System of one quadratic and two linear equations over the positive integers
Find all solutions $(x, y, z)$ of the system of equations
\begin{align*} x y + z &= 2019 ,\\ x − y + 2 z &= 1 , \\ x + y − 7 z &= 2 \end{align*}
in positive integers.
I am thinking of solving this system equation. The only problem I have is that I've not learned any method for solving $3 \times 3$ systems of equations. I am pretty amazed by any help!
hint
Equation $ 2 $ plus( +)equation $ 3$ gives $$2x-5z=3$$
Equation $ 3 $ minus (- )equation $ 2$ gives $$2y-9z=1$$
so $$\boxed{x=\frac{5z+3}{2}\;\;;\;\;y=\frac{9z+1}{2}}$$
what we replace in equation $ 1 $, to get
$$(5z+3)(9z+1)+4z=8076$$ or
$$45z^2+36z-8073=0$$
$$\iff 5z^2+4z-897=0$$
thus $$z=\frac{-2\pm 67}{5}$$ One solution is $$z=\frac{65}{5}=\color{red}{13}\;\;x=\frac{5z+3}{2}=\color{red}{34}$$ $$\;y=\frac{9z+1}{2}=\color{red}{59}$$
the second is $$z=\color{blue}{\frac{-69}{5}}\;\;, x=...,y=...$$
To start you take the equations
XY + Z = 2019 X - Y + 2z = 1 X + Y - 7z = 2
Solve for x
XY + Z = 2019
Subtract z
XY = -z + 2019
Divide by y
X = (-z + 2019)/y
Solve for y
X - Y + 2z = 1
Add y
X + 2z = Y + 1
Subtract 2z
X = -2z + Y + 1
Now the second equation
X + Y - 7z = 2
Add 7z
X + Y = 7z + 2
Subtract Y
X = 7z - Y + 2
Now solve for Y
7z - Y + 2 = -2z + Y + 1
Add 2z
9z - Y + 2 = Y + 1
Add Y
9z + 2 = 2y + 1
Subtract 1
9z + 1 = 2y
Divide by 2
Y = 4.5z + 0.5
Bring back the equation for x
X = (-z + 2019)/y
Input Y
X = (-z + 2019)/(4.5z + 0.5)
Simplify
X = 1009.5/-4.5
X = 224 1/3
Input answer to X back into equation
224 1/3 = (-z + 2019)/y
Multiply by Y
Y • 224 1/3 = -z + 2019
Subtract 2019
Y • 224 1/3 - 2019 = -z
Multiply by negative 1
Z = -y • 224 1/3 + 2019
Solve for Z again
-2z + Y + 1 = (-z + 2019)/Y
Multiply by Y
Yˆ2 - 2ZY + Y = -Z + 2019
Subtract 2019
-Z = Yˆ2 - 2ZY + Y - 2019
Divide by -1
Z = -Yˆ2 + 2ZY - Y + 2019
Now solve for Y
-yˆ2 + 2ZY - Y + 2019 = -224 1/3Y + 2019
Subtract 2019
-Yˆ2 + 2ZY - Y = -224 1/3Y
Divide by -Y
Y - 2Z + 1 = 224 1/3
Add 1
Y - 2Z = 225 1/3
Subtract Y
2Z = -Y + 225 1/3
Divide by 2
Z = -1/2Y + 112 2/3
Solve for Y using the 2 equation for Z
-1/2Y + 112 2/3 = -224 1/3Y + 2019
Subtract 112 2/3
-1/2Y = -224 1/3Y + 1906 1/3
Add 224 1/3Y
223 5/6Y = 1906 1/3
Divide by 223 5/6
Y = 8 694/1343
Now solve for Z
224 1/3 = (-z + 2019)/(8 694/1343)
Multiply by 8 694/1343
1910 2384/4029 = -Z + 2019
Subtract 2019
-108 1645/4029 = -Z
Divide by -1
Z = 108 1645/4029
So in the end you get
X = 224 1/3
Y = 8 694/1543
Z = 108 1645/4029