Problem 49 - Elementary Algebra Within Reach

Question

Solve the equation. Round your answer to two decimal places. $$5.6=1.1 x-1.2$$

Step-by-Step Solution

Answer

The solution to the equation \(5.6=1.1x-1.2\) is approximately \(x = 6.18\) when rounded to two decimal places.

Step 1 - Isolate the x term

Begin by removing the -1.2 from the right side of the equation. You can do this by adding 1.2 to both sides of the equation. This will give us a new equation \[5.6 + 1.2 = 1.1x\]. When the left side of the equation is simplified, we get \[6.8 = 1.1x\].

Step 2 - Solve for x

Now that we have isolated terms, let's solve for \(x\). Divide both sides of the equation \[6.8 = 1.1x\] by 1.1 to solve for \(x\). This will give us a new equation \[x = \frac{6.8}{1.1}\]. By solving the right side of the equation, we get \(x\) approximately equals 6.18, rounding to two decimal places.

Key Concepts

Algebraic Operations

Understanding the basics of algebraic operations is essential to solving linear equations. Think of these operations as a set of tools we use to manipulate equations, such as addition, subtraction, multiplication, and division. In algebra, we perform these operations to both sides of an equation to maintain balance, as if we were working with scales.

Let's take our original problem: \[5.6 = 1.1x - 1.2\]. Here, to begin isolating the variable, we use addition to move the constant term from one side of the equation to the other, ensuring we do the same to both sides to keep the equation balanced. By adding 1.2 to both sides, we are using an algebraic operation to systematically transform the equation.

This basic principle enables us to move things around in an equation to reach a simpler form that we can solve. Always remember, whatever you do to one side, do to the other!

Isolating Variables

Isolating the variable, often 'x', is a crucial step in solving linear equations. Our goal is to get the variable by itself on one side of the equation so that we can see what it equals. To do this, we often have to 'undo' whatever operations are being performed on it.

In our example equation \[6.8 = 1.1x\], the variable x is being multiplied by 1.1. To isolate x, we need to eliminate this multiplication. How? By doing the opposite operation, which is division.

Dividing Both Sides

We divide both sides by 1.1, effectively cancelling out the multiplication on the right side, leaving us with \[x = \frac{6.8}{1.1}\]. This step illustrates the importance of understanding and correctly applying inverse operations to isolate variables.

Rounding Decimals

In algebra, we frequently encounter numbers that extend indefinitely, leading us to round them to make them easier to work with. Rounding decimals involves looking at the digits and determining which is the nearest value at the precision we need.

For instance, with our solution \(x = \frac{6.8}{1.1} = 6.18181818...\), we want to round to two decimal places. We look at the third decimal place, which is a '1' in this case. Since it's less than 5, we keep the second decimal place as it is: 6.18. If the third decimal place were 5 or higher, we would round up the second decimal place by one.

Why Round?

Rounding gives us a practical approximation that's easier to use and understand, especially when exact values are not needed, which is often the case in real-world scenarios.