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Matemáticas 3 · Tema 5

Determinantes 2×2

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Ejemplo

{5x−3y=212x+y=4\begin{cases} 5x - 3y = 21 \\ 2x + y = 4 \end{cases}
Calculamos xx
x=∣21−341∣∣5−321∣=21+125+6=3311x = \dfrac{\begin{vmatrix} 21 & -3 \\ 4 & 1 \end{vmatrix}}{\begin{vmatrix} 5 & -3 \\ 2 & 1 \end{vmatrix}} = \dfrac{21 + 12}{5 + 6} = \dfrac{33}{11} x=3x = 3
Calculamos yy
y=∣52124∣∣5−321∣=20−425+6=−2211y = \dfrac{\begin{vmatrix} 5 & 21 \\ 2 & 4 \end{vmatrix}}{\begin{vmatrix} 5 & -3 \\ 2 & 1 \end{vmatrix}} = \dfrac{20 - 42}{5 + 6} = \dfrac{-22}{11} y=−2y = -2

Comprobación

5(3)−3(−2)=2115+6=2121=21\begin{aligned} 5(3) - 3(-2) &= 21 \\ 15 + 6 &= 21 \\ 21 &= 21 \end{aligned}
2(3)+(−2)=46−2=44=4\begin{aligned} 2(3) + (-2) &= 4 \\ 6 - 2 &= 4 \\ 4 &= 4 \end{aligned}

Ejercicios

Ejercicio 1
{2x+4y=−103x−2y=−3\begin{cases} 2x + 4y = -10 \\ 3x - 2y = -3 \end{cases}
Ejercicio 2
{5x−2y=43x−4y=−6\begin{cases} 5x - 2y = 4 \\ 3x - 4y = -6 \end{cases}
Ejercicio 3
{x+3y=12x−5y=−9\begin{cases} x + 3y = 1 \\ 2x - 5y = -9 \end{cases}
Ejercicio 4
{2x+3y=−2−2x−6y=−4\begin{cases} 2x + 3y = -2 \\ -2x - 6y = -4 \end{cases}

Respuestas

▼
Ejercicio 1
{2x+4y=−103x−2y=−3\begin{cases} 2x + 4y = -10 \\ 3x - 2y = -3 \end{cases}
Calculamos xx
x=∣−104−3−2∣∣243−2∣=20+12−4−12=32−16x = \dfrac{\begin{vmatrix} -10 & 4 \\ -3 & -2 \end{vmatrix}}{\begin{vmatrix} 2 & 4 \\ 3 & -2 \end{vmatrix}} = \dfrac{20 + 12}{-4 - 12} = \dfrac{32}{-16} x=−2x = -2
Calculamos yy
y=∣2−103−3∣∣243−2∣=−6+30−4−12=24−16y = \dfrac{\begin{vmatrix} 2 & -10 \\ 3 & -3 \end{vmatrix}}{\begin{vmatrix} 2 & 4 \\ 3 & -2 \end{vmatrix}} = \dfrac{-6 + 30}{-4 - 12} = \dfrac{24}{-16} y=−1.5y = -1.5

Comprobación

2(−2)+4(−1.5)=−10−4−6=−10−10=−10\begin{aligned} 2(-2) + 4(-1.5) &= -10 \\ -4 - 6 &= -10 \\ -10 &= -10 \end{aligned}
3(−2)−2(−1.5)=−3−6+3=−3−3=−3\begin{aligned} 3(-2) - 2(-1.5) &= -3 \\ -6 + 3 &= -3 \\ -3 &= -3 \end{aligned}
Ejercicio 2
{5x−2y=43x−4y=−6\begin{cases} 5x - 2y = 4 \\ 3x - 4y = -6 \end{cases}
Calculamos xx
x=∣4−2−6−4∣∣5−23−4∣=−16−12−20+6=−28−14x = \dfrac{\begin{vmatrix} 4 & -2 \\ -6 & -4 \end{vmatrix}}{\begin{vmatrix} 5 & -2 \\ 3 & -4 \end{vmatrix}} = \dfrac{-16 - 12}{-20 + 6} = \dfrac{-28}{-14} x=2x = 2
Calculamos yy
y=∣543−6∣∣5−23−4∣=−30−12−20+6=−42−14y = \dfrac{\begin{vmatrix} 5 & 4 \\ 3 & -6 \end{vmatrix}}{\begin{vmatrix} 5 & -2 \\ 3 & -4 \end{vmatrix}} = \dfrac{-30 - 12}{-20 + 6} = \dfrac{-42}{-14} y=3y = 3

