A start-up company opened with 8 employees. The company’s growth plan assumes that 2 new employees will be hired each quarter (every 3 months) for the first 5 years. If an equation is written in the form y = ax + b to represent the number of employees, y, employed by the company x quarters after the company opened, what is the value of b ?

Explanation

To solve the problem, we need to determine the value of \( b \) in the equation \( y = ax + b \), where \( y \) represents the number of employees and \( x \) represents the number of quarters after the company opened.

### Step-by-step Solution:

1. **Identify Initial Conditions**:
The company starts with 8 employees. This means that at \( x = 0 \) (the time when the company opened), the number of employees \( y \) is 8.

**Hint**: Remember that the initial number of employees is the value of \( y \) when \( x = 0 \).

2. **Determine the Rate of Hiring**:
The company hires 2 new employees every quarter. This means that for every increase of 1 in \( x \) (which represents one quarter), \( y \) increases by 2. Therefore, the slope \( a \) of the equation is 2.

**Hint**: The slope \( a \) indicates how much \( y \) changes for each unit increase in \( x \).

3. **Formulate the Equation**:
We can now write the equation in the form \( y = ax + b \). Substituting the values we have:
\[ y = 2x + b \]

**Hint**: The equation format \( y = ax + b \) is a linear equation where \( a \) is the rate of change and \( b \) is the y-intercept.

4. **Substitute Known Values**:
We know that when \( x = 0 \), \( y = 8 \). We can substitute these values into the equation to find \( b \):
\[ 8 = 2(0) + b \]
Simplifying this gives:
\[ 8 = 0 + b \implies b = 8 \]

**Hint**: When substituting \( x = 0 \), the term \( 2(0) \) will vanish, leaving you with \( b \) directly equal to \( y \).

5. **Conclusion**:
The value of \( b \) is 8, which represents the initial number of employees when the company opened.

**Final Answer**: \( b = 8 \)