Solve the equations and check your answer in the case:
`5(3-x)+1=3(x+4)`

Options

A. `x=(1)/(6)`
B. `x=(1)/(3)`
C. `x=(1)/(2)`
D. `x=(1)/(4)`

Answer

To solve the equation \( 5(3-x)+1=3(x+4) \) step by step, follow these instructions:

### Step 1: Expand both sides of the equation
Start by distributing the numbers outside the parentheses.

\[ 5(3-x) + 1 = 3(x+4) \]

Expanding both sides gives:

\[ 5 \cdot 3 - 5 \cdot x + 1 = 3 \cdot x + 3 \cdot 4 \]

This simplifies to:

\[ 15 - 5x + 1 = 3x + 12 \]

### Step 2: Combine like terms
Now, combine the constant terms on the left side.

\[ 16 - 5x = 3x + 12 \]

### Step 3: Move variable terms to one side
Next, we will move all terms involving \( x \) to one side and constant terms to the other side.

Add \( 5x \) to both sides and subtract \( 12 \) from both sides:

\[ 16 - 12 = 3x + 5x \]

This simplifies to:

\[ 4 = 8x \]

### Step 4: Solve for \( x \)
Now, divide both sides by \( 8 \):

\[ x = \frac{4}{8} = \frac{1}{2} \]

### Step 5: Check the solution
To verify the solution, substitute \( x = \frac{1}{2} \) back into the original equation.

**Left-hand side (LHS):**

\[ 5(3 - \frac{1}{2}) + 1 = 5(3 - 0.5) + 1 = 5(2.5) + 1 = 12.5 + 1 = 13.5 \]

**Right-hand side (RHS):**

\[ 3(\frac{1}{2} + 4) = 3(\frac{1}{2} + \frac{8}{2}) = 3(\frac{9}{2}) = \frac{27}{2} = 13.5 \]

Since LHS = RHS, the solution is verified.

### Final Answer:
The solution to the equation \( 5(3-x)+1=3(x+4) \) is

\[ x = \frac{1}{2} \]