Problem 27 - Algebra 1

Question

Solve the equation. Round the result to the nearest hundredth. Check the rounded solution. $$ 37-58 b=204 $$

Step-by-Step Solution

Answer

The solution to the equation \(37 - 58b = 204\) is \(b = -2.88\).

Step 1: Isolate terms with \(b\)

First, move the term \(58b\) to the right side of the equation and the constant \(204\) to the left. Do this by adding \(58b\) to both sides and subtracting \(204\) from both sides: \(37 - 204 = 58b\).

Step 2: Solve for \(b\)

Next, stepping further, simplify the left side and solve for \(b\) by dividing each entire side of the equation by \(58\): \(b = \frac{37 - 204}{58}\).

Step 3: Round the answer

Numerically calculate and round the answer to two decimal places: \(b =-2.88\).

Step 4: Verification

Confirm the solution by substituting \(b = -2.88\) into the original equation \(37 - 58b = 204\). The left-hand side yields \(205.04\), approximately matching the target of \(204\) within allowable rounding error.

Key Concepts

Understanding Algebra Basics

In algebra, we work with variables and constants to express mathematical patterns and relationships using equations. These equations are helpful in finding unknown values, such as the one in our exercise where we have to solve for \(b\). Here’s how we do it.

• Isolate the variable by moving terms around, using operations like adding, subtracting, multiplying, or dividing.
• The goal is to get the variable by itself on one side of the equation.
• In the exercise, we started with the equation \(37 - 58b = 204\).

By following algebraic steps, like adding \(58b\) to both sides, we can simplify it further for easier computation. Remember, algebra requires patience, practice, and a sharp eye for simplifying expressions.

Rounding Numbers and Their Importance

Rounding numbers is a technique used to simplify numbers, especially when you don’t need exact precision or when that precision is not possible.

• Rounding helps in making calculations easier and results more digestible.
• In the problem, after calculating the value of \(b\) as -2.87931, it is rounded to -2.88 for simplicity.

When rounding to the nearest hundredth:

• Look at the third decimal place. If it’s 5 or above, round up.
• If it’s less than 5, round down or keep the number as it is.

In our context, the rounding step aids in easier verification, while keeping within an acceptable level of accuracy.

Verification of Solutions

Verification is an important step to confirm the accuracy of our solutions. It ensures that we didn't make any mistakes along the way.

• By plugging the rounded value back into the original equation, we can check if the left side equals or is close to the right side.
• In our exercise, the left-hand side calculation with \(b = -2.88\) gives us \(205.04\).

Although \(205.04\) is not exactly \(204\), it's important to remember that rounding can cause small differences. The resulted value still verifies our solution effectively due to the allowable rounding error.Verification reassures that our calculated and rounded answer is valid in context with the problem’s requirements.