Problem 32 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$x+7-9=6$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 8\).

Step 1: Simplify the Equation

Begin by simplifying the terms on the left side of the equation. The equation is initially given as \(x + 7 - 9 = 6\). Combine like terms \(+7\) and \(-9\). This results in \(x - 2 = 6\).

Step 2: Isolate the Variable

To isolate \(x\), you need to get rid of \(-2\) from the left side. Do this by adding \(2\) to both sides of the equation. This results in: \(x - 2 + 2 = 6 + 2\), which simplifies to \(x = 8\).

Step 3: Check the Solution

To ensure the solution is correct, substitute \(x = 8\) back into the original equation: \(8 + 7 - 9 = 6\). Simplifying the left side, \(8 + 7 = 15\) and then \(15 - 9 = 6\), which matches the right side, confirming that \(x = 8\) is indeed correct.

Key Concepts

Equation Simplification

Simplifying an equation is often the first step in solving it. This involves combining like terms and reducing the equation to its simplest form. In our given problem, \(x + 7 - 9 = 6\), notice that the left side of the equation has constant terms: \(+7\) and \(-9\).

To simplify, you will add these constants together:

• Combine \(+7\) and \(-9\) to get \(-2\).
• This transforms the equation to \(x - 2 = 6\).

Now, the equation is easier to handle as it focuses only on isolating the variable \(x\). Simplification reduces potential errors in further steps and makes the equation clearer.

Algebraic Isolation

Once the equation is simplified, the next crucial step is to isolate the variable you are solving for. The goal is to get the variable on one side of the equation by itself. In our example, the equation \(x - 2 = 6\) already puts us in a good position to do this.

To isolate \(x\), follow these steps:

• Add \(2\) to both sides of the equation to counteract the \(-2\) that's with \(x\).
• This balances the equation and keeps it true: \(x - 2 + 2 = 6 + 2\).
• Simplify the result to \(x = 8\).

Through these actions, you successfully determine the value of \(x\) while maintaining equality throughout. This process is a fundamental skill in algebra, allowing us to find unknown values systematically.

Solution Verification

After isolating the variable and finding its value, it is important to confirm that this solution is correct. Verification involves substituting the found value back into the original equation to see if both sides of the equation still balance.

For our equation, we substitute \(x = 8\) into the original problem, \(x + 7 - 9 = 6\):

• Insert \(8\) for \(x\), giving \(8 + 7 - 9\).
• First calculate \(8 + 7 = 15\).
• Then, compute \(15 - 9\), which equals \(6\).

Since both sides match, we confirm that our solution, \(x = 8\), is correct. Solution verification ensures that the steps taken were correct and reinforces your confidence in algebraic manipulation.