How many solutions does the system have?

Question

12x−15y=18
4x−5y=6

Answer 1

To determine the number of solutions for this system of equations, we can use the method of elimination.

Multiply the first equation by 4 and the second equation by 3 to make the coefficients of x match.

First equation: 12x - 15y = 18 --> (4)(12x) - (4)(15y) = (4)(18) --> 48x - 60y = 72
Second equation: 4x - 5y = 6 --> (3)(4x) - (3)(5y) = (3)(6) --> 12x - 15y = 18

Now, subtract the second equation from the first equation:

(48x - 60y) - (12x - 15y) = 72 - 18
36x - 45y = 54

This new equation is equivalent to the original equations, but with eliminated x terms.

We can see that both the original equations and the new equation have the same number of variables (x and y), meaning they have the same number of solutions.

Therefore, the system of equations has the same number of solutions as the new equation, which is infinitely many solutions.