Problem 59 - Algebra 1: Concepts and Skills

Question

Solve the equation. Round the result to the nearest hundredth. Check the rounded solution. $$11 k+12=-9$$

Step-by-Step Solution

Answer

The solution rounded to the nearest hundredth is \(k = -1.91\).

Step 1: Isolate the variable

First we want to isolate k. As a first step, subtract 12 from both sides of the equation. This will leave the equation as \(11k = -9 - 12\).

Step 2: Solve for k

Now, we should simplify the right side of the equation. By subtracting 12 from -9, we get -21. So, the equation becomes \(11k = -21\). Now divide both sides by 11 to finally isolate k. Therefore, we have \(k = -21/11\).

Step 3: Round the result

Now, we need to round the resultant fraction to the nearest hundredth. So that, \(k = -1.91\).

Step 4: Check the Solution

In this final step, substitute k=-1.91 back into the original equation to ensure that it holds true. If the left side equals the right side, then the solution is correct.

Key Concepts

Solving Linear Equations

Linear equations are equations that involve a linear combination of variables. Solving them means finding the value for these variables that makes the equation true.
To solve a linear equation like the one in the exercise, 11k + 12 = -9, the first step is to isolate the variable. This means getting the variable by itself on one side of the equation. Here, you need to subtract 12 from both sides of the equation:

• Start with the equation: \(11k + 12 = -9\).
• Subtract 12 from both sides to eliminate the +12: \(11k = -9 - 12\).
• Simplify the right side: \(11k = -21\).

After isolating the variable on one side, solve for it by performing arithmetic operations. In this case, divide both sides by 11 to solve for \(k\):

• \(k = -21/11\).

This operation gives the exact solution. To get a more practical number, often we round it, as detailed in the next section.

Rounding Numbers

Rounding numbers helps to make figures easier to work with and interpret. In algebra, you often round results to a certain decimal place, depending on the instructions. In this exercise, after calculating \(k = -21/11\), we want to round this to the nearest hundredth.
Follow these steps to round a number:

• First, convert \(-21/11\) into a decimal. You can do this by dividing 21 by 11, giving you approximately \(-1.909090...\).
• To round to the nearest hundredth, look at the thousandth place. This is the third number after the decimal point.
• If this number is 5 or greater, increase the hundredth digit by 1. If it's less than 5, leave the hundredth place as it is. In this case, the number is 9, so we increase \(-1.90\) to \(-1.91\).

Rounding the solution provides \(k = -1.91\). This rounded result is easier to interpret and use in further calculations.

Checking Solutions

After solving and rounding an equation, it is crucial to check if our result satisfies the original equation. This step ensures that the calculations were correct and that any rounding hasn't led to significant deviations.
Here's how to check a solution:

• Start with the rounded solution \(k = -1.91\) and substitute it back into the original equation \(11k + 12 = -9\).
• Calculate the left side of the equation: \(11 \times -1.91 + 12\).
• This equals approximately \(-21.01 + 12\). Simplify it to \(-9.01\).
• The right side of the original equation is \(-9\).

Though \(-9.01\) is very close to \(-9\), the small difference is due to rounding, indicating the original solution was correct before rounding. Checking solutions confirms the accuracy of your work and gives confidence in the results.