If `3x+3y=1`, what is the value of `x=y-6`?

Answer

To solve the problem, we have the equation \(3x + 3y = 1\) and we need to find the value of \(x\) given that \(x = y - 6\).

### Step 1: Simplify the given equation
We start with the equation:
\[ 3x + 3y = 1 \]
We can simplify this by dividing every term by 3:
\[ x + y = \frac{1}{3} \]

**Hint:** To simplify equations, you can divide or multiply both sides by the same non-zero number.

### Step 2: Substitute the expression for \(x\)
We know from the problem that \(x = y - 6\). We can substitute this expression into the simplified equation:
\[ (y - 6) + y = \frac{1}{3} \]

**Hint:** When substituting, make sure to replace the variable completely to avoid mistakes.

### Step 3: Combine like terms
Now, we combine the terms on the left side:
\[ 2y - 6 = \frac{1}{3} \]

**Hint:** When combining like terms, ensure that you keep track of all variables and constants.

### Step 4: Isolate \(y\)
Next, we add 6 to both sides to isolate the term with \(y\):
\[ 2y = \frac{1}{3} + 6 \]
To add \(6\) to \(\frac{1}{3}\), we convert \(6\) into a fraction with a denominator of \(3\):
\[ 6 = \frac{18}{3} \]
So,
\[ 2y = \frac{1}{3} + \frac{18}{3} = \frac{19}{3} \]

**Hint:** When adding fractions, make sure they have the same denominator.

### Step 5: Solve for \(y\)
Now, divide both sides by 2 to solve for \(y\):
\[ y = \frac{19}{3} \div 2 = \frac{19}{6} \]

**Hint:** Dividing by a number is the same as multiplying by its reciprocal.

### Step 6: Find \(x\)
Now that we have \(y\), we can find \(x\) using the equation \(x = y - 6\):
\[ x = \frac{19}{6} - 6 \]
Convert \(6\) into a fraction:
\[ 6 = \frac{36}{6} \]
So,
\[ x = \frac{19}{6} - \frac{36}{6} = \frac{19 - 36}{6} = \frac{-17}{6} \]

**Hint:** When subtracting fractions, subtract the numerators while keeping the denominator the same.

### Final Answer
The value of \(x\) is:
\[ \boxed{\frac{-17}{6}} \]