Solve for x : `(7x + 3)/(4) + (9x-5)/(8) = (16x-3)/(16)`

Options

A. `(1)/(6)`
B. `(5)/(3)`
C. `-(3)/(5)`
D. `-(1)/(6)`

Answer

To solve the equation

\[ \frac{7x + 3}{4} + \frac{9x - 5}{8} = \frac{16x - 3}{16} \]

we will follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
The denominators are 4, 8, and 16. The LCM of these numbers is 16.

**Hint:** To find the LCM, look for the smallest number that all denominators can divide into without leaving a remainder.

### Step 2: Multiply through by the LCM
Multiply every term in the equation by 16 to eliminate the denominators:

\[ 16 \cdot \frac{7x + 3}{4} + 16 \cdot \frac{9x - 5}{8} = 16 \cdot \frac{16x - 3}{16} \]

This simplifies to:

\[ 4(7x + 3) + 2(9x - 5) = 16x - 3 \]

**Hint:** Distributing the LCM across each term helps to eliminate the fractions.

### Step 3: Distribute
Now distribute the constants:

\[ 28x + 12 + 18x - 10 = 16x - 3 \]

**Hint:** Remember to multiply each term inside the parentheses by the number outside.

### Step 4: Combine like terms
Combine the like terms on the left side:

\[ (28x + 18x) + (12 - 10) = 16x - 3 \]

This simplifies to:

\[ 46x + 2 = 16x - 3 \]

**Hint:** Grouping similar terms together makes it easier to simplify the equation.

### Step 5: Move all terms involving x to one side
Subtract \(16x\) from both sides:

\[ 46x - 16x + 2 = -3 \]

This simplifies to:

\[ 30x + 2 = -3 \]

**Hint:** To isolate the variable, move all terms involving x to one side of the equation.

### Step 6: Move constant to the other side
Subtract 2 from both sides:

\[ 30x = -3 - 2 \]

This simplifies to:

\[ 30x = -5 \]

**Hint:** Always perform the same operation on both sides of the equation to maintain equality.

### Step 7: Solve for x
Divide both sides by 30:

\[ x = \frac{-5}{30} \]

This simplifies to:

\[ x = \frac{-1}{6} \]

**Hint:** Simplifying fractions can help in finding the simplest form of the solution.

### Final Answer
Thus, the solution for \(x\) is:

\[ x = -\frac{1}{6} \]