How many solutions do the linear equations x-y=2 -2x+2y=-1 have?
x - y = 2. m = -A/B = -1/-1 = 1.
-2x + 2y = -1. m = 2/2 = 1.
The lines have equal slopes. Therefore,
they are parallel and do not intersect.
So there are no solutions.
To find the number of solutions for a system of linear equations, we need to consider the slopes and intercepts of the lines represented by the equations.
For the given system of equations:
1. x - y = 2
2. -2x + 2y = -1
First, we rearrange equation 1 to solve for y:
1. x - y = 2
-y = -x + 2
y = x - 2
Now we can examine the slopes of the lines. The coefficient of x in each equation represents the slope. In equation 1, the slope is 1, and in equation 2, the slope is -1.
If the slopes are different, the lines will intersect at a single point, resulting in a unique solution.
However, if the slopes are equal, the lines will either be parallel or coincide (be the same line) depending on whether the y-intercepts are different or equal.
To check the y-intercepts of the lines, we evaluate the equations when x = 0:
For equation 1:
y = 0 - 2
y = -2
For equation 2:
-2(0) + 2y = -1
2y = -1
y = -1/2
Since the y-intercepts (-2 and -1/2) are different, the lines are not parallel and will intersect at a single point. Therefore, there is exactly one solution for this system of linear equations.