To solve the system of linear equations by graphing, we start with the given equation:
- Equation: \( y = x + 3 \)
To graph this equation, we can identify some key points:
-
When \( x = 0 \): \[ y = 0 + 3 = 3 \quad \text{(point: (0, 3))} \]
-
When \( x = -3 \): \[ y = -3 + 3 = 0 \quad \text{(point: (-3, 0))} \]
-
When \( x = 3 \): \[ y = 3 + 3 = 6 \quad \text{(point: (3, 6))} \]
Next, we can plot these points on a graph:
- (0, 3)
- (-3, 0)
- (3, 6)
Now we can draw a line through these points. The line will slope upwards because the coefficient of \( x \) is positive, indicating a positive relationship between \( x \) and \( y \).
Description of the Graph
The line appears to be straight and crosses the y-axis at (0, 3). It also crosses the x-axis at (-3, 0). As you move from left to right along the x-axis, the graph rises, indicating that as \( x \) increases, \( y \) also increases.
Interpretation of the Solution
Since this is a single equation, it defines a line on the graph. Any point that lies on this line is a solution to the equation \( y = x + 3 \).
If there are no other equations provided, we cannot explicitly identify a single solution. However, if you were comparing this line with another line or set of equations, the point of intersection would represent the solution to the system of equations.
In the case of multiple lines:
- One intersection point: There’s one unique solution.
- No intersection: The lines are parallel; hence no solution exists (inconsistent system).
- Infinite intersections: The lines are coincident; hence there are infinitely many solutions (dependent system).
Without the additional equation, we can only conclude that the current system consists of one linear equation that represents an infinite set of solutions along the line \( y = x + 3 \).