Solve for x 1/3(x-6)=1/9x-5/3

Solution:

Step 1: Simplify the term $\frac{1}{3}(x - 6)$.

• Step 1.1: Express the equation as $\frac{1}{3}(x - 6) = \frac{1}{9}x - \frac{5}{3}$ by adding zeros if necessary.

• Step 1.2: Apply the distributive property to $\frac{1}{3}(x - 6)$.

• Step 1.3: Combine like terms to get $\frac{x}{3} - 2 = \frac{1}{9}x - \frac{5}{3}$.

Step 2: Isolate terms with $x$ on one side.

• Step 2.1: Subtract $\frac{1}{9}x$ from both sides to get $\frac{x}{3} - \frac{1}{9}x - 2 = -\frac{5}{3}$.

• Step 2.2: Find a common denominator for the $x$ terms, which is $9$, and combine them.

Step 3: Simplify the equation.

• Step 3.1: Combine the $x$ terms to get $\frac{2x}{9} - 2 = -\frac{5}{3}$.

• Step 3.2: Move constant terms to the other side by adding $2$ to both sides.

Step 4: Solve for $x$.

• Step 4.1: Convert $2$ to a fraction with a denominator of $3$ to combine with $-\frac{5}{3}$.

• Step 4.2: Simplify the right side to get $\frac{2x}{9} = \frac{1}{3}$.

• Step 4.3: Multiply both sides by the reciprocal of $\frac{2}{9}$ to solve for $x$.

Step 5: Finalize the solution.

• Step 5.1: Simplify both sides to find $x = \frac{3}{2}$.

• Step 5.2: Present the solution in various forms: exact, decimal, and mixed number.

Exact Form: $x = \frac{3}{2}$ Decimal Form: $x = 1.5$ Mixed Number Form: $x = 1 \frac{1}{2}$

Knowledge Notes:

To solve the equation $\frac{1}{3}(x-6)=\frac{1}{9}x-\frac{5}{3}$, we follow a systematic approach:

1. Distributive Property: This property allows us to remove parentheses by multiplying each term inside the parentheses by the factor outside. For example, $\frac{1}{3}(x-6)$ becomes $\frac{1}{3}x - \frac{1}{3} \cdot 6$.

2. Combining Like Terms: We combine terms with the same variable and the same power. For instance, $\frac{1}{3}x$ and $-\frac{1}{9}x$ can be combined because they both contain the variable $x$ to the first power.

3. Common Denominators: When combining fractions, it's necessary to have a common denominator. In this problem, we use the common denominator of $9$ for the terms involving $x$.

4. Isolation of the Variable: To solve for $x$, we need to get all the terms with $x$ on one side of the equation and the constants on the other. This often involves adding or subtracting terms from both sides of the equation.

5. Multiplication by Reciprocal: To isolate $x$, we multiply both sides of the equation by the reciprocal of the coefficient of $x$. In this case, we multiply by $\frac{9}{2}$ to cancel out the $\frac{2}{9}$ that is multiplied by $x$.

6. Simplification: After isolating $x$, we simplify the equation to find the value of $x$. This involves canceling out common factors and performing any necessary arithmetic.

7. Solution Representation: The solution can be presented in different forms, such as an exact fraction, a decimal, or a mixed number, depending on the context or preference.