Tambayoyi Misali Game da Tsarin Daidaito Mai Layi da Rashin Daidaito
Tsarin lissafin layi da rashin daidaito muhimmin batu ne a fannin lissafi wanda ake amfani da shi sosai a fannoni daban-daban, kamar tattalin arziki, kimiyya, da injiniyanci. A cikin wannan labarin, za mu tattauna misalan matsalolin da suka shafi tsarin lissafin layi da rashin daidaito da kuma yadda za a magance su dalla-dalla.
Ma'anar Tsarin Daidaito Mai Layi
Tsarin lissafin layi ya ƙunshi lissafin layi biyu ko fiye waɗanda ke da alaƙa da juna. Misalai sune:
\[
\begin{cases}
2x + 3y = 5 \\
4x-y = 1
\ƙarshen
\]
Manufar warware wannan tsarin ita ce nemo ƙimar \(x\) da \(y\) waɗanda suka gamsar da daidaiton biyu a lokaci guda.
Hanyoyi don Magance Tsarin Lissafin Layi
Akwai hanyoyi da dama don warware tsarin lissafin layi, ciki har da:
1. Hanyar Sauyawa
2. Hanyar Kawarwa
3. Hanyar Matrix (Juyawa ko Gauss-Jordan)
Misali Tambaya ta 1: Hanyar Sauyawa
Bari mu warware tsarin mai zuwa ta amfani da hanyar maye gurbin:
\[
\begin{cases}
x + 2y = 10 \\
3x-y = 5
\ƙarshen
\]
Matakai:
1. Ware ɗaya daga cikin masu canji a cikin ɗaya daga cikin lissafin.
Daga lissafin farko, mun ware \(x\):
\[
x = 10 – 2y
\]
2. A maye gurbin furucin da aka samo a cikin wani lissafi.
Sauya \(x = 10 – 2y\) cikin lissafi na biyu:
\[
3(10 – 2y) – y = 5
\]
Warware don \(y\):
\[
Shekaru 30 – 6 – y = 5
\]
\[
Shekaru 30 – 7y = 5
\]
\[
-7y = -25
\]
\[
y = \frac{25}{7}
\]
3. Yi amfani da ƙimar da aka samo don nemo wasu masu canji.
Sauya \(y = \frac{25}{7}\) zuwa cikin bayanin \(x\):
\[
x = 10 – 2\left(\frac{25}{7}\right)
\]
\[
x = 10 – \frac{50}{7}
\]
\[
x = \frac{70}{7} – \frac{50}{7}
\]
\[
x = \frac{20}{7}
\]
Don haka, mafita ga tsarin sune \( x = \frac{20}{7} \) da \( y = \frac{25}{7} \).
Misali Tambaya ta 2: Hanyar Kawar da Mugunta
Na gaba, bari mu yi amfani da hanyar kawarwa don magance tsarin da ke tafe:
\[
\begin{cases}
2x + 3y = 12 \\
4x + 6y = 24
\ƙarshen
\]
A wannan yanayin, mun ga cewa lissafi na biyu sau ɗaya ne na lissafi na farko. Don rage tsarin, za mu iya ninka lissafi na farko da 2 sannan mu cire shi daga lissafi na biyu:
1. A ninka lissafin farko da 2:
\[
2(2x + 3y) = 2 \cdot 12
\]
\[
4x + 6y = 24
\]
2. Cire lissafin farko da aka ninka daga lissafin na biyu:
\[
(4x + 6y) – (4x + 6y) = 24 – 24
\]
\[
= 0 0
\]
Wannan yana ba da \(0 = 0\), wanda ke nuna cewa tsarin yana da mafita marasa iyaka kuma waɗannan daidaiton sun dogara ne akan su.
Misali Tambaya ta 3: Rashin daidaiton layi
Rashin daidaiton layi yana bin ƙa'idodi iri ɗaya da daidaiton layi, amma ya haɗa da alamun rashin daidaito kamar \(<, \leq, >, \geq\). Bari mu dubi wani misali mai sauƙi:
\[
\begin{cases}
3x – y < 7 \\ 2x + y \geq 4 \end{cases} \] Matakai: 1. Muna amfani da hanyar zane don tantance yankin mafita na wannan tsarin. Zana kowane rashin daidaito. 2. Maida rashin daidaito zuwa lissafi don tantance layin iyaka: Ga \(3x - y < 7\), layin iyaka shine \(3x - y = 7\)
