Determine whether the system of linear equations has one and only one solution, infinitely many solutions, or no solution.
3x − y = −3
6x − 2y = 8
a) one and only one solution
b) infinitely many solutions
c) no solution
can you please show the work
To determine whether the system of linear equations has one and only one solution, infinitely many solutions, or no solution, we need to solve the system. Let's start by using the method of substitution:
From the first equation, we can isolate y:
3x - y = -3
-y = -3 - 3x
y = 3x + 3
Now, we can substitute this expression for y into the second equation:
6x - 2(3x + 3) = 8
6x - 6x - 6 = 8
-6 = 8
Since the equation -6 = 8 is false, there are no values of x and y that satisfy both equations simultaneously. Therefore, the system has no solution. Hence, the correct answer is c) no solution.
To determine whether the system of linear equations has one and only one solution, infinitely many solutions, or no solution, you can use the method of solving by elimination or substitution. Let's use the method of solving by elimination to work through this problem.
Given equations:
1) 3x - y = -3
2) 6x - 2y = 8
Let's start by multiplying equation 1) by 2:
2 * (3x - y) = 2 * (-3)
6x - 2y = -6
Now, we can compare this new equation to equation 2):
6x - 2y = -6
6x - 2y = 8
Upon comparing the two equations, we observe that they are inconsistent because the coefficients of x and y are the same, but the constant terms on the right side are different (-6 and 8, respectively).
Since there is a contradiction and the two lines will never intersect, the system of linear equations has no solution.
Therefore, the answer is c) no solution.
double the 1st equation and you have
6x-2y = -6
It is clear to see that since -6 ≠ -8 the two equations cannot have a single solution. The lines are parallel, and do not intersect.