System of one quadratic and two linear equations over the positive integers

Question

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.

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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!

Answer 1

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 $$z=\color{blue}{\frac{-69}{5}}\;\;, x=...,y=...$$

Answer 2

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