How many solutions does the system of equations have? x – 4y = 12 and 5x – 20y = 60 (1 point) Responses one one two two infinitely many infinitely many none
To determine the number of solutions, we can rewrite the first equation as x = 4y + 12 and substitute it into the second equation:
5x – 20y = 60
5(4y + 12) – 20y = 60
20y + 60 – 20y = 60
60 = 60
The equation simplifies to 60 = 60, which means that the equation is true regardless of the value of y. Therefore, the system of equations has infinitely many solutions.
To find the number of solutions of the system of equations, we can use the concept of linear independence or dependence.
The given system of equations is:
1) x – 4y = 12
2) 5x – 20y = 60
We can simplify equation 2 by dividing both sides by 5:
2') x - 4y = 12
Comparing equations 1 and 2' reveals that they are exactly the same equation. Therefore, equations 1 and 2' are linearly dependent, meaning they are redundant and provide the same information.
In this case, we have infinitely many solutions. The system of equations represents multiple expressions of the same line in a coordinate plane. Thus, the answer is: "infinitely many" solutions.
To find the number of solutions of a system of equations, we need to determine if the two equations are independent, dependent, or inconsistent.
Let's start by rewriting the given system of equations:
Equation 1: x - 4y = 12
Equation 2: 5x - 20y = 60
To determine the number of solutions, we can use a few different methods, such as substitution, elimination, or graphing. Let's use the elimination method in this case.
First, let's multiply Equation 1 by 5 to make the coefficient of x the same as in Equation 2:
5(x - 4y) = 5(12)
=> 5x - 20y = 60
As we can see, Equation 1 is equivalent to Equation 2, which means these two equations represent the same line in the coordinate plane. Therefore, the system of equations has infinitely many solutions.
To check this, graphing the equations on the coordinate plane would reveal that the lines are coincident or overlapping. This confirms that there are infinitely many solutions.
In conclusion, the system of equations has infinitely many solutions.