Issolvi 20^2+left(1.5tright)^2-left(12+1.5tright)^2/60t=frac{34^2+{left(1.5tright)}^2-{left(30+1.5tright)}^2}{-102t}

$$\frac{ { 20 }^{ 2 } + { \left(1.5t \right) }^{ 2 } - { \left(12+1.5t \right) }^{ 2 } }{ 60t } = \frac{ { 34 }^{ 2 } + { \left(1.5t \right) }^{ 2 } - { \left(30+1.5t \right) }^{ 2 } }{ -102t }$$

Solvi għal t

$$t = \frac{32}{7} = 4\frac{4}{7} \approx 4.571428571$$

Passi biex Issolvi l-Ekwazzjoni Lineari

Il-varjabbli t ma jistax ikun ugwali għal 0 billi d-diviżjoni b'żero mhux iddefinit. Immultiplika ż-żewġ naħat tal-ekwazzjoni b'1020t, l-inqas denominatur komuni ta' 60t,-102t.

$$17\left(20^{2}+\left(1.5t\right)^{2}-\left(12+1.5t\right)^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Ikkalkula 20 bil-power ta' 2 u tikseb 400.

$$17\left(400+\left(1.5t\right)^{2}-\left(12+1.5t\right)^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Espandi \left(1.5t\right)^{2}.

$$17\left(400+1.5^{2}t^{2}-\left(12+1.5t\right)^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Ikkalkula 1.5 bil-power ta' 2 u tikseb 2.25.

$$17\left(400+2.25t^{2}-\left(12+1.5t\right)^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Uża teorema binomjali \left(a+b\right)^{2}=a^{2}+2ab+b^{2} biex tespandi \left(12+1.5t\right)^{2}.

$$17\left(400+2.25t^{2}-\left(144+36t+2.25t^{2}\right)\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Biex issib l-oppost ta' 144+36t+2.25t^{2}, sib l-oppost ta' kull terminu.

$$17\left(400+2.25t^{2}-144-36t-2.25t^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Naqqas 144 minn 400 biex tikseb 256.

$$17\left(256+2.25t^{2}-36t-2.25t^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Ikkombina 2.25t^{2} u -2.25t^{2} biex tikseb 0.

$$17\left(256-36t\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Uża l-propjetà distributtiva biex timmultiplika 17 b'256-36t.

$$4352-612t=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Ikkalkula 34 bil-power ta' 2 u tikseb 1156.

$$4352-612t=-10\left(1156+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Espandi \left(1.5t\right)^{2}.

$$4352-612t=-10\left(1156+1.5^{2}t^{2}-\left(30+1.5t\right)^{2}\right)$$

Ikkalkula 1.5 bil-power ta' 2 u tikseb 2.25.

$$4352-612t=-10\left(1156+2.25t^{2}-\left(30+1.5t\right)^{2}\right)$$

Uża teorema binomjali \left(a+b\right)^{2}=a^{2}+2ab+b^{2} biex tespandi \left(30+1.5t\right)^{2}.

$$4352-612t=-10\left(1156+2.25t^{2}-\left(900+90t+2.25t^{2}\right)\right)$$

Biex issib l-oppost ta' 900+90t+2.25t^{2}, sib l-oppost ta' kull terminu.

$$4352-612t=-10\left(1156+2.25t^{2}-900-90t-2.25t^{2}\right)$$

Naqqas 900 minn 1156 biex tikseb 256.

$$4352-612t=-10\left(256+2.25t^{2}-90t-2.25t^{2}\right)$$

Ikkombina 2.25t^{2} u -2.25t^{2} biex tikseb 0.

$$4352-612t=-10\left(256-90t\right)$$

Uża l-propjetà distributtiva biex timmultiplika -10 b'256-90t.

$$4352-612t=-2560+900t$$

Naqqas 900t miż-żewġ naħat.

$$4352-612t-900t=-2560$$

Ikkombina -612t u -900t biex tikseb -1512t.

$$4352-1512t=-2560$$

Naqqas 4352 miż-żewġ naħat.

$$-1512t=-2560-4352$$

Naqqas 4352 minn -2560 biex tikseb -6912.

$$-1512t=-6912$$

Iddividi ż-żewġ naħat b'-1512.

$$t=\frac{-6912}{-1512}$$

Naqqas il-frazzjoni \frac{-6912}{-1512} għat-termini l-aktar baxxi billi testratta u tikkanċella barra -216.

$$t=\frac{32}{7}$$