Question

Graph with a solid line touching two, seven and negative 1, 1 dotted line on the same graph with a DOT -1, 1 what is the system equations shown in the graph y equals 2x + 3 y = 3x + 4 y equals negative x + 1 y = 2x + 7 y = 2x - 3 = 3x - 4 = 3x + 2 y = 4x + 3

Answer 1

To interpret the information provided and determine the system of equations represented in the graph, let's first clarify the components mentioned:

1. **Solid line touching points (2, 7) and (-1, 1)**: This suggests that this line has at least those two points. We can find the equation of this line using two points.

2. **Dotted line that goes through the point (-1, 1)**: This line is represented as a dotted line, implying that it could be either a boundary or a non-inclusive line in the context of a system of equations.

Let's derive the equation for the solid line that touches the points (2, 7) and (-1, 1).

### Finding the equation of the solid line

1. **Calculate the slope** \( m \):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 1}{2 - (-1)} = \frac{6}{3} = 2
\]

2. **Using point-slope form** of the line equation, \( y - y_1 = m(x - x_1) \), we can choose one of the points, say (2, 7):
\[
y - 7 = 2(x - 2)
\]
\[
y - 7 = 2x - 4
\]
\[
y = 2x + 3
\]

So, the equation of the solid line is:
\[
y = 2x + 3
\]

### Considering the dotted line

The dotted line passes through the point (-1, 1) and could potentially be one of the other equations mentioned. We can check which of the given equations pass through (-1, 1):

1. **Test various equations**:
- **\( y = 3x + 4 \)**:
\[
y = 3(-1) + 4 = -3 + 4 = 1 \quad \text{(Passes through (-1, 1))}
\]

- **\( y = -x + 1 \)**:
\[
y = -(-1) + 1 = 1 + 1 = 2 \quad \text{(Does not pass)}
\]

- **\( y = 2x + 7 \)**:
\[
y = 2(-1) + 7 = -2 + 7 = 5 \quad \text{(Does not pass)}
\]

- **\( y = 2x - 3 \)**:
\[
y = 2(-1) - 3 = -2 - 3 = -5 \quad \text{(Does not pass)}
\]

- **\( y = 3x - 4 \)**:
\[
y = 3(-1) - 4 = -3 - 4 = -7 \quad \text{(Does not pass)}
\]

- **\( y = 3x + 2 \)**:
\[
y = 3(-1) + 2 = -3 + 2 = -1 \quad \text{(Does not pass)}
\]

- **\( y = 4x + 3 \)**:
\[
y = 4(-1) + 3 = -4 + 3 = -1 \quad \text{(Does not pass)}
\]

Only the equation \( y = 3x + 4 \) passes through the point (-1, 1).

### Summary of the system of equations

From the analysis, the system of equations shown in the graph likely involves:

1. **Solid Line**:
\[
y = 2x + 3
\]

2. **Dotted Line**:
\[
y = 3x + 4
\]

Therefore, the equations represented in the graph are:
\[
\begin{align*}
1. & \quad y = 2x + 3 \quad (\text{Solid Line}) \\
2. & \quad y = 3x + 4 \quad (\text{Dotted Line})
\end{align*}
\]