To solve the system of equations given by [tex]\( y = 2x + 6 \)[/tex] and [tex]\( 3y = 6x + 18 \)[/tex], follow these steps:
1. Write down the given equations:
- [tex]\( y = 2x + 6 \)[/tex]
- [tex]\( 3y = 6x + 18 \)[/tex]
2. Simplify the second equation to see if it matches the first equation:
Start with the second equation:
[tex]\[ 3y = 6x + 18 \][/tex]
Divide every term by 3 to simplify:
[tex]\[ y = 2x + 6 \][/tex]
3. Compare the simplified second equation with the first equation:
After simplification, the second equation is:
[tex]\[ y = 2x + 6 \][/tex]
Notice that this is identical to the first equation:
[tex]\[ y = 2x + 6 \][/tex]
This indicates that the equations represent the same line.
4. Understand the implications:
Since both equations are the same line, the system does not intersect in a single unique point, nor are they parallel (they are not different lines that never meet). Instead, every point on the line [tex]\( y = 2x + 6 \)[/tex] satisfies both equations simultaneously.
5. Conclusion:
Therefore, there are an infinite number of solutions because any point [tex]\((x, y)\)[/tex] that lies on the line [tex]\( y = 2x + 6 \)[/tex] will satisfy both equations.
Thus, the solution to the system is:
[tex]\[ \text{Infinite solutions} \][/tex]
