Problem 99 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ x+9=14 $$

Step-by-Step Solution

Answer

The solution is \(x = 5\).

Step 1: Isolate x

To solve for \(x\), we need to isolate it on one side of the equation. Begin by subtracting 9 from both sides of the equation: \[x + 9 - 9 = 14 - 9\]

Step 2: Simplify

Simplify both sides of the equation. \[x = 5\] This shows the solution for \(x\).

Key Concepts

Isolating the Variable

When solving linear equations, our goal is to find the value of the unknown variable. To do this, we need to isolate the variable. Isolating means getting the variable by itself on one side of the equation. This is a key step in solving equations as it allows us to see the solution clearly.
To isolate a variable, perform operations that remove other terms from the same side as the variable. In this particular example, we have the equation \(x + 9 = 14\). To isolate \(x\), subtract 9 from both sides.
This is because subtraction "undoes" the addition of 9 on the left side. So we write:

• Subtract 9 from the left side: \(x + 9 - 9\)
• Subtract 9 from the right side: \(14 - 9\)

Now, the equation is simplified to \(x = 5\). The key takeaway is to perform the same operation on both sides to maintain equality.

Simplifying Equations

Simplifying an equation means reducing it to its simplest form, where the variable is isolated, and all operations are minimized. In basic algebra, simplifications usually involve combining like terms and ensuring that the variable is clearly expressed.
Once we've subtracted 9 from both sides in our example, we simplify the equation:

• The left side simplifies because \(9\) is subtracted: \(x + 9 - 9 = x\).
• The right side simplifies because the subtraction gives: \(14 - 9 = 5\).

This turns the equation into \(x = 5\), and this simple expression makes it easy to understand that \(x\) is equal to 5. Remember, simplifying helps us reach the solution directly by reducing complex operations into straightforward ones.

Basic Algebra Steps

Solving linear equations involves understanding some fundamental steps in algebra. By familiarizing yourself with these steps, solving equations becomes a straightforward process.
The first step is always to identify the equation you need to solve and the variable to isolate. With our equation \(x + 9 = 14\), the task is to solve for \(x\).
Here’s a basic rundown of the steps:

• Identify the operation associated with the variable: Here, it's addition \(+9\).
• Perform the inverse operation: Subtract 9 from both sides to counteract the addition.
• Simplify the results: Clear away numerical values to find \(x\).

These fundamental steps - identifying, performing the inverse operation, and simplifying - form the backbone of solving most linear equations. These steps need to be followed carefully to arrive at the simplest form of the equation where the solution is apparent.