Problem 54 - Elementary Algebra

Question

Kathy is drawing up the blueprints for a house she is designing. She wants to have four square windows of equal size in the living room, with a total area of 64 square feet. Solve the equation \(4 s^{2}=64\) to find \(s\), the length of the sides of the windows.

Step-by-Step Solution

Answer

The length of the sides of the windows is 4 feet.

Step 1: Understand the Equation

Kathy needs to find the length of the sides of four equal square windows with a total area of 64 square feet. The equation given is: \[4 s^{2} = 64\]

Step 2: Isolate the Variable

Start by isolating the term with the variable. Divide both sides of the equation by 4: \[s^{2} = \frac{64}{4}\]

Step 3: Simplify the Equation

Perform the division on the right-hand side to simplify the equation: \[s^{2} = 16\]

Step 4: Solve for the Variable

Find the square root of both sides of the equation to solve for \(s\): \[s = \sqrt{16}\]

Step 5: Simplify the Square Root

Calculate the square root of 16 which gives: \[s = 4\]

Key Concepts

isolating variables

When solving equations, particularly quadratic equations like the one in Kathy's blueprint problem, the first step is often isolating the variable you need to solve for. In Kathy's case, it's the length of the side of the square window, denoted as \(s\). The original equation given is \[4s^{2} = 64\], where the multiplier of the term \(s^{2}\) is 4. To isolate \(s^{2}\), you need to remove the coefficient 4 by performing the same operation on both sides of the equation.

You achieve this by dividing both sides of the equation by 4, leading to:
\[s^{2} = \frac{64}{4}\].

After this step, \(s^{2}\) stands alone, making it easier to solve for \(s\). This is the essence of isolating variables: transforming the equation in a way that leaves the variable of interest by itself on one side.

simplifying equations

Once you've successfully isolated the variable, the next step in solving a quadratic equation is simplifying the equation. Using Kathy's problem as an example, after dividing both sides by 4, we get:

\[ s^{2} = \frac{64}{4} = 16 \].

This step involves performing basic arithmetic to simplify the equation further. Simplification helps highlight the relationship between the variable and the constants in the equation, making it easier to solve.

Always remember:

• Perform arithmetic operations systematically.
• Double-check your calculations for accuracy.

In this case, dividing 64 by 4 directly provides the result \(16\), thus simplifying the equation significantly.

square roots

The final concept to understand when solving such equations is taking the square root. To solve for \(s\) in the simplified equation \[ s^{2} = 16 \], you need to find the square root of 16:

\( s = \sqrt{16} \).

Taking the square root undoes the squaring of \(s\). Square roots find a number which, when multiplied by itself, gives the original number. For 16, we know:

\( \sqrt{16} = 4 \).

This means \( s = 4 \).

Remember, there can typically be both a positive and a negative square root. In this context, however, negative lengths don't make sense, so we only consider \( s = 4 \). Taking square roots and understanding their properties is crucial in solving quadratic equations and many other mathematical problems.

In summary:

• Find the square root of both sides to solve for the isolated variable.
• Consider the context to determine whether both positive and negative roots are applicable.