Solve the following equations and verify the results.
`2/9(x+3)=4x-7`

Answer

To solve the equation \( \frac{2}{9}(x + 3) = 4x - 7 \) step by step, follow these instructions:

### Step 1: Eliminate the fraction
Multiply both sides of the equation by 9 to eliminate the fraction:

\[ 9 \cdot \frac{2}{9}(x + 3) = 9 \cdot (4x - 7) \]

This simplifies to:

\[ 2(x + 3) = 36x - 63 \]

**Hint:** Multiplying by the denominator helps to clear the fraction from the equation.

### Step 2: Distribute on the left side
Distribute the 2 on the left side:

\[ 2x + 6 = 36x - 63 \]

**Hint:** Distributing means multiplying the term outside the parentheses by each term inside the parentheses.

### Step 3: Rearrange the equation
Now, rearrange the equation to get all terms involving \(x\) on one side and constant terms on the other side. Subtract \(2x\) from both sides:

\[ 6 = 36x - 2x - 63 \]

This simplifies to:

\[ 6 = 34x - 63 \]

**Hint:** Moving terms across the equals sign involves changing their sign.

### Step 4: Isolate the variable
Add 63 to both sides to isolate the term with \(x\):

\[ 6 + 63 = 34x \]

This gives:

\[ 69 = 34x \]

**Hint:** Isolating the variable helps to solve for \(x\).

### Step 5: Solve for \(x\)
Now, divide both sides by 34 to solve for \(x\):

\[ x = \frac{69}{34} \]

**Hint:** Dividing both sides by the coefficient of \(x\) allows you to find its value.

### Step 6: Verification
To verify the solution, substitute \(x = \frac{69}{34}\) back into the original equation.

**Left Hand Side (LHS)**:
\[ LHS = \frac{2}{9}\left(\frac{69}{34} + 3\right) \]

Convert 3 to a fraction with a common denominator:
\[ 3 = \frac{102}{34} \]

So,
\[ LHS = \frac{2}{9}\left(\frac{69 + 102}{34}\right) = \frac{2}{9}\left(\frac{171}{34}\right) \]

Now simplify:
\[ LHS = \frac{342}{306} = \frac{19}{17} \]

**Right Hand Side (RHS)**:
\[ RHS = 4x - 7 = 4\left(\frac{69}{34}\right) - 7 \]

Calculating:
\[ RHS = \frac{276}{34} - 7 = \frac{276}{34} - \frac{238}{34} = \frac{38}{34} = \frac{19}{17} \]

Since \(LHS = RHS\), our solution is verified.

**Final Answer**:
\[ x = \frac{69}{34} \]