Problem: Solve the equation 6x - 2 / 9 + 3x + 5 / 18 = 1 / 3.

Solution:

To solve this equation, we need to simplify and combine like terms. Let's break it down step by step.

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

To simplify the expression, we need to simplify each term separately.

Term 1: 6x

There are no like terms to combine with 6x, so we leave it as it is.

Term 2: -2 / 9

To simplify -2 / 9, we divide -2 by 9, which gives us -2/9.

Term 3: 3x

Again, there are no like terms to combine with 3x, so we leave it as it is.

Term 4: 5 / 18

To simplify 5 / 18, we divide 5 by 18, which gives us 5/18.

Step 2: Combine the simplified terms.

Now that we have simplified each term, we can combine them together.

6x - 2/9 + 3x + 5/18 = 1/3

Combining like terms, we get:

(6x + 3x) - (2/9 + 5/18) = 1/3

Simplifying further:

9x - (4/18 + 5/18) = 1/3

9x - 9/18 = 1/3

9x - 1/2 = 1/3

Step 3: Isolate the variable.

To isolate the variable x, we need to get rid of the constant term on the left side of the equation. To do that, we can add 1/2 to both sides of the equation.

9x - 1/2 + 1/2 = 1/3 + 1/2

The 1/2 on the left side cancels out, leaving us with:

9x = 1/3 + 1/2

Step 4: Combine the fractions on the right side.

To combine 1/3 and 1/2, we need to find a common denominator. The least common multiple of 3 and 2 is 6. So, we can rewrite the equation as:

9x = 2/6 + 3/6

Simplifying the fractions:

9x = 5/6

Step 5: Solve for x.

To solve for x, we need to divide both sides of the equation by 9:

(9x)/9 = (5/6)/9

The 9 on the left side cancels out, leaving us with:

x = 5/54

Final Answer: The solution to the equation 6x - 2 / 9 + 3x + 5 / 18 = 1 / 3 is x = 5/54.