Problem 32 - Algebra 1

Question

Solve the equation. Round the result to the nearest hundredth. $$ 12.67+42.35 x=5.34 x+26.58 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x=0.38\).

Step 1: Arrange terms with variable \(x\) on one side

Subtract \(5.34x\) from both sides of the equation, yielding \(42.35x-5.34x=26.58-12.67\).

Step 2: Simplify both sides of the equation

Combine like terms on each side of the equation. This results in \(37.01x=13.91\).

Step 3: Solve for \(x\)

Divide both sides of the equation by 37.01 to isolate \(x\), yielding \(x=13.91/37.01\).

Step 4: Round the result

Calculate the value of \(x\) and round it to the nearest hundredth. Thus, \(x=0.38\)

Key Concepts

Solving Equations

When working with linear equations, the goal is to determine the value of the variable that makes the equation true. A linear equation is typically in the form of \( ax + b = cx + d \). Here, \( a \), \( b \), \( c \), and \( d \) are constants, and \( x \) is the variable. The main task is to solve this equation by isolating \( x \).

• Start by moving all the terms involving \( x \) to one side of the equation. This may involve adding or subtracting terms from each side to combine like terms.
• Once the variable terms are on one side, the constant terms should be moved to the opposite side.
• The equation is then simplified to find \( x \), making sure to apply inverse operations correctly at each stage.

By meticulously following these steps, the solution to the equation can be easily revealed. Always check your work by substituting the value back into the original equation to ensure no mistakes were made.

Step-by-Step Solution

A structured approach towards solving equations is essential for gaining clarity and accuracy. Let's take a closer look at this specific example where the equation is \( 12.67 + 42.35x = 5.34x + 26.58 \):
First, isolate the variable terms. Subtract \( 5.34x \) from both sides to gather all terms with \( x \) on one side, getting \( 42.35x - 5.34x = 26.58 - 12.67 \). This step eliminates the \( x \) on the right side and cancels it out on the left.
Next, simplify by combining like terms. The left side becomes \( 37.01x \) while the right becomes \( 13.91 \).
To find the value of \( x \), divide both sides by \( 37.01 \). This step is crucial as it isolates \( x \) to provide the solution. As a result, \( x = 13.91 / 37.01 \).
This detailed approach ensures each stage of the solution is manageable for students, building a strong foundation in algebraic problem-solving.

Rounding Numbers

Rounding numbers is a necessary step to present your final answer in a manageable and meaningful format. In mathematics, rounding is often to a specified decimal place, such as to the nearest hundredth. This method is used to simplify numbers while maintaining a suitable level of precision for practical use.
Here's a quick guide to rounding:

• Identify the place value to which you are rounding. For example, rounding to the nearest hundredth, focus on the third decimal place.
• If the digit in this place is 5 or greater, round up, which means increasing the digit in the rounding place by one.
• If the digit is less than 5, simply retain the digit in the rounding place.

By applying these rules, the calculated value of \( x \), which initially equals \( 0.375 \), gets rounded to \( 0.38 \) when rounded to the nearest hundredth. Practicing rounding ensures precision in your calculations.