Mister Exam
−35x+10y=78; 21x+5y=46
The solution
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-35*x + 10*y = 78
$$- 35 x + 10 y = 78$$
21*x + 5*y = 46
$$21 x + 5 y = 46$$
21*x + 5*y = 46
Detail solution
Given the system of equations
$$- 35 x + 10 y = 78$$
$$21 x + 5 y = 46$$
Let's express from equation 1 x
$$- 35 x + 10 y = 78$$
Let's move the summand with the variable y from the left part to the right part performing the sign change
$$- 35 x = 78 - 10 y$$
$$- 35 x = 78 - 10 y$$
Let's divide both parts of the equation by the multiplier of x
$$\frac{\left(-1\right) 35 x}{-35} = \frac{78 - 10 y}{-35}$$
$$x = \frac{2 y}{7} - \frac{78}{35}$$
Let's try the obtained element x to 2-th equation
$$21 x + 5 y = 46$$
We get:
$$5 y + 21 \left(\frac{2 y}{7} - \frac{78}{35}\right) = 46$$
$$11 y - \frac{234}{5} = 46$$
We move the free summand -234/5 from the left part to the right part performing the sign change
$$11 y = 46 + \frac{234}{5}$$
$$11 y = \frac{464}{5}$$
Let's divide both parts of the equation by the multiplier of y
$$\frac{11 y}{11} = \frac{464}{5 \cdot 11}$$
$$y = \frac{464}{55}$$
Because
$$x = \frac{2 y}{7} - \frac{78}{35}$$
then
$$x = - \frac{78}{35} + \frac{2 \cdot 464}{7 \cdot 55}$$
$$x = \frac{2}{11}$$
The answer:
$$x = \frac{2}{11}$$
$$y = \frac{464}{55}$$
$$- 35 x + 10 y = 78$$
$$21 x + 5 y = 46$$
Let's express from equation 1 x
$$- 35 x + 10 y = 78$$
Let's move the summand with the variable y from the left part to the right part performing the sign change
$$- 35 x = 78 - 10 y$$
$$- 35 x = 78 - 10 y$$
Let's divide both parts of the equation by the multiplier of x
$$\frac{\left(-1\right) 35 x}{-35} = \frac{78 - 10 y}{-35}$$
$$x = \frac{2 y}{7} - \frac{78}{35}$$
Let's try the obtained element x to 2-th equation
$$21 x + 5 y = 46$$
We get:
$$5 y + 21 \left(\frac{2 y}{7} - \frac{78}{35}\right) = 46$$
$$11 y - \frac{234}{5} = 46$$
We move the free summand -234/5 from the left part to the right part performing the sign change
$$11 y = 46 + \frac{234}{5}$$
$$11 y = \frac{464}{5}$$
Let's divide both parts of the equation by the multiplier of y
$$\frac{11 y}{11} = \frac{464}{5 \cdot 11}$$
$$y = \frac{464}{55}$$
Because
$$x = \frac{2 y}{7} - \frac{78}{35}$$
then
$$x = - \frac{78}{35} + \frac{2 \cdot 464}{7 \cdot 55}$$
$$x = \frac{2}{11}$$
The answer:
$$x = \frac{2}{11}$$
$$y = \frac{464}{55}$$
Rapid solution
$$x_{1} = \frac{2}{11}$$
=
$$\frac{2}{11}$$
=
$$y_{1} = \frac{464}{55}$$
=
$$\frac{464}{55}$$
=
=
$$\frac{2}{11}$$
=
0.181818181818182
$$y_{1} = \frac{464}{55}$$
=
$$\frac{464}{55}$$
=
8.43636363636364
Gaussian elimination
Given the system of equations
$$- 35 x + 10 y = 78$$
$$21 x + 5 y = 46$$
We give the system of equations to the canonical form
$$- 35 x + 10 y = 78$$
$$21 x + 5 y = 46$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}-35 & 10 & 78\\21 & 5 & 46\end{matrix}\right]$$
In 1 -th column
$$\left[\begin{matrix}-35\\21\end{matrix}\right]$$
let’s convert all the elements, except
1 -th element into zero.
- To do this, let’s take 1 -th line
$$\left[\begin{matrix}-35 & 10 & 78\end{matrix}\right]$$
,
and subtract it from other lines:
From 2 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}21 - - -21 & 5 - \frac{\left(-3\right) 10}{5} & 46 - \frac{\left(-3\right) 78}{5}\end{matrix}\right] = \left[\begin{matrix}0 & 11 & \frac{464}{5}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}-35 & 10 & 78\\0 & 11 & \frac{464}{5}\end{matrix}\right]$$
In 2 -th column
$$\left[\begin{matrix}10\\11\end{matrix}\right]$$
let’s convert all the elements, except
2 -th element into zero.
