Question

Question 2: Infinite Solutions
Consider the two equations you circled, which form a system with infinite solutions.
Graph them together on the following coordinate grid.
What do you notice about these graphs? What does it mean for a system to have
infinite solutions? Explain what it means graphically to have infinite solutions.
Plug in (3,10) and (0,1) and check to show that the system of equations has
more than one solution. Explain your work.

y=3x+1
2y=6x+2

Answer 1

To analyze the system of equations you provided, let's first rewrite both equations clearly:

1. \( y = 3x + 1 \)
2. \( 2y = 6x + 2 \)

Now, let's simplify the second equation by dividing everything by 2:

\( y = 3x + 1 \)

Now we see that both equations are the same, which indicates that they represent the same line in the coordinate plane. Thus, the system has infinite solutions, since every point on this line satisfies both equations.

### Graphical Analysis

To graph these equations, we can start by plotting the equation \( y = 3x + 1 \):

1. **Y-Intercept:** When \( x = 0 \), \( y = 3(0) + 1 = 1 \). So, the point \( (0, 1) \) is on the graph.
2. **Another Point:** When \( x = 1 \), \( y = 3(1) + 1 = 4 \). So, the point \( (1, 4) \) is on the graph.

Connecting these points and extending the line will yield a slope of 3, meaning for every 1 unit you move to the right (in the x-direction), you move 3 units up (in the y-direction).

Since we established that both equations are equivalent, they will produce the same graph, which is a straight line.

### Interpretation of Infinite Solutions

When we say that a system of equations has infinite solutions, we mean that there are countless points along the line that satisfy both equations, which visually means that the two lines completely overlap.

### Checking Specific Points

Now let's check the points \( (3, 10) \) and \( (0, 1) \) to demonstrate:

1. **For \( (3, 10) \)**:
- Substitute \( x = 3 \) into \( y = 3x + 1 \):
\[
y = 3(3) + 1 = 9 + 1 = 10
\]
So, \( (3, 10) \) lies on the line.

2. **For \( (0, 1) \)**:
- Substitute \( x = 0 \):
\[
y = 3(0) + 1 = 0 + 1 = 1
\]
So, \( (0, 1) \) also lies on the line.

### Conclusion

This demonstrates that \( (3, 10) \) is indeed a solution to the system, and since the two equations are the same line, we can conclude that there are infinitely many points (solutions) along that line, confirming that the system has infinite solutions.

Overall, infinite solutions in a system means that the equations represent the same line in the coordinate space, indicating that every point on that line is a solution to the system.