 Sangeetha Pulapaka
1

Divide the original equation by 5 to get  2x+y = 3.

1.A equation is said to have exactly one solution with this, if the set of equations meet at a point, or have a unique point of intersection. This only happens when the coefficient of x and the coefficient of y are different in the new equation. An equation which has exactly one solution in common with this will be 3x+2y = 6.

Notice that the coefficients of both the equations are different. Now solve them to find the common solution.Solving these we get the point (0,3). This is our common solution. So, the equation which has exactly one solution with Mai's equation is 3x+2y = 6

2..An equation is said to have no solutions when the coefficients of both x and y are the same but the y-intercepts are different. For example we have an equation 2x+y=6 This equation has the same coefficients of Mai's equation, but the y-intercepts are different. Solving this we get 3 = 6 which is not true since 3 \neq 6. This means that 2x+y = 6 has no solutions in common with Mai's equation.

3.An equation is said to have infinitely many solutions if the result will be 0=0 or 1 = 1or x=x. For this to happen the coefficients of x, y and the y-intercepts in both the equations should be equal to each other. Consider the equation 2x+y = 3. This equation has no solutions in common with Mai's equation.

This is because Mai's equation is 10x+5y = 15 which is 2x+y = 3 after getting divided with 5. Solving 2x+y = 3 and 2x+y = 3, we get  0 = 0. This means that 2x+y = 3 will be our equation with infinitely many solutions in common with Mai's equation.