Problem 30 - Algebra 1: Concepts and Skills

Question

Solve the equation. Round the result to the nearest hundredth. $$ 9.47 x=7.45 x-8.81 $$

Step-by-Step Solution

Answer

\(x = -4.36\)

Step 1: Group the x terms

Begin by moving \(7.45x\) from the right side of the equation to the left, which will look like this: \(9.47x - 7.45x = -8.81\)

Step 2: Simplify the equation

Simplify the left side of the equation by subtracting the \(x\) terms: \(2.02x = -8.81\)

Step 3: Solve for x

Solve for \(x\) by dividing both sides of the equation by 2.02, so the equation becomes \(x = -8.81 / 2.02\)

Step 4: Round the answer

After calculation, you get x approximately equals to -4.36. However, we need to round the answer to the nearest hundredth, which gives \(x = -4.36\)

Key Concepts

Rounding Numbers

Rounding numbers is a technique used to simplify numbers, making them easier to work with or understand, especially in cases where only an approximate value is needed. When rounding to the nearest hundredth, we look at the number in the thousandth place to determine whether to round up or down. For example, if you have a number like 5.678, you look at the 8 in the thousandth place. Since it is 5 or greater, you round up the number in the hundredth place, resulting in 5.68.

• If the digit is 0-4, keep the hundredth number.
• If the digit is 5-9, increase the hundredth number by one.

This simple rule helps maintain consistency across calculations, especially when presenting final results in a clear manner.

Combining Like Terms

Combining like terms is an essential part of solving linear equations. It involves merging terms with the same variable to simplify expressions. Look at terms within an equation that have the same variable part; if they are on opposite sides of the equation, move them to one side. For instance, in the equation \(9.47x = 7.45x - 8.81\), you start by transferring \(7.45x\) to the left side, changing it to its negative counterpart, yielding \(9.47x - 7.45x = -8.81\).

• Identify like terms with the same base and exponent.
• Add or subtract coefficients to combine them.
• This simplification makes equations easier to solve.

Combining terms simplifies the problem, helping to reach the solution more efficiently.

Simplifying Equations

Simplifying equations is an important step in solving them efficiently. Once like terms are combined, you can work towards isolating the variable. In our example, after combining like terms, the equation \(2.02x = -8.81\) remains. To further simplify, solve for the variable by dividing both sides by the coefficient of \(x\), which is 2.02. This operation yields the solution \(x = -8.81 / 2.02\).

• Simplify complex expressions into manageable parts.
• Perform arithmetic operations to isolate the variable.
• Main goal: transform the equation to \(x = \text{{number}}\).

This step-by-step simplification makes problem-solving straightforward and reduces the chance of errors, providing a clear path to finding the solution.