Problem 40 - Prealgebra

Question

Solve each equation. Round to the nearest tenth, if necessary. $$190.5=1.5 b^{2}$$

Step-by-Step Solution

Answer

The solution is approximately \( b \approx 11.3 \).

Step 1: Isolate the variable term

First, we want to isolate the term with the variable by dividing both sides of the equation by 1.5 to get rid of the coefficient in front of the variable. This gives us: \[ b^2 = \frac{190.5}{1.5} \]

Step 2: Simplify the division

Calculate \( \frac{190.5}{1.5} \) to simplify the equation. After performing the division, we find: \[ b^2 = 127 \]

Step 3: Solve for the variable

Take the square root of both sides to solve for \( b \). This gives us: \[ b = \sqrt{127} \]

Step 4: Calculate the square root

Compute \( \sqrt{127} \) to find the value of \( b \). Rounding to the nearest tenth, we find: \( b \approx 11.3 \).

Key Concepts

Isolating the Variable

When solving equations, isolating the variable is one of the first steps to simplify and solve the problem. This involves rearranging the equation so that the variable you're solving for is alone on one side. In the given exercise, the initial equation is:

• 190.5 = 1.5b^2

To isolate the term with the variable, divide both sides by 1.5, which is the coefficient of the variable term, \( b^2 \). This step clears the equation of any extra numbers besides the variable term:

• \( b^2 = \frac{190.5}{1.5} \)

By isolating the variable, we make it easier to move towards finding its value. This technique is often used as a foundation for solving various equations.

Simplifying Equations

Simplification is a crucial step to make an equation easier to solve. Once you've isolated the variable term, you'll often need to simplify the equation further. In this problem, after isolating the variable term, the next step is to simplify:

• \( b^2 = \frac{190.5}{1.5} \)

Perform the division to get:

• \( b^2 = 127 \)

Simplifying involves reducing the equation to its most manageable form. It allows you to more easily carry out subsequent operations, like taking a square root. Always perform arithmetic carefully and double-check your calculations to ensure accuracy. Simplification is about making the math proceed more smoothly!

Square Roots

Once the equation is simplified, solving for the variable might involve taking roots. Specifically, in this case, we take the square root—since the variable was isolated in squared form:

• \( b^2 = 127 \)

To find \( b \), compute:

• \( b = \sqrt{127} \)

Taking a square root reverses the squaring process. Because squaring a number results in a positive number, the square root of a positive number will have both a positive and a negative solution. However, in practical terms for equations like these, we often deal with only the positive root unless specified otherwise. Make use of a calculator for obtaining precise values when the square root isn't a neat whole number or simple fraction.

Rounding Numbers

Lastly, considering the instruction to round the solution, you must round numbers to a specific decimal place to present the final answer. After calculating the square root as follows:

• \( b = \sqrt{127} \approx 11.26942767 \)

The variable is expressed almost fully, but the task asks to round to the nearest tenth. Look at the digit in the hundredths place:

• If it's 5 or more, round up
• Otherwise, round down

Since 11.27 has a 2 in the hundredths place, round down, making it:

• \( b \approx 11.3 \)

Rounding gives the final answer in a more readable and usable form. Always remember, presenting results in the required format is just as important as the calculation itself.