How many solutions does the system have?
12x−15y=18 4x−5y=6 (1 point)
The system of equations has infinitely many solutions.
This can be determined by multiplying the first equation by 4 and the second equation by 3:
48x - 60y = 72
12x - 15y = 18
These two equations represent the same line, so every point on this line satisfies both equations, resulting in an infinite number of solutions.
To find the number of solutions for the given system of equations, we can use the concept of determinant.
The given system of equations is:
12x - 15y = 18 (Equation 1)
4x - 5y = 6 (Equation 2)
Let's represent the coefficients of x, y, and the constants in matrix form:
| 12 -15 | | x | | 18 |
| 4 -5 | * | y | = | 6 |
The determinant of the coefficient matrix is given by:
D = (12 * -5) - (-15 * 4)
Simplifying, we get:
D = -60 - (-60)
D = -60 + 60
D = 0
The determinant D is equal to 0.
When the determinant of the coefficient matrix is equal to 0, the system of equations either has infinite solutions or no solution.
To determine which case applies, we can calculate the determinant of the constant matrix from the original system of equations.
Let's represent the constants in matrix form:
| 18 |
| 6 |
The determinant of the constant matrix is given by:
D1 = (18 * -5) - (-15 * 6)
D1 = -90 + 90
D1 = 0
Since the determinant of the constant matrix is also 0, the system of equations has infinite solutions.
So, the given system of equations has infinitely many solutions.
To determine the number of solutions for this system of equations, we can use the method of elimination.
Step 1: Multiply the second equation by 3 to make the coefficients of x in both equations the same:
12x - 15y = 18
12x - 15y = 18
Step 2: Subtract the second equation from the first equation:
(12x - 15y) - (12x - 15y) = 18 - 18
0 = 0
Step 3: Simplify the equation:
0 = 0
Since the resulting equation is always true (0 = 0), this means that the two equations are equivalent and coincide with each other. Therefore, the system of equations has infinitely many solutions.
The answer is that the system has infinitely many solutions.