Problem 56 - Algebra 1: Concepts and Skills

Question

Solve the equation. Round the result to the nearest hundredth. Check the rounded solution. $$3 y-5=11$$

Step-by-Step Solution

Answer

The solution to the equation \( 3y - 5 = 11 \) rounded to the nearest hundredth is \( y ≈ 5.33 \)

Step 1: Isolate y

Begin by isolating \( y \) in the equation \[ 3 y - 5 = 11 \] The first step is to get rid of the constant term on the left side, so, add 5 to both sides of the equation: \[ 3 y = 11 + 5 \]

Step 2: Perform addition

Combine like terms, 11 + 5, to obtain 16: \[ 3 y = 16 \]

Step 3: Finalize isolating y

Divide both sides of the equation by 3 to isolate \( y \): \[ y = 16 / 3 \]

Step 4: Round the solution

Round the obtained value of \( y \) to the nearest hundredth: \[ y ≈ 5.33 \]

Step 5: Check the solution

Substitute the rounded value of \( y \) in the original equation to verify the solution: \[ 3(5.33) - 5 ≈ 11 \] The left side of the equation is approximately 11 which confirms that the solution is correct.

Key Concepts

Solving Equations

In mathematics, solving a linear equation is like untangling a ball of string. The goal is to isolate the variable—usually represented by a letter such as \( y \)—on one side of the equation. In the given exercise, the equation was \( 3y - 5 = 11 \). The process begins by eliminating the constant term that sits alongside the variable. Here's how it's done: - **Add 5** to both sides to get rid of the \(-5\) on the left, simplifying the equation to \(3y = 16\). - **Isolate \( y \):** Divide both sides of the equation by 3, so the variable \( y \) stands alone, resulting in \( y = \frac{16}{3}\). Breaking it down into these small steps ensures clarity and accuracy when solving equations, aiding you in arriving at the correct answer. Remember, you always do the same operation to both sides of the equation to maintain the balance.

Rounding Decimals

Rounding decimals is a common mathematical practice that allows us to simplify numbers while keeping their value close to the original. This is particularly useful when dealing with long decimal numbers that are challenging to work with. In this case, after solving the equation, we found \( y = \frac{16}{3} \), which is 5.3333... when rounded. To round a number to the nearest hundredth, remember the following tips:

• Identify the hundredth place; it is the second digit after the decimal point.
• Check the digit immediately after the hundredth place. If this digit is 5 or greater, you'll round up the hundredth digit by one.
• If the digit is less than 5, you keep the hundredth digit as is.

Using these steps, 5.3333... becomes approximately 5.33. This makes the value easier to work with while maintaining accuracy.

Checking Solutions

Once you've solved an equation and possibly rounded your answer, the final step is to check your work. This step is crucial because it confirms the solution is both accurate and applicable to the original equation. Here's how to check your solution:

• Substitute the rounded value back into the original equation. In our example, substitute \( y = 5.33 \) into \( 3y - 5 = 11 \).
• Calculate the new left side of the equation. If the left side equals the right side (or is very close, considering rounding), then the solution is verified.

With our example, substituting \( y = 5.33 \) yields \( 3(5.33) - 5 = 15.99 \), which is approximately 11. This confirms that the rounded solution is valid.