Write an equation that forms a system of equations with x + y = 4, so that the system has:
a) No solution b) Infinitely many solutions c) One solution
I know how to figure out if two linear systems have a solution, but I don't understand how you could figure out if just one linear system has a solution. Help please?

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1. (a) no solution

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2. (a) no solution
recall that two linear equations have no solution if they are parallel (solution means they meet at a certain point),, thus the two equations, in order to be parallel must have the SAME SLOPE.

to determine the slope of the given equation, we rewrite it in this form:
y = mx + b
where m = slope and b = y-intercept,, so x + y = 4 becomes:
y = -x + 4
where m = -1
now that we know the slope, we can now formulate a new equation with slope=1,, thus we only change the value of b in order to have a parallel equation,, an example would be:
y = -x + 1

(b) infinitely many solutions
here, it means that the two linear equations coincide meaning they have the SAME SLOPE and Y-INTERCEPT,, this means that the equations are equal, and most of the time, they are just the multiple of each other.
for the given equation, we can just multiply any number to both sides of the equation to formulate the coinciding equation,, for instance we multiply it by 2:
2(x + y = 4)
2x + 2y = 8

(c) one solution
it means that the two equation meet at EXACTLY ONE POINT,, thus the other equation must have at least, a DIFFERENT SLOPE,, an example would be:
x + 2y = 4
where they meet at point (4,0)

hope this helps~ :)

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3. sorry for the first post by the way~ ^^;
i just accidentally hit the enter button~ ^^;

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