How many solutions does the system have%3F 12x−15y%3D18 4x−5y%3D6(1 point) Responses one solution one solution no solution no solution infinitely many solutions infinitely many solutions two solutions
The given system of equations is:
12x - 15y = 18
4x - 5y = 6
To find the number of solutions, we can compare the slopes of the two equations. The slopes of the two equations are:
12x - 15y = 18 --> slope = 12/15 = 4/5
4x - 5y = 6 --> slope = 4/5
Since the slopes are equal, the two lines are parallel and will never intersect. Therefore, the system of equations has:
Answer: no solution.
To determine the number of solutions for the system of equations, we need to solve them simultaneously. The given system of equations is:
12x - 15y = 18 ...(Equation 1)
4x - 5y = 6 ...(Equation 2)
To solve this system, we can use the method of substitution or elimination. Let's use the elimination method:
Multiply Equation 2 by 3:
3 * (4x - 5y) = 3 * 6
12x - 15y = 18 ...(Equation 3)
Comparing Equation 1 and Equation 3, we can see that they are identical. This means that Equation 1 and Equation 3 represent the same line.
Since the two equations represent the same line, the system of equations has infinitely many solutions.
Therefore, the correct response is: infinitely many solutions.
To determine the number of solutions a system of equations has, we can solve the system and analyze the result.
Given the system of equations:
12x - 15y = 18 (Equation 1)
4x - 5y = 6 (Equation 2)
We can use the method of substitution or elimination to solve the system. Let's use the method of elimination here.
First, multiply Equation 2 by 3 to make the coefficients of x in both equations equal:
12x - 15y = 18 (Equation 1)
12x - 15y = 18 (3 * Equation 2)
Now, subtract Equation 2 from Equation 1:
(12x - 15y) - (12x - 15y) = 18 - 18
0 = 0
The resulting equation, 0 = 0, indicates that the two equations are equivalent. This means that both equations represent the same line on the coordinate plane. In other words, they are just different algebraic representations of the same line.
Since the equations represent the same line, there are infinitely many solutions to this system. Infinitely many because every point on the line satisfies both equations simultaneously.