Problem 20 - Algebra 1: Concepts and Skills

Question

Solve the equation. Round the result to the nearest hundredth. Check the rounded solution. $$ 13 x-7=27 $$

Step-by-Step Solution

Answer

The rounded solution to the equation is \(x = 2.62\).

Step 1: Isolate the term with the variable

We begin by isolating the term with the variable x by adding 7 to both sides of the equation. It's crucial to do identical operations to both sides of the equation to maintain equality. The equation will then become: \(13x = 27 + 7\) or \(13x = 34\).

Step 2: Find the value of x

Now our variable x is multiplied by 13 on one side of the equation. To isolate x, we'll divide the entire equation by 13: \(\frac{13x}{13} = \frac{34}{13}\) simplifying to \(x = \frac{34}{13}\).

Step 3: Round the result

Now we need to round the result to the nearest hundredth as per the instruction. The division gives 2.61538461538, but rounding this result we get \(x = 2.62\).

Key Concepts

Isolating Variables

To solve a linear equation, one of the primary steps is isolating the variable. This means we want the variable, like \( x \) in our equation, to be by itself on one side of the equation. Here’s a simple breakdown of how it works:

• First, identify the term that includes the variable.
• Perform operations to both sides of the equation to move all other terms away from the variable.
• For instance, in the equation \(13x - 7 = 27\), we see that \(x\) is tied up with other numbers, specifically worked into a term with \(13\) and alongside \(-7\).

In this problem, we handle the \(-7\) by doing the opposite operation: adding 7. Always remember to do the same operation to both sides to keep the equation balanced, which transforms it to \(13x = 34\). Once isolated, we focus on \(x\) alone, here divided by 13, giving us \(x = \frac{34}{13}\). This step is essential because it directly leads us to the solution of the equation.

Rounding to Nearest Hundredth

When you're working with a solution like \(x = \frac{34}{13}\), the result can be a long decimal. To make computations easier and fit specific problem requirements, we often round the number. Rounding involves adjusting the number to its nearest place value, which simplifies the number while keeping it as accurate as possible.

• In this case, divide 34 by 13 to get the approximate value \(2.61538461538\).
• We need to round this to the nearest hundredth, which is the second digit to the right of the decimal point.
• Look at the third digit: if it's 5 or more, round the second digit up.

In our equation, the number becomes 2.62 after rounding. Rounding is a common step to prepare numbers for practical applications, making them easier to interpret.

Checking Solutions

After finding and rounding the solution to an equation, it’s crucial to check your work. Checking ensures that the rounded solution satisfies the original equation. This step acts like a verification process where you plug the value back into the original equation to see if it holds true.

• Take the rounded solution, which is \(x = 2.62\), and substitute it back into the original equation \(13x - 7 = 27\).
• Compute the left-hand side: \(13 \times 2.62 - 7\).
• Ensure that this value is approximately the same as the right-hand side (27). Small differences could occur due to rounding.

By performing this check, you verify that your process was done correctly, reaffirming your confidence in your algebraic skills.