How many unique solutions are there to the system of equations below?

2x + 4y = 10

6x + 12y = 30 

Group of answer choices

Infinite Solutions

zero

two

one

Next 


To determine the number of unique solutions for this system of linear equations, let's analyze the equations:

1. The first equation is:
   \[
   2x + 4y = 10
   \]

2. The second equation is:
   \[
   6x + 12y = 30
   \]

We can simplify each equation by dividing both sides by their respective coefficients:

1. Dividing the first equation by 2:
   \[
   x + 2y = 5
   \]

2. Dividing the second equation by 6:
   \[
   x + 2y = 5
   \]

After simplifying, both equations are equivalent to:
\[
x + 2y = 5
\]

This means the two equations are **dependent**; they represent the same line. Since they are the same line, there are infinitely many solutions (any point on this line is a solution to the system).

### Answer:
The correct answer is: **Infinite Solutions**

Would you like further details or have any questions?

Here are 5 additional questions for further understanding:
1. How do we determine if a system of equations has no solutions?
2. What does it mean for a system of equations to be "consistent"?
3. How would this analysis change if the equations were not multiples of each other?
4. What graphical interpretation represents an infinite solution in a system of equations?
5. Can a system of linear equations have exactly two solutions? Why or why not?

**Tip:** For linear equations, if simplifying both equations gives the same line, the system has infinitely many solutions.

How many unique solutions are there to the system of equations below?

2x + 4y = 10
6x + 12y = 30

Group of answer choices:
1. Infinite Solutions
2. Zero
3. Two
4. One

Explore how to determine the number of unique solutions for the system of equations 2x + 4y = 10 and 6x + 12y = 30. This problem involves simplifying equations and understanding the concept of dependent and consistent systems. Suitable for students in Grades 8-10.

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