Evaluate 3(x-6)=100+x

Solution:

Step 1: Expand the expression on the left side of the equation.

• Step 1.1: Start with the original equation. $0 + 0 + 3(x - 6) = 100 + x$
• Step 1.2: Recognize that adding zero does not change the value. $3(x - 6) = 100 + x$
• Step 1.3: Distribute the $3$ across the parentheses. $3x - 3 \cdot 6 = 100 + x$
• Step 1.4: Perform the multiplication. $3x - 18 = 100 + x$

Step 2: Isolate the variable terms on one side.

• Step 2.1: Subtract $x$ from both sides to start grouping $x$ terms. $3x - x - 18 = 100$
• Step 2.2: Combine like terms. $2x - 18 = 100$

Step 3: Move the constant term to the opposite side.

• Step 3.1: Add $18$ to both sides to isolate the variable term. $2x = 100 + 18$
• Step 3.2: Combine the constants on the right side. $2x = 118$

Step 4: Solve for the variable.

• Step 4.1: Divide the equation by the coefficient of $x$ to solve for $x$. $\frac{2x}{2} = \frac{118}{2}$

• Step 4.2: Simplify the equation by canceling out the $2$s on the left side.

  • Step 4.2.1: Cancel the $2$s. $\frac{\cancel{2}x}{\cancel{2}} = \frac{118}{2}$
  • Step 4.2.1.1: Simplify the left side. $x = \frac{118}{2}$

• Step 4.3: Simplify the right side by dividing. $x = 59$

Knowledge Notes:

The problem-solving process involves solving a linear equation in one variable. The key steps are:

1. Distributive Property: This property states that $a(b + c) = ab + ac$. It's used to expand expressions like $3(x - 6)$.

2. Combining Like Terms: This involves simplifying expressions by adding or subtracting terms that have the same variable to the same power. For instance, $3x - x$ simplifies to $2x$.

3. Isolating the Variable: To solve for $x$, we need to get $x$ by itself on one side of the equation. This often involves moving terms from one side of the equation to the other by performing inverse operations (e.g., if there's a $+x$ on one side, we subtract $x$ from both sides).

4. Inverse Operations: These are operations that reverse the effect of another operation. For example, addition is the inverse of subtraction, and multiplication is the inverse of division.

5. Solving Basic Algebraic Equations: The goal is to isolate the variable on one side to find its value. This usually involves a series of steps including distributing, combining like terms, and using inverse operations.

6. Checking Solutions: After finding the value of $x$, you can check your solution by substituting it back into the original equation to see if it makes a true statement.

In this particular problem, we applied these principles to solve for $x$ in the equation $3(x - 6) = 100 + x$.