Question

x
=
7



x
=
|
7
y
|


y
=
7


y
2
=
7
x

Answer 1

I’m not sure how to read what you pasted. Do you mean these four equations?

1) x = 7
2) x = |7y|
3) y = 7
4) y^2 = 7x

If that’s right, do you want to (a) solve them all simultaneously, (b) find intersections pairwise, or (c) something else? Below I assume that interpretation and give the quick possibilities.

Immediate observations
- All four cannot hold at once: x = 7 and x = |7y| together would force |7y| = 7 ⇒ |y| = 1, but y = 7 contradicts that.

Pairwise / small-system solutions (under the interpretation above)
- x = 7 and y = 7 ⇒ (7,7). This satisfies y^2 = 7x (49 = 49) but does NOT satisfy x = |7y| (because |7·7| = 49 ≠ 7).
- x = 7 and x = |7y| ⇒ |7y| = 7 ⇒ |y| = 1 ⇒ (7,1) and (7,−1).
- y = 7 and x = |7y| ⇒ x = |7·7| = 49 ⇒ (49,7).
- x = 7 and y^2 = 7x ⇒ y^2 = 49 ⇒ y = ±7 ⇒ (7,7) and (7,−7).
- y = 7 and y^2 = 7x ⇒ 49 = 7x ⇒ x = 7 ⇒ (7,7).

Solve x = |7y| together with y^2 = 7x (both simultaneously)
- Substitute x = |7y| into y^2 = 7x: y^2 = 7|7y| = 49|y|.
Let t = |y| ≥ 0: t^2 = 49t ⇒ t(t − 49) = 0 ⇒ t = 0 or t = 49.
So |y| = 0 ⇒ y = 0 gives x = |7·0| = 0 ⇒ (0,0).
Or |y| = 49 ⇒ y = ±49 gives x = |7·±49| = 343 ⇒ (343,49) and (343,−49).
Thus the solutions of {x = |7y|, y^2 = 7x} are (0,0), (343,49), (343,−49).

If this wasn’t what you meant, please clarify the intended equations/formatting and what you want done (solve, graph, check which pairs satisfy which equations, etc.).