The equation x + y = 1 can be viewed as a linear system of one equation in two unknowns. Express a general solution of this equation as a particular solution plus a general solution of the associated homogeneous system.
Then, give a geometric interpretation of the result.
To express a general solution of the equation x + y = 1 as a particular solution plus a general solution of the associated homogeneous system, we first need to identify the associated homogeneous system.
The associated homogeneous system is obtained by setting the right-hand side of the equation to zero. So, in this case, the associated homogeneous system is x + y = 0.
To find the particular solution of the original equation x + y = 1, we can choose any arbitrary values for one of the variables (let's say x or y) and solve for the other variable. Let's choose x = 1. Substituting this into the equation, we get 1 + y = 1. Solving for y, we find y = 0. Therefore, a particular solution is (1, 0).
Now, to find the general solution of the associated homogeneous system x + y = 0, we need to solve this equation without any specific values. Rearranging the equation, we have y = -x. This tells us that for any value of x that we choose, y will be the negation of x. So, the general solution can be expressed as (x, -x), where x can take on any real value.
Combining the particular solution (1, 0) and the general solution (x, -x) of the associated homogeneous system, we get the general solution of the original equation x + y = 1 as (1 + x, -x).
Geometric interpretation:
The equation x + y = 1 represents a straight line in a two-dimensional Cartesian coordinate system. The particular solution (1, 0) represents a specific point on this line, while the general solution (1 + x, -x) represents all the points along this line.
The associated homogeneous system x + y = 0, which corresponds to the equation x + y = 0, represents another straight line passing through the origin. The general solution (x, -x) represents all the points along this line.
Therefore, the general solution of the original equation as a particular solution plus a general solution of the associated homogeneous system represents all the points on the original line (x + y = 1) along with all the points on the associated homogeneous line (x + y = 0) passing through the origin.
beats me but maybe:
y = b + m x
for any m, y = b when x = 0
then
y = 1 + m x is family of straight lines of different slopes through (0,1)
in this particular case
y = 1 -1 x, slope = -1