Rozwiąż układy równań metodą wyznaczników
a) 5x + 2y = 10
15x + 6y = 15
b) 10x - 11y = 23
33y - 30x = 25
c) 2y - 3x = 4
3y - 2x = 5
d) 8x - 7y = 12
22y - 21x = 36
a) 5x + 2y = 10
15x + 6y = 15
b) 10x - 11y = 23
33y - 30x = 25
c) 2y - 3x = 4
3y - 2x = 5
d) 8x - 7y = 12
22y - 21x = 36
Mathēmatikós
![a) \\ \\ \begin{cases} 5x+2y = 10 \\ 15x + 6y = 15 \end{cases} \\ \\ \\ W= \left[\begin{array}{ccc}5&2\\15&6\\\end{array}\right] \\ \\ \\ W_{x} = \left[\begin{array}{ccc}10&2\\15&6\\\end{array}\right] \\ \\ \\ W_{y} = \left[\begin{array}{ccc}5&10\\15&15\\\end{array}\right] \\ \\ det(W) = 5*6 - 2*15 = 30-30 = 0 \\ \\ det(W_{x}) =10*6 - 15*2 = 60 - 30 = 30 \\ \\ det(W_{y}) = 5*15 - 15*10 = 75 - 150 = -75](https://tex.z-dn.net/?f=a%29+%5C%5C++%5C%5C+%5Cbegin%7Bcases%7D+5x%2B2y+%3D+10+%5C%5C+15x+%2B+6y+%3D+15+%5Cend%7Bcases%7D++%5C%5C++%5C%5C++%5C%5C+W%3D+%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D5%262%5C%5C15%266%5C%5C%5Cend%7Barray%7D%5Cright%5D++%5C%5C++%5C%5C+%5C%5C+W_%7Bx%7D+%3D++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D10%262%5C%5C15%266%5C%5C%5Cend%7Barray%7D%5Cright%5D++%5C%5C++%5C%5C++%5C%5C+W_%7By%7D+%3D++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D5%2610%5C%5C15%2615%5C%5C%5Cend%7Barray%7D%5Cright%5D++++%5C%5C++%5C%5C++det%28W%29+%3D+5%2A6+-+2%2A15+%3D+30-30+%3D+0+%5C%5C++%5C%5C+det%28W_%7Bx%7D%29+%3D10%2A6+-+15%2A2+%3D+60+-+30+%3D+30+%5C%5C++%5C%5C+det%28W_%7By%7D%29+%3D+5%2A15++-+15%2A10+%3D+75+-+150+%3D+-75 "a) \\ \\ \begin{cases} 5x+2y = 10 \\ 15x + 6y = 15 \end{cases} \\ \\ \\ W= \left[\begin{array}{ccc}5&2\\15&6\\\end{array}\right] \\ \\ \\ W_{x} = \left[\begin{array}{ccc}10&2\\15&6\\\end{array}\right] \\ \\ \\ W_{y} = \left[\begin{array}{ccc}5&10\\15&15\\\end{array}\right] \\ \\ det(W) = 5*6 - 2*15 = 30-30 = 0 \\ \\ det(W_{x}) =10*6 - 15*2 = 60 - 30 = 30 \\ \\ det(W_{y}) = 5*15 - 15*10 = 75 - 150 = -75")
![b) \\ \\ \begin{cases} 10x - 11y = 23 \\ -30x + 33y = 25 \end{cases} \\ \\ \\W= \left[\begin{array}{ccc}10&-11\\-30&33\\\end{array}\right] \\ \\ \\ W_{x} = \left[\begin{array}{ccc}23&-11\\25&33\\\end{array}\right] \\ \\ \\ W_{y} =\left[\begin{array}{ccc}10&23\\-30&33\\\end{array}\right] \\ \\ \\ det(W) = 10*33 - (-30) * (-11) = 330 - 330 = 0 \\ \\ det(W_{x}) = 23 * 33 - 25 * (-11) = 759 + 275 = 1034 \\ \\ det(W_{y}) = 10 * 33 - (-30) * 23 = 330 + 690 =1020 \\ \\](https://tex.z-dn.net/?f=b%29+%5C%5C++%5C%5C++%5Cbegin%7Bcases%7D+10x+-+11y+%3D+23+%5C%5C+-30x+%2B+33y+%3D+25+%5Cend%7Bcases%7D+%5C%5C++%5C%5C++%5C%5CW%3D+%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D10%26-11%5C%5C-30%2633%5C%5C%5Cend%7Barray%7D%5Cright%5D++%5C%5C++%5C%5C++%5C%5C+W_%7Bx%7D+%3D+%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D23%26-11%5C%5C25%2633%5C%5C%5Cend%7Barray%7D%5Cright%5D+%5C%5C++%5C%5C++%5C%5C+W_%7By%7D+%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D10%2623%5C%5C-30%2633%5C%5C%5Cend%7Barray%7D%5Cright%5D++%5C%5C++%5C%5C++%5C%5C+det%28W%29+%3D+10%2A33+-+%28-30%29+%2A+%28-11%29+%3D+330+-+330+%3D+0+%5C%5C++%5C%5C+det%28W_%7Bx%7D%29+%3D+23+%2A+33+-+25+%2A+%28-11%29+%3D+759+%2B+275+%3D+1034+%5C%5C++%5C%5C+det%28W_%7By%7D%29+%3D+10+%2A+33+-+%28-30%29+%2A+23+%3D+330+%2B+690+%3D1020+%5C%5C++%5C%5C+ "b) \\ \\ \begin{cases} 10x - 11y = 23 \\ -30x + 33y = 25 \end{cases} \\ \\ \\W= \left[\begin{array}{ccc}10&-11\\-30&33\\\end{array}\right] \\ \\ \\ W_{x} = \left[\begin{array}{ccc}23&-11\\25&33\\\end{array}\right] \\ \\ \\ W_{y} =\left[\begin{array}{ccc}10&23\\-30&33\\\end{array}\right] \\ \\ \\ det(W) = 10*33 - (-30) * (-11) = 330 - 330 = 0 \\ \\ det(W_{x}) = 23 * 33 - 25 * (-11) = 759 + 275 = 1034 \\ \\ det(W_{y}) = 10 * 33 - (-30) * 23 = 330 + 690 =1020 \\ \\")
![c) \\ \\ \begin{cases}-3x+2y= 4 \\ -2x + 3y = 5 \end{cases} \\ \\ \\ W= \left[\begin{array}{ccc}-3&2\\-2&3\\\end{array}\right] \\ \\ \\ W_{x} = \left[\begin{array}{ccc}4&2\\5&3\\\end{array}\right] \\ \\ \\ W_{y} = \left[\begin{array}{ccc}-3&-2\\4&5\\\end{array}\right] \\ \\ det(W) = -3 * 3 -(-2) * 2 = -9 + 4 = -5 \\ \\ det(W_{x}) = 4*3 - 2*5 =12 - 10 = 2 \\ \\ det(W_{y}) = -3 * 5 - 4*(-2) = -15 + 8 = -7 \\ \\](https://tex.z-dn.net/?f=c%29+%5C%5C++%5C%5C+%5Cbegin%7Bcases%7D-3x%2B2y%3D+4++%5C%5C+-2x+%2B+3y+%3D+5+%5Cend%7Bcases%7D++%5C%5C++%5C%5C++%5C%5C+W%3D+%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-3%262%5C%5C-2%263%5C%5C%5Cend%7Barray%7D%5Cright%5D++%5C%5C++%5C%5C++%5C%5C+W_%7Bx%7D+%3D++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D4%262%5C%5C5%263%5C%5C%5Cend%7Barray%7D%5Cright%5D+%5C%5C++%5C%5C++%5C%5C+W_%7By%7D+%3D++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-3%26-2%5C%5C4%265%5C%5C%5Cend%7Barray%7D%5Cright%5D++%5C%5C++%5C%5C++det%28W%29+%3D+-3+%2A+3+-%28-2%29+%2A+2+%3D+-9+%2B+4+%3D+-5+%5C%5C++%5C%5C+det%28W_%7Bx%7D%29+%3D+4%2A3+-+2%2A5+%3D12+-+10+%3D+2+%5C%5C++%5C%5C+det%28W_%7By%7D%29+%3D+-3+%2A+5+-+4%2A%28-2%29+%3D+-15+%2B+8+%3D+-7+%5C%5C++%5C%5C+ "c) \\ \\ \begin{cases}-3x+2y= 4 \\ -2x + 3y = 5 \end{cases} \\ \\ \\ W= \left[\begin{array}{ccc}-3&2\\-2&3\\\end{array}\right] \\ \\ \\ W_{x} = \left[\begin{array}{ccc}4&2\\5&3\\\end{array}\right] \\ \\ \\ W_{y} = \left[\begin{array}{ccc}-3&-2\\4&5\\\end{array}\right] \\ \\ det(W) = -3 * 3 -(-2) * 2 = -9 + 4 = -5 \\ \\ det(W_{x}) = 4*3 - 2*5 =12 - 10 = 2 \\ \\ det(W_{y}) = -3 * 5 - 4*(-2) = -15 + 8 = -7 \\ \\")
![d) \\ \\ \begin{cases} 8x - 7y = 12 \\ -21x + 22y = 36 \end{cases} \\ \\ \\ W=\left[\begin{array}{ccc}8&-7\\-21&22\\\end{array}\right] \\ \\ \\ W_{x} = \left[\begin{array}{ccc}12&-7\\36&22\\\end{array}\right] \\ \\ \\ W_{y} = \left[\begin{array}{ccc}8&12\\-21&36\\\end{array}\right] \\ \\ det(W) = 8*22 - (-21) * (-7) = 176 - 147 = 29 \\ \\ det(W_{x}) = 12 * 22 - 36* (-7) =264 + 252 = 516 \\ \\ det(W_{y}) = 8*36 - (-21) * 12 =288 +252 = 540](https://tex.z-dn.net/?f=d%29+%5C%5C++%5C%5C+%5Cbegin%7Bcases%7D+8x+-+7y+%3D+12+%5C%5C+-21x+%2B+22y+%3D+36+%5Cend%7Bcases%7D++%5C%5C++%5C%5C++%5C%5C+W%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D8%26-7%5C%5C-21%2622%5C%5C%5Cend%7Barray%7D%5Cright%5D++%5C%5C++%5C%5C++%5C%5C+W_%7Bx%7D+%3D+%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D12%26-7%5C%5C36%2622%5C%5C%5Cend%7Barray%7D%5Cright%5D++%5C%5C++%5C%5C++%5C%5C+W_%7By%7D+%3D+%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D8%2612%5C%5C-21%2636%5C%5C%5Cend%7Barray%7D%5Cright%5D++%5C%5C++%5C%5C+det%28W%29+%3D+8%2A22+-+%28-21%29+%2A+%28-7%29+%3D+176+-+147+%3D+29+%5C%5C++%5C%5C+det%28W_%7Bx%7D%29+%3D+12+%2A+22+-+36%2A+%28-7%29+%3D264+%2B++252+%3D++516+%5C%5C++%5C%5C+det%28W_%7By%7D%29+%3D+8%2A36+-+%28-21%29+%2A+12+%3D288+%2B252+%3D+540 "d) \\ \\ \begin{cases} 8x - 7y = 12 \\ -21x + 22y = 36 \end{cases} \\ \\ \\ W=\left[\begin{array}{ccc}8&-7\\-21&22\\\end{array}\right] \\ \\ \\ W_{x} = \left[\begin{array}{ccc}12&-7\\36&22\\\end{array}\right] \\ \\ \\ W_{y} = \left[\begin{array}{ccc}8&12\\-21&36\\\end{array}\right] \\ \\ det(W) = 8*22 - (-21) * (-7) = 176 - 147 = 29 \\ \\ det(W_{x}) = 12 * 22 - 36* (-7) =264 + 252 = 516 \\ \\ det(W_{y}) = 8*36 - (-21) * 12 =288 +252 = 540")
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