Mister Exam
x+(10000+y)=11000; 11000-x=10000+y
The solution
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x + 10000 + y = 11000
$$x + \left(y + 10000\right) = 11000$$
11000 - x = 10000 + y
$$11000 - x = y + 10000$$
11000 - x = y + 10000
Detail solution
Given the system of equations
$$x + \left(y + 10000\right) = 11000$$
$$11000 - x = y + 10000$$
Let's express from equation 1 x
$$x + \left(y + 10000\right) = 11000$$
Let's move the summand with the variable y from the left part to the right part performing the sign change
$$x + 10000 = 11000 - y$$
$$x + 10000 = 11000 - y$$
We move the free summand 10000 from the left part to the right part performing the sign change
$$x = \left(11000 - y\right) - 10000$$
$$x = 1000 - y$$
Let's try the obtained element x to 2-th equation
$$11000 - x = y + 10000$$
We get:
$$11000 - \left(1000 - y\right) = y + 10000$$
so
is always executed
$$x + \left(y + 10000\right) = 11000$$
$$11000 - x = y + 10000$$
Let's express from equation 1 x
$$x + \left(y + 10000\right) = 11000$$
Let's move the summand with the variable y from the left part to the right part performing the sign change
$$x + 10000 = 11000 - y$$
$$x + 10000 = 11000 - y$$
We move the free summand 10000 from the left part to the right part performing the sign change
$$x = \left(11000 - y\right) - 10000$$
$$x = 1000 - y$$
Let's try the obtained element x to 2-th equation
$$11000 - x = y + 10000$$
We get:
$$11000 - \left(1000 - y\right) = y + 10000$$
so
is always executed
Rapid solution
$$x_{1} = 1000 - y$$
=
$$1000 - y$$
=
=
$$1000 - y$$
=
1000 - y
Gaussian elimination
Given the system of equations
$$x + \left(y + 10000\right) = 11000$$
$$11000 - x = y + 10000$$
We give the system of equations to the canonical form
$$x + y = 1000$$
$$- x - y = -1000$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}1 & 1 & 1000\\-1 & -1 & -1000\end{matrix}\right]$$
In 1 -th column
$$\left[\begin{matrix}1\\-1\end{matrix}\right]$$
let’s convert all the elements, except
1 -th element into zero.
- To do this, let’s take 1 -th line
$$\left[\begin{matrix}1 & 1 & 1000\end{matrix}\right]$$
,
and subtract it from other lines:
From 2 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}-1 - -1 & -1 - -1 & -1000 - \left(-1\right) 1000\end{matrix}\right] = \left[\begin{matrix}0 & 0 & 0\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 1 & 1000\\0 & 0 & 0\end{matrix}\right]$$
In 1 -th column
$$\left[\begin{matrix}1\\0\end{matrix}\right]$$
let’s convert all the elements, except
1 -th element into zero.
- To do this, let’s take 1 -th line
$$\left[\begin{matrix}1 & 1 & 1000\end{matrix}\right]$$
,
and subtract it from other lines:
It is almost ready, all we have to do is to find variables, solving the elementary equations:
$$x_{1} + x_{2} - 1000 = 0$$
$$0 - 0 = 0$$
We get the answer:
$$x_{1} = 1000 - x_{2}$$
where x2 - the free variables
$$x + \left(y + 10000\right) = 11000$$
$$11000 - x = y + 10000$$
We give the system of equations to the canonical form
$$x + y = 1000$$
$$- x - y = -1000$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}1 & 1 & 1000\\-1 & -1 & -1000\end{matrix}\right]$$
In 1 -th column
$$\left[\begin{matrix}1\\-1\end{matrix}\right]$$
let’s convert all the elements, except
1 -th element into zero.
- To do this, let’s take 1 -th line
$$\left[\begin{matrix}1 & 1 & 1000\end{matrix}\right]$$
,
and subtract it from other lines:
From 2 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}-1 - -1 & -1 - -1 & -1000 - \left(-1\right) 1000\end{matrix}\right] = \left[\begin{matrix}0 & 0 & 0\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 1 & 1000\\0 & 0 & 0\end{matrix}\right]$$
In 1 -th column
$$\left[\begin{matrix}1\\0\end{matrix}\right]$$
let’s convert all the elements, except
1 -th element into zero.
- To do this, let’s take 1 -th line
$$\left[\begin{matrix}1 & 1 & 1000\end{matrix}\right]$$
,
and subtract it from other lines:
It is almost ready, all we have to do is to find variables, solving the elementary equations:
$$x_{1} + x_{2} - 1000 = 0$$
$$0 - 0 = 0$$
We get the answer:
$$x_{1} = 1000 - x_{2}$$
where x2 - the free variables