Question

1
The population growth of two towns over a period of five years is represented by the system of equations below, both algebraically and graphically. Y= x+6 Y=2x+2

Answer 1

To analyze the population growth of the two towns represented by the system of equations:

1. **Equations**:
- Town A: \( y = x + 6 \)
- Town B: \( y = 2x + 2 \)

### Step 1: Algebraic Solution

To find the point where the two towns' populations are equal, we need to set the two equations equal to each other:

\[
x + 6 = 2x + 2
\]

### Step 2: Solve for \(x\)

1. Rearranging the equation, we get:
\[
6 - 2 = 2x - x
\]
\[
4 = x
\]

2. Now, substitute \(x = 4\) back into either of the original equations to find \(y\):

Using Town A's equation:
\[
y = 4 + 6 = 10
\]

### Step 3: Intersection Point

The intersection point of the two lines occurs at:
\[
(x, y) = (4, 10)
\]

### Step 4: Graphical Representation

To graph the equations:

- For Town A (\(y = x + 6\)):
- When \(x = 0\), \(y = 6\)
- When \(x = -6\), \(y = 0\)

- For Town B (\(y = 2x + 2\)):
- When \(x = 0\), \(y = 2\)
- When \(x = -1\), \(y = 0\)

#### Plotting
1. Plot points for both equations:
- For \(y = x + 6\): (0, 6), (-6, 0)
- For \(y = 2x + 2\): (0, 2), (-1, 0)

2. Draw lines through these points.

3. Identify the intersection at (4, 10).

### Summary
The towns' populations will be equal after 4 years, at which point both towns will have a population of 10.

If you require further analysis or specific graphing instructions, let me know!