make it so It has infinitely many solutions

To create a system of equations that has infinitely many solutions, the equations must be dependent, meaning one equation is a multiple of the other. This ensures that both equations represent the same line, meaning they have the same solution for all points on that line.

For a system of two linear equations in the form:

\[
a_1x + b_1y = c_1
\]
\[
a_2x + b_2y = c_2
\]

To have infinitely many solutions, the two equations must be proportional. Specifically, the following must hold:

\[
\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}
\]

### Example:
Let's start with a simple equation:

\[
2x + 3y = 6
\]

Now, create another equation that is a multiple of the first equation, say multiplying by 2:

\[
4x + 6y = 12
\]

This system:

\[
2x + 3y = 6
\]
\[
4x + 6y = 12
\]

has infinitely many solutions because the second equation is just a scaled version of the first, meaning both represent the same line.

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Would you like to explore more examples or variations? 

Here are 5 related questions to deepen understanding:
1. How can we verify that two equations represent the same line?
2. What happens if two equations are not proportional?
3. Can a system of more than two equations have infinitely many solutions?
4. How does this concept extend to three variables?
5. How do you solve systems of equations graphically to identify infinitely many solutions?

**Tip**: When you encounter a system of equations, checking proportionality between the coefficients can quickly tell you if it has no, one, or infinitely many solutions.

Discover how to create a system of linear equations that has infinitely many solutions. Learn about the proportionality rule for coefficients and how dependent equations represent the same line. This guide is suitable for students in Grades 8-10.

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