How many solutions does the system have? 12x−15y=18 4x−5y=6(1 point) Responses no solution no solution infinitely many solutions infinitely many solutions one solution one solution two solutions
To determine the number of solutions for the given system of equations, we can use the method of elimination or substitution. Let's use the method of substitution:
We are given the following equations:
1) 12x - 15y = 18
2) 4x - 5y = 6
We can rearrange equation 2 to solve for x:
4x - 5y = 6
4x = 5y + 6
x = (5y + 6)/4
Now substitute this value of x in equation 1:
12((5y + 6)/4) - 15y = 18
(60y + 72)/4 - 15y = 18
15y + 18 - 15y = 18
18 = 18
We have obtained an identity (18 = 18), meaning that the two equations are equivalent. This tells us that the two equations represent the same line. Since the equations are identical, they intersect at infinitely many points, which means the system has infinitely many solutions.
Therefore, the correct response is: infinitely many solutions.
To determine the number of solutions for the given system of equations:
12x - 15y = 18 ...(equation 1)
4x - 5y = 6 ...(equation 2)
Method 1: Graphing
One way is to graph the two equations on a coordinate plane. If the lines are parallel, they will never intersect, indicating no solution. If the lines are coincident (overlapping), they will intersect at every point on the line, indicating infinitely many solutions. If the lines intersect at a single point, there is one solution.
Method 2: Solving algebraically
To solve algebraically, we can start by manipulating one equation to solve for a variable, and then substitute that value into the other equation.
Let's solve equation 2 for x:
4x - 5y = 6
4x = 5y + 6
x = (5y + 6)/4 ...(equation 3)
Now substitute equation 3 into equation 1:
12[(5y + 6)/4] - 15y = 18
(60y + 72)/4 - 15y = 18
60y + 72 - 60y = 72
72 = 72
The equation simplifies to 72 = 72, which is always true. This means that both equations are equivalent and represent the same line. Therefore, the system of equations has infinitely many solutions.
Hence, the answer is: infinitely many solutions.
The given system of equations is:
12x - 15y = 18 (Equation 1)
4x - 5y = 6 (Equation 2)
To determine the number of solutions, we can solve the system using any method. Let's use the method of substitution.
From Equation 2, we can express x in terms of y:
4x - 5y = 6
4x = 5y + 6
x = (5y + 6)/4 (Equation 3)
Now substitute Equation 3 into Equation 1:
12x - 15y = 18
12((5y + 6)/4) - 15y = 18
(30y + 36)/4 - 15y = 18
30y + 36 - 60y = 72
-30y = 36
y = -36/30
y = -6/5
Substitute the value of y into Equation 3 to find x:
x = (5(-6/5) + 6)/4
x = (-6 + 6)/4
x = 0/4
x = 0
Therefore, the system has one solution, x = 0 and y = -6/5.