Problem 4 - Elementary Algebra

Question

Solve each of the equations. $$x+11.8=17.1$$

Step-by-Step Solution

Answer

The solution is \(x = 5.3\).

Step 1: Isolate the Variable

The equation given is \(x + 11.8 = 17.1\). The goal is to find the value of \(x\) by isolating it. To do this, subtract 11.8 from both sides of the equation to cancel out the 11.8 on the left side. This gives: \(x = 17.1 - 11.8\).

Step 2: Perform the Subtraction

Now, perform the subtraction on the right side of the equation: \(17.1 - 11.8 = 5.3\).

Step 3: Write the Solution

The value of \(x\) is found to be 5.3. So, the solution to the equation is \(x = 5.3\).

Key Concepts

Isolating the Variable

In algebra, one of the primary techniques for solving equations is to isolate the variable. This means we need to get the variable, often represented as "\( x \)", by itself on one side of the equation. Consider the equation given in the exercise: \( x + 11.8 = 17.1 \). Here, the term \( 11.8 \) is added to \( x \), which means that our task is to "undo" or cancel out the addition of \( 11.8 \) in order to solve for \( x \).
To do this, we perform the opposite operation: subtraction. By subtracting \( 11.8 \) from both sides of the equation, we can effectively remove it from the left side, leaving \( x \) isolated. The subtraction rule states that you need to do the same mathematical operation on both sides of the equation to maintain balance.

• Step 1: The equation is \( x + 11.8 = 17.1 \).
• Step 2: Subtract \( 11.8 \) from both sides: \( x = 17.1 - 11.8 \).

By isolating \( x \), the equation becomes much simpler to solve.

Subtraction in Equations

Subtraction is a crucial operation when solving linear equations, especially when it comes to isolating terms. In the context of the given exercise, we use subtraction to "remove" a number that is added to the variable. This helps in isolating the variable to find its value.
When performing the subtraction \( 17.1 - 11.8 \), we do a straightforward calculation:

• Align the numbers: Setting them as \( 17.1\) (top) and \( 11.8\) (bottom) allows for easy subtraction.
• Subtract each digit vertically: Starting with the rightmost digit of the decimal portion, you would perform the subtraction normally.

In this context, subtraction shows the change from the original situation (\( x + 11.8 = 17.1 \)) to the new situation (\( x = 5.3 \)), where all necessary operations have been taken care of.

Elementary Algebra Concepts

Understanding elementary algebra concepts is essential for solving equations like the one in our exercise. These are the building blocks for more complex mathematics. Algebraic equations involve mathematical symbols and operations that represent problems we aim to solve.
Here are a few basic terms:

• Variable: A symbol (usually a letter) that represents an unknown number, for example, \( x \).
• Operations: Actions such as addition, subtraction, multiplication, and division used to solve equations.
• Balance: An equation stays true when the same operation is applied to both sides.

Understanding these concepts allows you to approach problems methodically, using techniques like isolating variables and balancing equations to find solutions efficiently. This knowledge forms a foundation for tackling algebraic tasks in higher-level classes and real-world applications.