Problem 29 - Algebra 1: Concepts and Skills

Question

Solve the equation. Round the result to the nearest hundredth. Check the rounded solution. $$ 42=23 x-9 $$

Step-by-Step Solution

Answer

The solution to the equation \(42 = 23x - 9\) rounded to nearest hundredths place is \(x = 2.22\).

Step 1: Isolate the term with x

Rearrange the equation such that terms with x are isolated on the right side in the first step, while the constant remains on the left side. You do this by adding 9 to both sides of the equation, that will yield: \(42+9 = 23x\).

Step 2: Solve for x

The next step is to figure out \(x\). You can find the value of \(x\) by dividing both sides of the equation by 23. This gives: \(x = \frac{42+9}{23}\).

Step 3: Round the answer

By calculating the value, we have \(x = \frac{51}{23} = 2.2174\)However, the question requires the solution to be rounded to the nearest hundredth:\(x = 2.22\).

Step 4: Check the solution

Finally, you can check your solution by substitifying x = 2.22 in the original equation. Calculation gives \(42 = 23 * 2.22 - 9\) which verifies that our solution is correct as both sides are balanced.

Key Concepts

Understanding Algebraic Equations

An algebraic equation is a mathematical statement that asserts the equality of two expressions. It is composed of variables, coefficients, constants, and arithmetic operations. A fundamental concept when working with algebraic equations is the use of equality - that is, both sides of the equation must have the same value.

To solve an algebraic equation, one needs to find the values for the variable that make the equation true. For example, with the given exercise \(42=23x-9\), the goal is to find the value of \(x\) that satisfies this equation. Think of the equation as a balance scale where your job is to keep it level while you manipulate the parts.

Isolating Variables

The step towards solving an equation for a variable, known as isolating the variable, involves rearranging the equation to get the unknown variable by itself on one side and everything else on the other side. This is achieved through a series of algebraic operations that are applied equally to both sides so the balance (equality) of the equation is maintained.

In our textbook exercise, isolating \(x\) begins with moving the constant term on the other side of the equation. We add 9 to both sides, leading to \(42 + 9 = 23x\). Next, algebra guides us to divide every term by 23 to successfully isolate \(x\), resulting in the expression \(x = \frac{42+9}{23}\). Always carry out the same operation on both sides to preserve the 'balance' of the equation.

Rounding Decimals

After solving for the variable, one might need to round the answer to a desired degree of accuracy, often due to the context of the problem or instruction given. Rounding decimals is a method of approximating a number to a certain level of precision. The textbook exercise requires rounding to the nearest hundredth, which means that we want our answer to have two digits after the decimal point.

To round \(2.2174\) to the nearest hundredth, we look at the third digit (1 in this case) which is less than 5. Therefore, the second digit (7) remains the same, and the final rounded solution is \(2.22\). Rounding can impact the accuracy of the solution, so it's crucial to always check if the problem asks for a rounded answer.