to isolate a variable:

• Identify the operations involving the variable you want to isolate.
• Apply the inverse operation to both sides of the equation.

For our specific problem:

• Division reverses multiplication, so divide both sides by \(L\) resulting in \(W = \frac{A}{L}\).

Once isolated, substitute known values to find a specific solution easily.

Area Formula

The area formula of a rectangle \(A = LW\) connects area \(A\) with the rectangle's length \(L\) and width \(W\). Understanding this formula helps in practical scenarios such as designing spaces or calculating material needs. The formula originates from the concept that multiplying length by width gives the space within.
Key points to remember:

• The area gives a measure of how much space is enclosed within the rectangle.
• Knowing two dimensions lets you find the third, if needed.

In our problem, we solved for \(W\) by rearranging the formula. Given that \(A = 26\) and \(L = 2\), substituting gives: \(W = \frac{26}{2} = 13\).
This means the width of the rectangle is 13 units when the area is 26 square units.