Question

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Which is an accurate conclusion regarding the system of equations below?

x + y = 4
y = -x + 4

A) There is no solution, since both equations have the same slope.
B) There are infinitely many solutions, since the same line represents both equations.
C) The only solution is the ordered pair (4, 0).
D) The only solution is the ordered pair (0, 4).

Answer 1

Answer: B) Infinitely many solutions; both equations are equivalent

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Work Shown:

x+y = 4 ... start with the first equation

x + (-x+4) = 4 ... replace y with (-x+4)

x-x+4 = 4

0x+4 = 4

0+4 = 4

4 = 4 ... this is a true statement regardless of what x you pick

So there are infinitely many solutions. Each solution (x,y) is of the form (x,-x+4). All solutions fall on the line y = -x+4 which is equivalent to x+y = 4. Note how we add x to both sides.

Or you could start with x+y = 4 and subtract x from both sides to get y = -x+4. Either way, we're dealing with the same equation which is why they both graph out the same line.