How many solutions does the system have?
12x−15y=18
4x−5y=6 (1 point)
Responses
no solution
no solution
one solution
one solution
infinitely many solutions
infinitely many solutions
two solutions
The system has one solution.
To determine how many solutions the system has, we can use the method of elimination or substitution. Let's first use the method of elimination to solve the system of equations:
We are given the following equations:
1) 12x - 15y = 18
2) 4x - 5y = 6
To eliminate one variable, we can multiply equation 2 by 3:
3) 3(4x - 5y) = 3(6)
12x - 15y = 18
By comparing equation 1 and equation 3, we can see that both have the same coefficients for x and y. This means that the two equations represent the same line or are overlapping.
Since the two equations are identical, they have infinitely many solutions. Therefore, the answer is:
infinitely many solutions
To determine how many solutions a system of equations has, we can use a method called elimination or substitution. In this case, we have the following system of equations:
12x - 15y = 18 (Equation 1)
4x - 5y = 6 (Equation 2)
Let's use the elimination method to solve this system:
First, we need to multiply Equation 2 by 3 to make the coefficients of x the same in both equations. This gives us:
12x - 15y = 18 (Equation 1)
12x - 15y = 18 (Equation 2)
Notice that both equations are identical. This means that the two equations represent the same line in the coordinate plane. When two equations represent the same line, it means that they have infinitely many solutions. Therefore, the system has infinitely many solutions.
Therefore, the correct answer is:
- Infinitely many solutions.