Question

Solve the following simultaneous equations using Cramer's rule : 4x + 3y = 5; 6x + 5y = 9.​

Answer 1

Question:

Solve the following simultaneous equations using Cramer's rule :

• 4x + 3y = 5
• 6x + 5y = 9

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Answer:

The value of 'x' and 'y' are – 1 and 3 respectively.

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Step-by-step explaination:

The pair of equations given are :

• 4x + 3y = 5
• 6x + 5y = 9

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Let's compare these equations to ax + by = c :

• a₁ = 4
• a₂ = 6
• b₁ = 3
• b₂ = 5
• c₁ = 5
• c₂ = 9

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Let's find D firstly :

$\tt{:\implies~~ D=\left[\begin{array}{cc}a_1&b_1\\ a_2&b_2\end{array}\right]}$

$\tt{:\implies~~D= \left[\begin{array}{cc}4&3\\6&5\end{array}\right]}$

$\tt:\implies~~D= (4\times 5)-(6\times 3)$

$\tt:\implies~~ D=20-18$

$\tt:\implies~~ D=2$

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Now, finding Dx :

$\tt{:\implies~~ D_x=\left[\begin{array}{cc}c_1&b_1\\c_2&b_2\end{array}\right]}$

$\tt{:\implies~~ D_x=\left[\begin{array}{cc}5&3\\9&5\end{array}\right]}$

$\tt:\implies~~ D_x=(5\times 5)-(9\times3)$

$\tt:\implies~~ D_x = 25-27$

$\tt:\implies~~ D_x=-2$

$\:$

Finally, finding Dy :

$\tt{:\implies~~ D_y=\left[\begin{array}{cc}a_1&c_1\\a_2&c_2\end{array}\right]}$

$\tt{:\implies~~ D_y=\left[\begin{array}{cc}4&5\\6&9\end{array}\right]}$

$\tt:\implies~~ D_y=(4\times 9)-(6\times5)$

$\tt:\implies~~ D_y=36-30$

$\tt:\implies~~ D_y=6$

$\:$

From Cramer's rule, we get :

$\tt:\implies~~ x=\dfrac{D_x}{D},~ y=\dfrac{D_y}{D}$

$\tt:\implies~~ x=\dfrac{-2}{2},~ y=\dfrac{6}{2}$

$\tt :\implies~~ (x=-1,~ y=3)$