To determine the number of solutions for the given system of equations, we can analyze both equations.
We have:
- Equation 1: 2x + 6y = 5
- Equation 2: x + 3y = 2
We can represent these equations in a standard form. Let’s first manipulate Equation 2 to express x in terms of y:
x = 2 – 3y
Next, we substitute this expression for x into Equation 1:
2(2 – 3y) + 6y = 5
Expanding this gives:
4 – 6y + 6y = 5
Simplifying it results in:
4 = 5
This statement is not true; hence, the equations do not intersect at any point. When we have a situation like this, where a contradiction arises from simplifying the equations, it indicates that the two lines are parallel.
Since the lines do not intersect, there are no solutions for the system of equations. In conclusion, the answer is that there are zero solutions for the system of equations 2x + 6y = 5 and x + 3y = 2.