Mister Exam
x-28*y=390335; y=x/46
The solution
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x - 28*y = 390335
$$x - 28 y = 390335$$
x
y = --
46
$$y = \frac{x}{46}$$
y = x/46
Detail solution
Given the system of equations
$$x - 28 y = 390335$$
$$y = \frac{x}{46}$$
Let's express from equation 1 x
$$x - 28 y = 390335$$
Let's move the summand with the variable y from the left part to the right part performing the sign change
$$x = 28 y + 390335$$
$$x = 28 y + 390335$$
Let's try the obtained element x to 2-th equation
$$y = \frac{x}{46}$$
We get:
$$y = \frac{28 y + 390335}{46}$$
$$y = \frac{14 y}{23} + \frac{390335}{46}$$
Let's move the summand with the variable y from the right part to the left part performing the sign change
$$- \frac{14 y}{23} + y = \frac{390335}{46}$$
$$\frac{9 y}{23} = \frac{390335}{46}$$
Let's divide both parts of the equation by the multiplier of y
$$\frac{\frac{9}{23} y}{\frac{9}{23}} = \frac{390335}{\frac{9}{23} \cdot 46}$$
$$y = \frac{390335}{18}$$
Because
$$x = 28 y + 390335$$
then
$$x = 390335 + \frac{28 \cdot 390335}{18}$$
$$x = \frac{8977705}{9}$$
The answer:
$$x = \frac{8977705}{9}$$
$$y = \frac{390335}{18}$$
$$x - 28 y = 390335$$
$$y = \frac{x}{46}$$
Let's express from equation 1 x
$$x - 28 y = 390335$$
Let's move the summand with the variable y from the left part to the right part performing the sign change
$$x = 28 y + 390335$$
$$x = 28 y + 390335$$
Let's try the obtained element x to 2-th equation
$$y = \frac{x}{46}$$
We get:
$$y = \frac{28 y + 390335}{46}$$
$$y = \frac{14 y}{23} + \frac{390335}{46}$$
Let's move the summand with the variable y from the right part to the left part performing the sign change
$$- \frac{14 y}{23} + y = \frac{390335}{46}$$
$$\frac{9 y}{23} = \frac{390335}{46}$$
Let's divide both parts of the equation by the multiplier of y
$$\frac{\frac{9}{23} y}{\frac{9}{23}} = \frac{390335}{\frac{9}{23} \cdot 46}$$
$$y = \frac{390335}{18}$$
Because
$$x = 28 y + 390335$$
then
$$x = 390335 + \frac{28 \cdot 390335}{18}$$
$$x = \frac{8977705}{9}$$
The answer:
$$x = \frac{8977705}{9}$$
$$y = \frac{390335}{18}$$
Rapid solution
$$x_{1} = \frac{8977705}{9}$$
=
$$\frac{8977705}{9}$$
=
$$y_{1} = \frac{390335}{18}$$
=
$$\frac{390335}{18}$$
=
=
$$\frac{8977705}{9}$$
=
997522.777777778
$$y_{1} = \frac{390335}{18}$$
=
$$\frac{390335}{18}$$
=
21685.2777777778
Gaussian elimination
Given the system of equations
$$x - 28 y = 390335$$
$$y = \frac{x}{46}$$
We give the system of equations to the canonical form
$$x - 28 y = 390335$$
$$- \frac{x}{46} + y = 0$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}1 & -28 & 390335\\- \frac{1}{46} & 1 & 0\end{matrix}\right]$$
In 1 -th column
$$\left[\begin{matrix}1\\- \frac{1}{46}\end{matrix}\right]$$
let’s convert all the elements, except
1 -th element into zero.
- To do this, let’s take 1 -th line
$$\left[\begin{matrix}1 & -28 & 390335\end{matrix}\right]$$
,
and subtract it from other lines:
From 2 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{1}{46} - - \frac{1}{46} & 1 - - \frac{-14}{23} & - \frac{\left(-1\right) 390335}{46}\end{matrix}\right] = \left[\begin{matrix}0 & \frac{9}{23} & \frac{390335}{46}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & -28 & 390335\\0 & \frac{9}{23} & \frac{390335}{46}\end{matrix}\right]$$
In 2 -th column
$$\left[\begin{matrix}-28\\\frac{9}{23}\end{matrix}\right]$$
let’s convert all the elements, except
2 -th element into zero.
