To find the number of solutions for this system of equations, we can solve the equations simultaneously to see if there is a unique solution, no solution, or infinitely many solutions.
We can use the method of elimination to solve this system. First, let's multiply the second equation by 2 to eliminate the y-term:
-3x - 2y = -1
2(-3x - 2y) = 2(-1)
-6x - 4y = -2
Now, let's add this equation to the first equation:
2x + 3y = 1
+ (-6x - 4y = -2)
-----------------
-4x - y = -1
Now, we have a system of two equations:
-4x - y = -1
2x + 3y = 1
To eliminate the y-term this time, let's multiply the first equation by 3 and the second equation by 4:
3(-4x - y = -1)
4(2x + 3y = 1)
This gives us:
-12x - 3y = -3
8x + 12y = 4
Now, let's add these two equations:
-12x - 3y = -3
+ (8x + 12y = 4)
----------------
-4x + 9y = 1
We now have a system of two equations:
-4x + 9y = 1
2x + 3y = 1
We can eliminate the x-term by multiplying the first equation by 2 and the second equation by -4:
2(-4x + 9y = 1)
-4(2x + 3y = 1)
This gives us:
-8x + 18y = 2
-8x - 12y = -4
Now, let's add these two equations:
-8x + 18y = 2
+ (-8x - 12y = -4)
------------------
0x + 6y = -2
Simplifying this equation gives us:
6y = -2
y = -2/6
y = -1/3
Now, let's substitute this value of y back into one of the original equations, such as the first equation:
2x + 3y = 1
2x + 3(-1/3) = 1
2x - 1 = 1
2x = 2
x = 1
Therefore, the solution to the system of equations is x = 1 and y = -1/3.
Since we found a unique solution (x = 1, y = -1/3), the system of equations has only one solution.
How many solutions does the following system have?
2x + 3y = 1; - 3x - 2y = - 1
1 answer
1 answer