Problem 24 - Algebra 1: Concepts and Skills

Question

Solve the equation. Round the result to the nearest hundredth. Check the rounded solution. $$ -7 x+17=-6 $$

Step-by-Step Solution

Answer

The solution to the equation \( -7x + 17 = -6 \) rounded to the nearest hundredth is \( x = 3.29 \).

Step 1: Subtract 17 from both sides

In the equation \( -7x + 17 = -6 \), we first need to move 17 to the other side. We do this by subtracting 17 from both sides. \( -7x + 17 - 17 = -6 - 17 \), which simplifies to \( -7x = -23 \).

Step 2: Divide both sides by -7

Next, we divide both sides of the equation \( -7x = -23 \) by -7. That is \( -7x / -7 = -23 / -7 \), which simplifies to \( x = 3.286... \)

Step 3: Round the result

Now we take the calculated variable \( x \) and round it to the nearest hundredth. This will give us \( x = 3.29 \)

Step 4: Check the solution

Lastly, we need to check the solution by substituting \( x = 3.29 \) back into the original equation. So, \( -7 * 3.29 + 17 = -6 \). This gives us a slight difference because of the rounded number, but it's close enough to ensure that our solution is likely correct.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is an essential part of solving linear equations. It involves rearranging the equation to isolate the variable and solve for it. This process usually includes operations such as adding, subtracting, multiplying, or dividing both sides of the equation by the same number to maintain equality.

For the given equation, \( -7x + 17 = -6 \), we start by eliminating the constant term from the side with the variable, allowing us to focus on the variable itself. This is done by subtracting 17 from both sides to get \( -7x = -23 \). Now, since the variable \( x \) is multiplied by -7, we reverse this operation by dividing both sides by -7, giving us the solution \( x = 3.286... \).

Throughout this process, the goal is to simplify the equation step by step, being careful with the algebraic operations to ensure accuracy before reaching the solution.

Equations Rounding

When solving equations in algebra, we often encounter numbers that continue indefinitely, like decimal expansions. Rounding these numbers can make them easier to work with and understand. However, rounding during intermediate steps can lead to inaccuracies in the final answer.

In our example, once we solve for \( x \) and obtain \( x = 3.286... \), we need to round this to the nearest hundredth as the problem instructs. This gives us \( x = 3.29 \). Rounding to the nearest hundredth means looking at the third decimal place, which is 6 in this case. Since it is 5 or higher, we round up the second decimal place to 9.

It's important to only round at the end of the calculation to maintain precision throughout the algebraic process. This prevents the rounding error from influencing the subsequent steps and leading to an incorrect solution.

Solution Verification

Solution verification is a critical step to ensure that the rounded answer still satisfies the original equation as closely as possible. After rounding, we must check that our solution makes sense in the context of the problem.

For the exercise in question, we substitute our rounded solution \( x = 3.29 \) back into the initial equation to verify. Plugging it in, we get \( -7(3.29) + 17 \), which should be approximately -6. Due to rounding, there might be a slight discrepancy, but the result should be close enough to the original equation's right-hand side value to confirm our solution's validity.

It's crucial to check even after rounding because approximated solutions can sometimes lead to substantially different outcomes, especially in real-world applications where precision is key. By completing this step, we build confidence in the rounded answer and our understanding of how to solve linear equations.