Comprobación

5(2)−2(3)=410−6=44=4\begin{aligned} 5(2) - 2(3) &= 4 \\ 10 - 6 &= 4 \\ 4 &= 4 \end{aligned}
3(2)−4(3)=−66−12=−6−6=−6\begin{aligned} 3(2) - 4(3) &= -6 \\ 6 - 12 &= -6 \\ -6 &= -6 \end{aligned}
Ejercicio 3
{x+3y=12x−5y=−9\begin{cases} x + 3y = 1 \\ 2x - 5y = -9 \end{cases}
Calculamos xx
x=∣13−9−5∣∣132−5∣=−5+27−5−6=22−11x = \dfrac{\begin{vmatrix} 1 & 3 \\ -9 & -5 \end{vmatrix}}{\begin{vmatrix} 1 & 3 \\ 2 & -5 \end{vmatrix}} = \dfrac{-5 + 27}{-5 - 6} = \dfrac{22}{-11} x=−2x = -2
Calculamos yy
y=∣112−9∣∣132−5∣=−9−2−5−6=−11−11y = \dfrac{\begin{vmatrix} 1 & 1 \\ 2 & -9 \end{vmatrix}}{\begin{vmatrix} 1 & 3 \\ 2 & -5 \end{vmatrix}} = \dfrac{-9 - 2}{-5 - 6} = \dfrac{-11}{-11} y=1y = 1

Comprobación

(−2)+3(1)=1−2+3=11=1\begin{aligned} (-2) + 3(1) &= 1 \\ -2 + 3 &= 1 \\ 1 &= 1 \end{aligned}
2(−2)−5(1)=−9−4−5=−9−9=−9\begin{aligned} 2(-2) - 5(1) &= -9 \\ -4 - 5 &= -9 \\ -9 &= -9 \end{aligned}
Ejercicio 4
{2x+3y=−2−2x−6y=−4\begin{cases} 2x + 3y = -2 \\ -2x - 6y = -4 \end{cases}
Calculamos xx
x=∣−23−4−6∣∣23−2−6∣=12+12−12+6=24−6x = \dfrac{\begin{vmatrix} -2 & 3 \\ -4 & -6 \end{vmatrix}}{\begin{vmatrix} 2 & 3 \\ -2 & -6 \end{vmatrix}} = \dfrac{12 + 12}{-12 + 6} = \dfrac{24}{-6} x=−4x = -4
Calculamos yy
y=∣2−2−2−4∣∣23−2−6∣=−8−4−12+6=−12−6y = \dfrac{\begin{vmatrix} 2 & -2 \\ -2 & -4 \end{vmatrix}}{\begin{vmatrix} 2 & 3 \\ -2 & -6 \end{vmatrix}} = \dfrac{-8 - 4}{-12 + 6} = \dfrac{-12}{-6} y=2y = 2

Comprobación

2(−4)+3(2)=−2−8+6=−2−2=−2\begin{aligned} 2(-4) + 3(2) &= -2 \\ -8 + 6 &= -2 \\ -2 &= -2 \end{aligned}
−2(−4)−6(2)=−48−12=−4−4=−4\begin{aligned} -2(-4) - 6(2) &= -4 \\ 8 - 12 &= -4 \\ -4 &= -4 \end{aligned}

Ejercicios extra

▼
Bloque 1
{3x+2y=4x−y=3\begin{cases} 3x + 2y = 4 \\ x - y = 3 \end{cases}
{4x−y=22x+3y=8\begin{cases} 4x - y = 2 \\ 2x + 3y = 8 \end{cases}
{5x+2y=−43x−4y=−18\begin{cases} 5x + 2y = -4 \\ 3x - 4y = -18 \end{cases}
{2x−3y=94x+y=11\begin{cases} 2x - 3y = 9 \\ 4x + y = 11 \end{cases}
{x+4y=43x−2y=−16\begin{cases} x + 4y = 4 \\ 3x - 2y = -16 \end{cases}
Bloque 2
{6x+y=92x−5y=−13\begin{cases} 6x + y = 9 \\ 2x - 5y = -13 \end{cases}
{3x−2y=−45x+4y=30\begin{cases} 3x - 2y = -4 \\ 5x + 4y = 30 \end{cases}
{2x+7y=14x−y=−13\begin{cases} 2x + 7y = 1 \\ 4x - y = -13 \end{cases}
{5x−3y=3x+2y=11\begin{cases} 5x - 3y = 3 \\ x + 2y = 11 \end{cases}
{4x+3y=−106x−5y=4\begin{cases} 4x + 3y = -10 \\ 6x - 5y = 4 \end{cases}
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