- To do this, let’s take 2 -th line
$$\left[\begin{matrix}0 & 11 & \frac{464}{5}\end{matrix}\right]$$
,
and subtract it from other lines:
From 1 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}-35 - \frac{0 \cdot 10}{11} & 10 - \frac{10 \cdot 11}{11} & 78 - \frac{10 \cdot 464}{5 \cdot 11}\end{matrix}\right] = \left[\begin{matrix}-35 & 0 & - \frac{70}{11}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}-35 & 0 & - \frac{70}{11}\\0 & 11 & \frac{464}{5}\end{matrix}\right]$$
It is almost ready, all we have to do is to find variables, solving the elementary equations:
$$\frac{70}{11} - 35 x_{1} = 0$$
$$11 x_{2} - \frac{464}{5} = 0$$
We get the answer:
$$x_{1} = \frac{2}{11}$$
$$x_{2} = \frac{464}{55}$$
$$- 35 x + 10 y = 78$$
$$21 x + 5 y = 46$$
We give the system of equations to the canonical form
$$- 35 x + 10 y = 78$$
$$21 x + 5 y = 46$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}-35 & 10 & 78\\21 & 5 & 46\end{matrix}\right]$$
In 1 -th column
$$\left[\begin{matrix}-35\\21\end{matrix}\right]$$
let’s convert all the elements, except
1 -th element into zero.
- To do this, let’s take 1 -th line
$$\left[\begin{matrix}-35 & 10 & 78\end{matrix}\right]$$
,
and subtract it from other lines:
From 2 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}21 - - -21 & 5 - \frac{\left(-3\right) 10}{5} & 46 - \frac{\left(-3\right) 78}{5}\end{matrix}\right] = \left[\begin{matrix}0 & 11 & \frac{464}{5}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}-35 & 10 & 78\\0 & 11 & \frac{464}{5}\end{matrix}\right]$$
In 2 -th column
$$\left[\begin{matrix}10\\11\end{matrix}\right]$$
let’s convert all the elements, except
2 -th element into zero.
- To do this, let’s take 2 -th line
$$\left[\begin{matrix}0 & 11 & \frac{464}{5}\end{matrix}\right]$$
,
and subtract it from other lines:
From 1 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}-35 - \frac{0 \cdot 10}{11} & 10 - \frac{10 \cdot 11}{11} & 78 - \frac{10 \cdot 464}{5 \cdot 11}\end{matrix}\right] = \left[\begin{matrix}-35 & 0 & - \frac{70}{11}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}-35 & 0 & - \frac{70}{11}\\0 & 11 & \frac{464}{5}\end{matrix}\right]$$
It is almost ready, all we have to do is to find variables, solving the elementary equations:
$$\frac{70}{11} - 35 x_{1} = 0$$
$$11 x_{2} - \frac{464}{5} = 0$$
We get the answer:
$$x_{1} = \frac{2}{11}$$
$$x_{2} = \frac{464}{55}$$
Cramer's rule
$$- 35 x + 10 y = 78$$
$$21 x + 5 y = 46$$
We give the system of equations to the canonical form
$$- 35 x + 10 y = 78$$
$$21 x + 5 y = 46$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}- 35 x_{1} + 10 x_{2}\\21 x_{1} + 5 x_{2}\end{matrix}\right] = \left[\begin{matrix}78\\46\end{matrix}\right]$$
- 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:
$$A = \operatorname{det}{\left(\left[\begin{matrix}-35 & 10\\21 & 5\end{matrix}\right] \right)} = -385$$
, 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 )
$$x_{1} = - \frac{\operatorname{det}{\left(\left[\begin{matrix}78 & 10\\46 & 5\end{matrix}\right] \right)}}{385} = \frac{2}{11}$$
$$x_{2} = - \frac{\operatorname{det}{\left(\left[\begin{matrix}-35 & 78\\21 & 46\end{matrix}\right] \right)}}{385} = \frac{464}{55}$$
$$21 x + 5 y = 46$$
We give the system of equations to the canonical form
$$- 35 x + 10 y = 78$$
$$21 x + 5 y = 46$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}- 35 x_{1} + 10 x_{2}\\21 x_{1} + 5 x_{2}\end{matrix}\right] = \left[\begin{matrix}78\\46\end{matrix}\right]$$
- 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:
$$A = \operatorname{det}{\left(\left[\begin{matrix}-35 & 10\\21 & 5\end{matrix}\right] \right)} = -385$$
, 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 )
$$x_{1} = - \frac{\operatorname{det}{\left(\left[\begin{matrix}78 & 10\\46 & 5\end{matrix}\right] \right)}}{385} = \frac{2}{11}$$
$$x_{2} = - \frac{\operatorname{det}{\left(\left[\begin{matrix}-35 & 78\\21 & 46\end{matrix}\right] \right)}}{385} = \frac{464}{55}$$
Numerical answer
[src]
x1 = 0.1818181818181818 y1 = 8.436363636363636
x1 = 0.1818181818181818 y1 = 8.436363636363636