Solve the following equations.Check your result in each case
`2/3x=3/8x+7/12`

Answer

To solve the equation \( \frac{2}{3}x = \frac{3}{8}x + \frac{7}{12} \), we will follow these steps:

### Step 1: Rearranging the Equation
We want to isolate the variable \( x \) on one side of the equation. We can do this by moving the \( \frac{3}{8}x \) term to the left side. This gives us:
\[ \frac{2}{3}x - \frac{3}{8}x = \frac{7}{12} \]

**Hint:** To move a term from one side to another, change its sign.

### Step 2: Finding a Common Denominator
Next, we need to combine the \( x \) terms on the left side. To do this, we find a common denominator for \( \frac{2}{3} \) and \( \frac{3}{8} \). The least common multiple (LCM) of 3 and 8 is 24.

**Hint:** The LCM helps us to add or subtract fractions easily.

### Step 3: Converting Fractions
Now we convert each fraction to have a denominator of 24:
\[ \frac{2}{3} = \frac{2 \times 8}{3 \times 8} = \frac{16}{24} \]
\[ \frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24} \]

### Step 4: Substituting Back into the Equation
Substituting these values back into the equation gives us:
\[ \frac{16}{24}x - \frac{9}{24}x = \frac{7}{12} \]

### Step 5: Combining Like Terms
Now we can combine the \( x \) terms:
\[ \frac{16 - 9}{24}x = \frac{7}{12} \]
\[ \frac{7}{24}x = \frac{7}{12} \]

**Hint:** When combining fractions, subtract the numerators while keeping the common denominator.

### Step 6: Isolating \( x \)
To isolate \( x \), we can multiply both sides by the reciprocal of \( \frac{7}{24} \), which is \( \frac{24}{7} \):
\[ x = \frac{7}{12} \times \frac{24}{7} \]

### Step 7: Simplifying
The \( 7 \) in the numerator and denominator cancels out:
\[ x = \frac{24}{12} = 2 \]

### Step 8: Checking the Solution
Now we will check our solution by substituting \( x = 2 \) back into the original equation:
\[ \frac{2}{3}(2) = \frac{3}{8}(2) + \frac{7}{12} \]
Calculating both sides:
Left side:
\[ \frac{4}{3} \]
Right side:
\[ \frac{6}{8} + \frac{7}{12} = \frac{3}{4} + \frac{7}{12} \]
Finding a common denominator for \( \frac{3}{4} \) and \( \frac{7}{12} \) (which is 12):
\[ \frac{3}{4} = \frac{9}{12} \]
Thus,
\[ \frac{9}{12} + \frac{7}{12} = \frac{16}{12} = \frac{4}{3} \]

Since both sides are equal, our solution is verified.

### Final Answer
Thus, the solution to the equation is:
\[ x = 2 \]

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