Step 1: Understanding the Concept:
This problem represents a standard linear equation in one variable. Solving an equation means finding the specific value for the unknown variable (in this case, \(x\)) that makes the statement true.
The primary logic used in algebra is the principle of "balance." An equation is like a weighing scale; whatever operation is performed on the left side (LHS) must be identically performed on the right side (RHS) to maintain the equality.
We use "Inverse Operations" to isolate the variable. Subtraction is countered with addition, and multiplication is countered with division. The goal is to move all numbers away from \(x\) until only \(x\) remains on one side.
Step 2: Key Formula or Approach:
For any linear equation of the form \(ax - b = c\):
1. Add \(b\) to both sides to eliminate the constant subtraction.
2. Divide both sides by the coefficient \(a\) to isolate the variable.
Step 3: Detailed Explanation:
Let's begin with the provided algebraic statement:
\[ 3x - 7 = 20 \]
Our first objective is to "unbundle" the subtraction of 7. To do this, we perform the inverse operation, which is adding 7 to both sides of the equation.
\[ 3x - 7 + 7 = 20 + 7 \]
This simplifies to:
\[ 3x = 27 \]
Now, the variable \(x\) is being multiplied by the coefficient 3. To isolate \(x\), we must perform the inverse of multiplication, which is division. We divide both sides of the equation by 3.
\[ \frac{3x}{3} = \frac{27}{3} \]
This gives us:
\[ x = 9 \]
To ensure our solution is accurate, we can perform a quick verification by plugging 9 back into the original expression:
\[ \text{LHS} = 3(9) - 7 = 27 - 7 = 20 \]
Since the LHS equals the RHS (20), our solution is confirmed to be perfectly accurate.
Step 4: Final Answer:
The value of \(x\) is 9, which matches option (c).