- To do this, let’s take 2 -th line
$$\left[\begin{matrix}0 & \frac{9}{23} & \frac{390335}{46}\end{matrix}\right]$$
,
and subtract it from other lines:
From 1 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}1 - \frac{\left(-644\right) 0}{9} & -28 - \frac{\left(-644\right) 9}{9 \cdot 23} & 390335 - \frac{\left(-644\right) 390335}{9 \cdot 46}\end{matrix}\right] = \left[\begin{matrix}1 & 0 & \frac{8977705}{9}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 0 & \frac{8977705}{9}\\0 & \frac{9}{23} & \frac{390335}{46}\end{matrix}\right]$$
It is almost ready, all we have to do is to find variables, solving the elementary equations:
$$x_{1} - \frac{8977705}{9} = 0$$
$$\frac{9 x_{2}}{23} - \frac{390335}{46} = 0$$
We get the answer:
$$x_{1} = \frac{8977705}{9}$$
$$x_{2} = \frac{390335}{18}$$
$$x - 28 y = 390335$$
$$y = \frac{x}{46}$$
We give the system of equations to the canonical form
$$x - 28 y = 390335$$
$$- \frac{x}{46} + y = 0$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}1 & -28 & 390335\\- \frac{1}{46} & 1 & 0\end{matrix}\right]$$
In 1 -th column
$$\left[\begin{matrix}1\\- \frac{1}{46}\end{matrix}\right]$$
let’s convert all the elements, except
1 -th element into zero.
- To do this, let’s take 1 -th line
$$\left[\begin{matrix}1 & -28 & 390335\end{matrix}\right]$$
,
and subtract it from other lines:
From 2 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{1}{46} - - \frac{1}{46} & 1 - - \frac{-14}{23} & - \frac{\left(-1\right) 390335}{46}\end{matrix}\right] = \left[\begin{matrix}0 & \frac{9}{23} & \frac{390335}{46}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & -28 & 390335\\0 & \frac{9}{23} & \frac{390335}{46}\end{matrix}\right]$$
In 2 -th column
$$\left[\begin{matrix}-28\\\frac{9}{23}\end{matrix}\right]$$
let’s convert all the elements, except
2 -th element into zero.
- To do this, let’s take 2 -th line
$$\left[\begin{matrix}0 & \frac{9}{23} & \frac{390335}{46}\end{matrix}\right]$$
,
and subtract it from other lines:
From 1 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}1 - \frac{\left(-644\right) 0}{9} & -28 - \frac{\left(-644\right) 9}{9 \cdot 23} & 390335 - \frac{\left(-644\right) 390335}{9 \cdot 46}\end{matrix}\right] = \left[\begin{matrix}1 & 0 & \frac{8977705}{9}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 0 & \frac{8977705}{9}\\0 & \frac{9}{23} & \frac{390335}{46}\end{matrix}\right]$$
It is almost ready, all we have to do is to find variables, solving the elementary equations:
$$x_{1} - \frac{8977705}{9} = 0$$
$$\frac{9 x_{2}}{23} - \frac{390335}{46} = 0$$
We get the answer:
$$x_{1} = \frac{8977705}{9}$$
$$x_{2} = \frac{390335}{18}$$
Cramer's rule
$$x - 28 y = 390335$$
$$y = \frac{x}{46}$$
We give the system of equations to the canonical form
$$x - 28 y = 390335$$
$$- \frac{x}{46} + y = 0$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}x_{1} - 28 x_{2}\\- \frac{x_{1}}{46} + x_{2}\end{matrix}\right] = \left[\begin{matrix}390335\\0\end{matrix}\right]$$
- 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:
$$A = \operatorname{det}{\left(\left[\begin{matrix}1 & -28\\- \frac{1}{46} & 1\end{matrix}\right] \right)} = \frac{9}{23}$$
, 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 )
$$x_{1} = \frac{23 \operatorname{det}{\left(\left[\begin{matrix}390335 & -28\\0 & 1\end{matrix}\right] \right)}}{9} = \frac{8977705}{9}$$
$$x_{2} = \frac{23 \operatorname{det}{\left(\left[\begin{matrix}1 & 390335\\- \frac{1}{46} & 0\end{matrix}\right] \right)}}{9} = \frac{390335}{18}$$
$$y = \frac{x}{46}$$
We give the system of equations to the canonical form
$$x - 28 y = 390335$$
$$- \frac{x}{46} + y = 0$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}x_{1} - 28 x_{2}\\- \frac{x_{1}}{46} + x_{2}\end{matrix}\right] = \left[\begin{matrix}390335\\0\end{matrix}\right]$$
- 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:
$$A = \operatorname{det}{\left(\left[\begin{matrix}1 & -28\\- \frac{1}{46} & 1\end{matrix}\right] \right)} = \frac{9}{23}$$
, 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 )
$$x_{1} = \frac{23 \operatorname{det}{\left(\left[\begin{matrix}390335 & -28\\0 & 1\end{matrix}\right] \right)}}{9} = \frac{8977705}{9}$$
$$x_{2} = \frac{23 \operatorname{det}{\left(\left[\begin{matrix}1 & 390335\\- \frac{1}{46} & 0\end{matrix}\right] \right)}}{9} = \frac{390335}{18}$$
Numerical answer
[src]
x1 = 997522.7777777778 y1 = 21685.27777777778
x1 = 997522.7777777778 y1 = 21685.27777777778