Question

Identify the number of solutions to the system of equation:
y = x + 3
3x + y = -1

How can you tell from the equations?

How can you tell from the graph?

Answer 1

To determine the number of solutions to the given system of equations, we can analyze it algebraically and graphically.

### Equations:
1. \( y = x + 3 \) (Equation 1)
2. \( 3x + y = -1 \) (Equation 2)

#### Algebraic Method:
1. **Substitute Equation 1 into Equation 2**:

We can substitute \( y \) from Equation 1 into Equation 2:
\[
3x + (x + 3) = -1
\]
Simplifying this:
\[
4x + 3 = -1
\]
\[
4x = -1 - 3
\]
\[
4x = -4
\]
\[
x = -1
\]

2. **Substitute \( x = -1 \) back into Equation 1 to find \( y \)**:
\[
y = -1 + 3 = 2
\]
So, the solution to the system is \( (x, y) = (-1, 2) \).

Since we found one unique solution, the system of equations has exactly **one solution**.

### Graphical Method:
1. **Graphing the equations**:
- For \( y = x + 3 \), the line has a slope of 1 and a y-intercept at (0, 3). It goes up one unit for each unit it goes to the right.
- For \( 3x + y = -1 \), we can rearrange it to slope-intercept form:
\[
y = -3x - 1
\]
The slope is -3, and the y-intercept is (0, -1). This line goes down three units for each unit it goes to the right.

2. **Analyzing the graphs**:
- When you graph both lines, you'll see that they intersect at the point (-1, 2). Since they intersect at a single point, this signifies that there is exactly one solution.

### Conclusion:
- **From the equations**: We found one unique solution by substituting and solving.
- **From the graph**: The two lines intersect at one point, indicating one solution.

Thus, the system of equations has exactly **one solution** at the point \((-1, 2)\).