Mister Exam
0.001х-0.0024y=0.004733+0,017453; -0.0024x+0.007467y=-0.0643
The solution
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x ---- - 0.0024*y = 0.022186 1000
$$\frac{x}{1000} - 0.0024 y = 0.022186$$
-0.0024*x + 0.007467*y = -0.0643
$$- 0.0024 x + 0.007467 y = -0.0643$$
-0.0024*x + 0.007467*y = -0.0643
Detail solution
Given the system of equations
$$\frac{x}{1000} - 0.0024 y = 0.022186$$
$$- 0.0024 x + 0.007467 y = -0.0643$$
Let's express from equation 1 x
$$\frac{x}{1000} - 0.0024 y = 0.022186$$
Let's move the summand with the variable y from the left part to the right part performing the sign change
$$\frac{x}{1000} = 0.0024 y + 0.022186$$
$$\frac{x}{1000} = 0.0024 y + 0.022186$$
Let's divide both parts of the equation by the multiplier of x
$$x = 2.4 y + 22.186$$
Let's try the obtained element x to 2-th equation
$$- 0.0024 x + 0.007467 y = -0.0643$$
We get:
$$0.007467 y - 0.0024 \left(2.4 y + 22.186\right) = -0.0643$$
$$0.001707 y - 0.0532464 = -0.0643$$
We move the free summand -0.0532464000000000 from the left part to the right part performing the sign change
$$0.001707 y = -0.0643 + 0.0532464$$
$$0.001707 y = -0.0110536$$
Let's divide both parts of the equation by the multiplier of y
$$\frac{0.001707 y}{0.001707} = - \frac{0.0110536}{0.001707}$$
$$1 y = -6.47545401288811$$
Because
$$x = 2.4 y + 22.186$$
then
$$x = \left(-6.47545401288811\right) 2.4 + 22.186$$
$$x = 6.64491036906854$$
The answer:
$$x = 6.64491036906854$$
$$1 y = -6.47545401288811$$
$$\frac{x}{1000} - 0.0024 y = 0.022186$$
$$- 0.0024 x + 0.007467 y = -0.0643$$
Let's express from equation 1 x
$$\frac{x}{1000} - 0.0024 y = 0.022186$$
Let's move the summand with the variable y from the left part to the right part performing the sign change
$$\frac{x}{1000} = 0.0024 y + 0.022186$$
$$\frac{x}{1000} = 0.0024 y + 0.022186$$
Let's divide both parts of the equation by the multiplier of x
/ x \ |----| \1000/ 0.022186 + 0.0024*y ------ = ------------------- 1/1000 1/1000
$$x = 2.4 y + 22.186$$
Let's try the obtained element x to 2-th equation
$$- 0.0024 x + 0.007467 y = -0.0643$$
We get:
$$0.007467 y - 0.0024 \left(2.4 y + 22.186\right) = -0.0643$$
$$0.001707 y - 0.0532464 = -0.0643$$
We move the free summand -0.0532464000000000 from the left part to the right part performing the sign change
$$0.001707 y = -0.0643 + 0.0532464$$
$$0.001707 y = -0.0110536$$
Let's divide both parts of the equation by the multiplier of y
$$\frac{0.001707 y}{0.001707} = - \frac{0.0110536}{0.001707}$$
$$1 y = -6.47545401288811$$
Because
$$x = 2.4 y + 22.186$$
then
$$x = \left(-6.47545401288811\right) 2.4 + 22.186$$
$$x = 6.64491036906854$$
The answer:
$$x = 6.64491036906854$$
$$1 y = -6.47545401288811$$
Rapid solution
$$x_{1} = 6.64491036906854$$
=
$$6.64491036906854$$
=
$$y_{1} = -6.47545401288811$$
=
$$-6.47545401288811$$
=
=
$$6.64491036906854$$
=
6.64491036906854
$$y_{1} = -6.47545401288811$$
=
$$-6.47545401288811$$
=
-6.47545401288811
Gaussian elimination
Given the system of equations
