Problem 25 - Algebra 1

Question

Solve the equation. Round the result to the nearest hundredth. Check the rounded solution. $$ 14 r+8=32 $$

Step-by-Step Solution

Answer

The solution to the equation, rounded to the nearest hundredth, is \(r = 1.71\). Substituting this into the original equation provides an approximate value of the right side of the equation, thus confirming the solution.

Step 1: Isolate the term with the variable

To do this, subtract 8 from both sides of the equation, which gives us \(14r = 32 - 8\), yielding \(14r = 24\).

Step 2: Solve for the variable

Divide both sides of the equation by 14 to isolate \(r\), thus \(r = 24 / 14\). This calculation gives \(r = 1.71\) when rounded to the nearest hundredth.

Step 3: Verify the solution

Substitute \(r\) back into the original equation to confirm. From \(14 r+8=32\), subtituting \(r = 1.71\) into the equation gives \(14(1.71) + 8 = 32\). After evaluating, the left side of the equation equals 31.94, which is approximately 32 when rounding off.

Key Concepts

Isolate Variables

Understanding how to isolate variables is a fundamental skill in algebra. It’s the process of getting the variable on one side of the equation so that you can solve for it. To make this concept crystal clear, let’s use the example from the exercise. To begin with, we have the equation \(14 r+8=32\). Our goal is to isolate the variable \(r\). This is achieved by performing the same operation on both sides of the equation to cancel out other terms.

In Step 1 of the solution, we subtract 8 from both sides, resulting in the simplified equation \(14r = 24\). Notice how the eight is eliminated on the left, leaving the term with \(r\) alone. The principle behind isolating the variable is that whatever you do to one side of the equation, you must do to the other. This maintains the balance of the equation, which is the cornerstone of solving linear equations.

Rounding Decimals

When working with decimals in algebra, you often need to round your result to make it more meaningful or adhere to the problem's requirements. Rounding is a way to simplify numbers, making them easier to use or understand. In our exercise, we're asked to round the result to the nearest hundredth, which means rounding to two decimal places.

In Step 2, after dividing both sides of the equation by 14, we get \(r = 24 / 14\), which results in a long decimal. However, we need to round this to the nearest hundredth. To do this effectively, look at the third decimal place. If it is 5 or higher, you increase the second decimal place by one. If it's less than 5, you leave it as is. Therefore, \(1.7142...\) is rounded to \(1.71\) because the third decimal (4) is less than 5.

Verifying Solutions

An essential step after solving any equation is to verify your solution. This step tests whether the answer you calculated really does work in the original equation. You can think of this as proof that you've found the correct solution.

Let's take Step 3 from our example as our guide. Once we've calculated and rounded \(r = 1.71\), we substitute it back into the original equation: \(14 r + 8 = 32\). Substituting \(r\) gives us \(14(1.71) + 8\), which simplifies to 31.94. This is very close to 32, which means our rounded value of \(r\) makes the equation balance, affirming our solution is accurate. However, if we hadn't rounded \(r\), the left side of the equation would have equaled exactly 32. Verifying helps us catch any mistakes and confirms that our rounding still provides an acceptable result for the problem.