Problem 32 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The solution to the problem, rounded to the nearest hundredth, is \(x = -0.34\)

Step 1: Rewrite the Equation

Rewrite the equation so that all terms with \(x\) are on one side and all constant terms are on the other: \(42.35x - 5.34x = -12.67\)

Step 2: Simplify the Equation

Combine like terms on both sides to simplify the equation: \(37.01x = -12.67\)

Step 3: Solve for \(x\)

Divide both sides of the equation by 37.01 to solve for \(x\): \(x = -12.67 / 37.01\)

Step 4: Round the Result

Finally, round the result to the nearest hundredth to get an approximate answer. So the solution would be \(x = -0.34\) when rounded to the nearest hundredth.

Key Concepts

Solving Equations

Solving equations like the one in the exercise involves isolating the variable, which in this case is \(x\). When you're presented with an equation, the goal is to get \(x\) by itself on one side of the equation. This process begins by moving terms around. For equations involving addition or subtraction, you will often add or subtract terms from both sides to keep the equation balanced. In this exercise, by subtracting \(5.34x\) from both sides, the equation becomes easier to handle, allowing you to collect all \(x\) terms on one side and all constant terms on the other. This results in the simplified form \(37.01x = -12.67\).

Next, to solve for \(x\), you'll perform the division step. Divide both sides by 37.01, which is the coefficient of \(x\). By doing this, you successfully isolate \(x\), achieving the solution \(x = -12.67 / 37.01\).

Solving each part of the equation in sequence helps keep your work organized and ensures that the end result is correct. Always check your solution by plugging it back into the original equation to verify its accuracy. This practice helps reinforce your understanding and confirms the validity of your solution.

Algebra Problems

In algebra, problems like this one give you a chance to apply fundamental laws and properties of numbers and operations. A critical skill is recognizing like terms, which are terms in an equation that contain the same variables raised to the same power. In the original equation, the terms \(42.35x\) and \(-5.34x\) are like terms because they both have \(x\).

Combining like terms simplifies the equation, making it easier to solve. This process relies on understanding basic arithmetic operations: addition, subtraction, multiplication, and division. By consistently applying these operations, you can reorganize and simplify complex algebra problems into manageable parts.

The exercise demonstrates the principle of balancing equations, a core concept in algebra. Whenever you perform an operation on one side of an equation, you must do the same to the other side to maintain balance. This principle is akin to the equality of a balanced scale. When solving algebra problems, carefully follow these steps:

• Identify and isolate like terms.
• Simplify the equation step-by-step.
• Maintain balance by applying operations equally on both sides of the equation.

This structured approach not only helps you arrive at the correct solution but also deepens your understanding of how algebra works.

Rounding Numbers

Rounding numbers is a useful technique in both everyday life and in mathematics. It allows you to simplify numbers, making them easier to work with, especially when precise calculations are not necessary. In this exercise, after calculating \(x = -12.67 / 37.01\), the result is a decimal fraction which needs to be rounded.

Rounding to the nearest hundredth means you look at the third decimal place. If it's 5 or higher, you increase the second decimal place by one; if it's 4 or lower, you leave the second decimal place as is.

Here’s a quick guide to rounding numbers:

• Identify the place you are rounding to. In this case, it's the hundredths place.
• Check the digit to the immediate right (the thousandths place).
• If this digit is 5 or greater, you round up by adding one to the hundredths place.
• If this digit is less than 5, the number in the hundredths place remains unchanged.

Applying this to our solution gives \(x = -0.34\) after rounding the computed division \(-12.67 / 37.01\). Rounding not only provides a cleaner answer but often matches the level of precision needed in practical applications.