$$\frac{x}{1000} - 0.0024 y = 0.022186$$
$$- 0.0024 x + 0.007467 y = -0.0643$$
We give the system of equations to the canonical form
$$\frac{x}{1000} - 0.0024 y = 0.022186$$
$$- 0.0024 x + 0.007467 y = -0.0643$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}0 & 0 & 0\\0 & 0 & - \frac{1}{10}\end{matrix}\right]$$
It is almost ready, all we have to do is to find variables, solving the elementary equations:
$$0 - 0 = 0$$
$$0 + 1/10 = 0$$
We get the answer:
$$\frac{x}{1000} - 0.0024 y = 0.022186$$
$$- 0.0024 x + 0.007467 y = -0.0643$$
We give the system of equations to the canonical form
$$\frac{x}{1000} - 0.0024 y = 0.022186$$
$$- 0.0024 x + 0.007467 y = -0.0643$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}0 & 0 & 0\\0 & 0 & - \frac{1}{10}\end{matrix}\right]$$
It is almost ready, all we have to do is to find variables, solving the elementary equations:
$$0 - 0 = 0$$
$$0 + 1/10 = 0$$
We get the answer:
Cramer's rule
$$\frac{x}{1000} - 0.0024 y = 0.022186$$
$$- 0.0024 x + 0.007467 y = -0.0643$$
We give the system of equations to the canonical form
$$\frac{x}{1000} - 0.0024 y = 0.022186$$
$$- 0.0024 x + 0.007467 y = -0.0643$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}0.001 x_{1} - 0.0024 x_{2}\\- 0.0024 x_{1} + 0.007467 x_{2}\end{matrix}\right] = \left[\begin{matrix}0.022186\\-0.0643\end{matrix}\right]$$
- 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:
$$A = \operatorname{det}{\left(\left[\begin{matrix}0.001 & -0.0024\\-0.0024 & 0.007467\end{matrix}\right] \right)} = 1.707 \cdot 10^{-6}$$
, 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 )
$$x_{1} = 585823.081429408 \operatorname{det}{\left(\left[\begin{matrix}0.022186 & -0.0024\\-0.0643 & 0.007467\end{matrix}\right] \right)} = 6.64491036906855$$
$$x_{2} = 585823.081429408 \operatorname{det}{\left(\left[\begin{matrix}0.001 & 0.022186\\-0.0024 & -0.0643\end{matrix}\right] \right)} = -6.4754540128881$$
$$- 0.0024 x + 0.007467 y = -0.0643$$
We give the system of equations to the canonical form
$$\frac{x}{1000} - 0.0024 y = 0.022186$$
$$- 0.0024 x + 0.007467 y = -0.0643$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}0.001 x_{1} - 0.0024 x_{2}\\- 0.0024 x_{1} + 0.007467 x_{2}\end{matrix}\right] = \left[\begin{matrix}0.022186\\-0.0643\end{matrix}\right]$$
- 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:
$$A = \operatorname{det}{\left(\left[\begin{matrix}0.001 & -0.0024\\-0.0024 & 0.007467\end{matrix}\right] \right)} = 1.707 \cdot 10^{-6}$$
, 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 )
$$x_{1} = 585823.081429408 \operatorname{det}{\left(\left[\begin{matrix}0.022186 & -0.0024\\-0.0643 & 0.007467\end{matrix}\right] \right)} = 6.64491036906855$$
$$x_{2} = 585823.081429408 \operatorname{det}{\left(\left[\begin{matrix}0.001 & 0.022186\\-0.0024 & -0.0643\end{matrix}\right] \right)} = -6.4754540128881$$
Numerical answer
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x1 = 6.644910369068552 y1 = -6.475454012888104
x1 = 6.644910369068552 y1 = -6.475454012888104