Question

Solve by using Crammer’s Rule: 2x + 3y =12 x – y = 1.

Answer 1

Question :-

Solve by using Crammer’s Rule: 2x + 3y =12 x – y = 1.

Solution :-

Given :

• 2x + 3y = 12
• x - y = 1

Need to find :

• find x and y by Crammer's rule.

By Crammer's rule,

D =

$\sf : \longmapsto{D = \begin{gathered} \sf \begin{bmatrix} 2 & 3 \\ 1 & - 1 \end{bmatrix} \end{gathered}} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf : \longmapsto{D =(2 \times - 1) - (1 \times 3)} \: \: \: \: \\ \\ \sf : \longmapsto{D = - 2 - 3} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf : \longmapsto{D = - 5} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

Dx =

$\sf : \longmapsto{Dx =</p><p>\begin{gathered} \sf \begin{bmatrix} 12 & 3 \\ 1 & - 1 \end{bmatrix} \end{gathered}} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf : \longmapsto{Dx} = (12 \times - 1) - (1 \times3) \: \\ \\ \sf : \longmapsto{Dx} = - 12 - 3 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf : \longmapsto{Dx} = - 15 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

Dy =

$\sf : \longmapsto{Dy =</p><p>\begin{gathered} \sf \begin{bmatrix} 2 & 12 \\ 1 & 1 \end{bmatrix} \end{gathered} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf : \longmapsto{Dy } =( 2 \times 1) - (1 \times 12) \\ \\ \sf : \longmapsto{Dy } = 2 - 12 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf : \longmapsto{Dy } = - 10 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

We know that,

$\sf : \longmapsto{x = \frac{Dx}{D}}$

Then,

$\sf : \longmapsto{x = \frac{ - 15}{ - 5}} \\ \\ \bf : \longmapsto \red{x =3} \: \: \:$

Also we know that,

$\sf : \longmapsto{y = \frac{Dy}{D}} \: \: \\ \\ \sf : \longmapsto{y = \frac{ - 10}{ - 5}} \\ \\ \bf : \longmapsto \red{y = 2} \: \:$

Answer :-

• x = 3
• y = 2

Additional Information :-

Let, Two equation are

• ax + by = e
• cx + dy = f

• D =

$\sf : \longmapsto{D = </p><p>\begin{gathered} \sf \begin{bmatrix} \sf a & \sf b\\ \sf c & \sf d \end{bmatrix} \end{gathered}}$

• Dx =

$\sf : \longmapsto{Dx = </p><p>\begin{gathered} \sf \begin{bmatrix} \sf e & \sf b\\ \sf f & \sf d \end{bmatrix} \end{gathered}}$

• Dy =

$\sf : \longmapsto{D y= </p><p>\begin{gathered} \sf \begin{bmatrix} \sf a & \sf e\\ \sf c & \sf f \end{bmatrix} \end{gathered}}$

• x =

$\sf:\longmapsto{x = \frac{Dx}{D}}$

• y =

$\sf:\longmapsto{y= \frac{Dy}{D}}$

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