How many Solutions does the following system have?
(3x+2y= 12
6x+4y = 24
a) Infinitely many solutions
b) Two Solutions
c) One Solution
d) No Solutions
The two equations can be simplified by dividing the second equation by 2:
3x + 2y = 12
3x + 2y = 12
Since the two equations are identical, they represent the same line. Therefore, there are infinitely many solutions.
The answer is: a) Infinitely many solutions
To determine the number of solutions for the given system of equations, we can use either substitution or elimination method. I will explain how to solve it using the elimination method.
First, let's rewrite the system of equations in standard form:
Equation 1: 3x + 2y = 12
Equation 2: 6x + 4y = 24
Now, we can multiply Equation 1 by 2, so that the coefficients of y for both equations become the same. This helps in eliminating the y variable by adding or subtracting the equations:
2(3x + 2y) = 2(12)
6x + 4y = 24 (same as Equation 2)
Now we can subtract Equation 1 from Equation 2:
(6x + 4y) - (6x + 4y) = 24 - 24
0 = 0
This tells us that 0 is equal to 0, which is always true. This means that the system of equations has infinitely many solutions.
Therefore, the correct answer is: a) Infinitely many solutions
To determine the number of solutions for the system of equations, we can start by simplifying the equations:
1. Multiply the first equation by 2 to make the coefficients of y the same in both equations:
6x + 4y = 24
2. Now, compare the simplified equations:
6x + 4y = 24
6x + 4y = 24
We can see that both equations are the same. Thus, these two equations represent the same line on a graph. In other words, they are identical equations.
Since the equations are identical, the system has infinitely many solutions.
Therefore, the answer is:
a) Infinitely many solutions