Question: determine all solutions to the diophantine equation 3x + 7y = 4

1 answer:

3 0

The difference between the coefficients 7 and 3 is 4, so one solution is
y = 1, x = -1.
Then the remaining solutions will be of the form
x = -7n-1
y = 3n+1
for some integer n.

_____
Some of the solutions are (for n=0, -1, -2, 1, 2)
(x, y) = (-1, 1), (6, -2), (13, -5), (-8, 4), (-15, 7)

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Please answer with explanation, I don't understand.<br><br> <img src="https://tex.z-dn.net/?f=%5Csqrt%7Bx%5E2-14x%2B49%7D%20%3Dx

Crank

Answer: x ≥ 7

Step-by-step explanation:

You have to square both sides of the equation. You can manipulate an equation or function any way imaginable as long as it is done equally and helps the problem become easier. If we square this, the radical will cancel and we will get two trinomials equal to each other.

$x^2-14x+49=(x-7)^2\\x^2-14x+49=x^2-14x+49$

We can see the two expressions are exactly equal to each other. All real numbers are solutions, besides a few. These few are when the radical is equal to a negative number. So lets set up an inequality stating that the radical must be greater than or equal to 0.

$x^2-14x+49\geq 0\\(x-7)^2 \geq 0\\x \geq 7$