Mister Exam
x+y=25; 200x+300y=65000
The solution
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x + y = 25
$$x + y = 25$$
200*x + 300*y = 65000
$$200 x + 300 y = 65000$$
200*x + 300*y = 65000
Detail solution
Given the system of equations
$$x + y = 25$$
$$200 x + 300 y = 65000$$
Let's express from equation 1 x
$$x + y = 25$$
Let's move the summand with the variable y from the left part to the right part performing the sign change
$$x = 25 - y$$
$$x = 25 - y$$
Let's try the obtained element x to 2-th equation
$$200 x + 300 y = 65000$$
We get:
$$300 y + 200 \left(25 - y\right) = 65000$$
$$100 y + 5000 = 65000$$
We move the free summand 5000 from the left part to the right part performing the sign change
$$100 y = -5000 + 65000$$
$$100 y = 60000$$
Let's divide both parts of the equation by the multiplier of y
$$\frac{100 y}{100} = \frac{60000}{100}$$
$$y = 600$$
Because
$$x = 25 - y$$
then
$$x = 25 - 600$$
$$x = -575$$
The answer:
$$x = -575$$
$$y = 600$$
$$x + y = 25$$
$$200 x + 300 y = 65000$$
Let's express from equation 1 x
$$x + y = 25$$
Let's move the summand with the variable y from the left part to the right part performing the sign change
$$x = 25 - y$$
$$x = 25 - y$$
Let's try the obtained element x to 2-th equation
$$200 x + 300 y = 65000$$
We get:
$$300 y + 200 \left(25 - y\right) = 65000$$
$$100 y + 5000 = 65000$$
We move the free summand 5000 from the left part to the right part performing the sign change
$$100 y = -5000 + 65000$$
$$100 y = 60000$$
Let's divide both parts of the equation by the multiplier of y
$$\frac{100 y}{100} = \frac{60000}{100}$$
$$y = 600$$
Because
$$x = 25 - y$$
then
$$x = 25 - 600$$
$$x = -575$$
The answer:
$$x = -575$$
$$y = 600$$
Rapid solution
$$x_{1} = -575$$
=
$$-575$$
=
$$y_{1} = 600$$
=
$$600$$
=
=
$$-575$$
=
-575
$$y_{1} = 600$$
=
$$600$$
=
600
Gaussian elimination
Given the system of equations
$$x + y = 25$$
$$200 x + 300 y = 65000$$
We give the system of equations to the canonical form
$$x + y = 25$$
$$200 x + 300 y = 65000$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}1 & 1 & 25\\200 & 300 & 65000\end{matrix}\right]$$
In 1 -th column
$$\left[\begin{matrix}1\\200\end{matrix}\right]$$
let’s convert all the elements, except
1 -th element into zero.
- To do this, let’s take 1 -th line
$$\left[\begin{matrix}1 & 1 & 25\end{matrix}\right]$$
,
and subtract it from other lines:
From 2 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}\left(-1\right) 200 + 200 & \left(-1\right) 200 + 300 & 65000 - 25 \cdot 200\end{matrix}\right] = \left[\begin{matrix}0 & 100 & 60000\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 1 & 25\\0 & 100 & 60000\end{matrix}\right]$$
In 2 -th column
$$\left[\begin{matrix}1\\100\end{matrix}\right]$$
let’s convert all the elements, except
2 -th element into zero.
- To do this, let’s take 2 -th line
$$\left[\begin{matrix}0 & 100 & 60000\end{matrix}\right]$$
,
and subtract it from other lines:
From 1 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}1 - \frac{0}{100} & 1 - \frac{100}{100} & 25 - \frac{60000}{100}\end{matrix}\right] = \left[\begin{matrix}1 & 0 & -575\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 0 & -575\\0 & 100 & 60000\end{matrix}\right]$$
It is almost ready, all we have to do is to find variables, solving the elementary equations:
$$x_{1} + 575 = 0$$
$$100 x_{2} - 60000 = 0$$
We get the answer:
$$x_{1} = -575$$
$$x_{2} = 600$$
$$x + y = 25$$
$$200 x + 300 y = 65000$$
We give the system of equations to the canonical form
$$x + y = 25$$
$$200 x + 300 y = 65000$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}1 & 1 & 25\\200 & 300 & 65000\end{matrix}\right]$$
In 1 -th column
$$\left[\begin{matrix}1\\200\end{matrix}\right]$$
let’s convert all the elements, except
1 -th element into zero.
