How many solutions does the system have?
12x−15y=18
4x−5y=6
To determine the number of solutions for this system of equations, we can use the method of elimination:
1) Multiply both sides of the first equation by 4:
48x - 60y = 72
2) Multiply both sides of the second equation by 3:
12x - 15y = 18
At this point, we can see that if we subtract the second equation from the first equation, the y terms will be eliminated:
(48x - 60y) - (12x - 15y) = 72 - 18
48x - 60y - 12x + 15y = 54x - 45y = 54
Simplifying, we get:
54x - 45y = 54
So, this system of equations can be simplified to a single equation: 54x - 45y = 54
Since there is only one equation, the system has infinite solutions, meaning every point on this line is a solution to the system of equations.
To determine the number of solutions for the system of equations:
First, we will rewrite the system in standard form:
12x - 15y = 18 ... (Equation 1)
4x - 5y = 6 ... (Equation 2)
Next, we will divide both equations by their respective coefficients of x:
Equation 1: (12/3)x - (15/3)y = 18/3
4x - 5y = 6 ... (Equation 3)
Therefore, Equation 1 simplifies to: 4x - 5y = 6
Since Equation 2 and Equation 3 are identical, we can conclude that the system is dependent (meaning infinitely many solutions) since the equations are the same.
Hence, the system of equations has infinitely many solutions.
To determine how many solutions a system of equations has, we need to solve the equations simultaneously and analyze the solution.
Let's solve the system of equations using the elimination method:
1) Multiply equation (2) by 3 to make the coefficients of y in both equations equal:
12x - 15y = 18
12x - 15y = 18
2) Subtract equation (2) from equation (1):
(12x - 15y) - (12x - 15y) = 18 - 18
0 = 0
The resulting equation, 0 = 0, is true regardless of the values of x and y. This means that the two equations represent the same line or are equivalent. Therefore, the system of equations has infinitely many solutions.
Explanation:
To determine the number of solutions, we used the elimination method, which involves manipulating the equations to eliminate one variable (in this case, y) by manipulating the coefficients. If the resulting equation is true, it means that the original equations are equivalent, and there are infinitely many solutions. Conversely, if the resulting equation is false (such as 0 = 1), it means that the original equations are inconsistent and have no solution.