- To do this, let’s take 1 -th line
$$\left[\begin{matrix}1 & 1 & 25\end{matrix}\right]$$
,
and subtract it from other lines:
From 2 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}\left(-1\right) 200 + 200 & \left(-1\right) 200 + 300 & 65000 - 25 \cdot 200\end{matrix}\right] = \left[\begin{matrix}0 & 100 & 60000\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 1 & 25\\0 & 100 & 60000\end{matrix}\right]$$
In 2 -th column
$$\left[\begin{matrix}1\\100\end{matrix}\right]$$
let’s convert all the elements, except
2 -th element into zero.
- To do this, let’s take 2 -th line
$$\left[\begin{matrix}0 & 100 & 60000\end{matrix}\right]$$
,
and subtract it from other lines:
From 1 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}1 - \frac{0}{100} & 1 - \frac{100}{100} & 25 - \frac{60000}{100}\end{matrix}\right] = \left[\begin{matrix}1 & 0 & -575\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 0 & -575\\0 & 100 & 60000\end{matrix}\right]$$
It is almost ready, all we have to do is to find variables, solving the elementary equations:
$$x_{1} + 575 = 0$$
$$100 x_{2} - 60000 = 0$$
We get the answer:
$$x_{1} = -575$$
$$x_{2} = 600$$
Cramer's rule
$$x + y = 25$$
$$200 x + 300 y = 65000$$
We give the system of equations to the canonical form
$$x + y = 25$$
$$200 x + 300 y = 65000$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}x_{1} + x_{2}\\200 x_{1} + 300 x_{2}\end{matrix}\right] = \left[\begin{matrix}25\\65000\end{matrix}\right]$$
- this is the system of equations that has the form
A*x = B
Let´s find a solution of this matrix equations using Cramer´s rule:
Since the determinant of the matrix:
$$A = \operatorname{det}{\left(\left[\begin{matrix}1 & 1\\200 & 300\end{matrix}\right] \right)} = 100$$
, then
The root xi is obtained by dividing the determinant of the matrix Ai. by the determinant of the matrix A.
( Ai we get it by replacement in the matrix A i-th column with column B )
$$x_{1} = \frac{\operatorname{det}{\left(\left[\begin{matrix}25 & 1\\65000 & 300\end{matrix}\right] \right)}}{100} = -575$$
$$x_{2} = \frac{\operatorname{det}{\left(\left[\begin{matrix}1 & 25\\200 & 65000\end{matrix}\right] \right)}}{100} = 600$$
$$200 x + 300 y = 65000$$
We give the system of equations to the canonical form
$$x + y = 25$$
$$200 x + 300 y = 65000$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}x_{1} + x_{2}\\200 x_{1} + 300 x_{2}\end{matrix}\right] = \left[\begin{matrix}25\\65000\end{matrix}\right]$$
- this is the system of equations that has the form
A*x = B
Let´s find a solution of this matrix equations using Cramer´s rule:
Since the determinant of the matrix:
$$A = \operatorname{det}{\left(\left[\begin{matrix}1 & 1\\200 & 300\end{matrix}\right] \right)} = 100$$
, then
The root xi is obtained by dividing the determinant of the matrix Ai. by the determinant of the matrix A.
( Ai we get it by replacement in the matrix A i-th column with column B )
$$x_{1} = \frac{\operatorname{det}{\left(\left[\begin{matrix}25 & 1\\65000 & 300\end{matrix}\right] \right)}}{100} = -575$$
$$x_{2} = \frac{\operatorname{det}{\left(\left[\begin{matrix}1 & 25\\200 & 65000\end{matrix}\right] \right)}}{100} = 600$$