Question

find the value of x and y from the given value of determinants ​

Answer 1

On solving :

-3y - (-4x) = -11

=> 4x - 3y = -11 ...(1) And

2x - 5y = -9 ...(2)

On multiplying by 2 and 4 in equation (1) and (2) respectively :

8x - 6y = -22 ... (3)

8x - 20y = -36 ...(4)

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On subtracting :

14y = 14

=> y = 1

On putting the value of y in equation (1) :

4x - 3(1) = -11

=> 4x = -11+3

=> 4x = -8

=> x = -2

So, Required values : x = -2 & y = 1

I hope it will be helpful for you ☺

Answer 2

Question:

Find the values of x and y from the given values of determinants.

$\left|\begin{array}{cc}\sf\:-\:3 & \sf\:x\\\sf\:-\:4 & \sf\:y\end{array}\right|\sf\:=\:-\:11\:\:,\:\:\left|\begin{array}{cc}\sf\:x & \sf\:y\\\sf\:5 & \sf\:2\end{array}\right|\sf\:=\:-\:9$

Answer:

The value of x is - 2.

The value of y is 1.

Step-by-step-explanation:

We have given two determinants.

We have to find the value of x and y in those determinants.

$\left|\begin{array}{cc}\sf\:-\:3 & \sf\:x\\\sf\:-\:4 & \sf\:y\end{array}\right|\sf\:=\:-\:11\\\\\implies\sf\:-\:3\:\times\:y\:-\:x\:\times\:(\:-\:4\:)\:=\:-\:11\\\\\implies\sf\:-\:3y\:+\:4x\:=\:-\:11\\\\\implies\sf\:4x\:-\:3y\:=\:-\:11\:\:\:-\:-\:-\:(\:1\:)$

Now,

$\left|\begin{array}{cc}\sf\:x & \sf\:y\\\sf\:5 & \sf\:2\end{array}\right|\sf\:=\:-\:9\\\\\implies\sf\:x\:\times\:2\:-\:y\:\times\:5\:=\:-\:9\\\\\implies\sf\:2x\:-\:5y\:=\:-\:9\:\:\:-\:-\:-\:(\:2\:)\\\\\implies\sf\:4x\:-\:10y\:=\:-\:18\:\:\:-\:-\:(\:3\:)\:[\:Multiplying\:by\:2\:]$

Now, by subtracting equation ( 3 ) from equation ( 1 ), we get,

$\sf\:\cancel{4x}\:-\:3y\:=\:-\:11\:\:\:-\:-\:-\:(\:1\:)\\\\\underline{\sf\:\cancel{4x}\:-\:10y\:=\:-\:18}\sf\:\:\:-\:-\:(\:2\:)\\\\\implies\sf\:\cancel{-}\:7y\:=\:\cancel{-}\:7\\\\\implies\sf\:y\:=\:\cancel{\frac{7}{7}}\\\\\implies\boxed{\red{\sf\:y\:=\:1}}$

By substituting y = 1 in equation ( 2 ), we get,

$\sf\:2x\:-\:5y\:=\:-\:9\:\:\:-\:-\:-\:(\:2\:)\\\\\implies\sf\:2x\:-\:5\:\times\:1\:=\:-\:9\\\\\implies\sf\:2x\:-\:5\:=\:-\:9\\\\\implies\sf\:2x\:=\:-\:9\:+\:5\\\\\implies\sf\:2x\:=\:-\:4\\\\\implies\sf\:x\:=\:-\:\cancel{\frac{4}{2}}\\\\\implies\boxed{\red{\sf\:x\:=\:-\:2}}$

Additional Information:

Determinant Method ( Carmer's Rule ):

1. Determinant method is one of the methods of solving simultaneous equations.

2. This method was introduced by a mathematician Gabriel Cramer. So, it is also known as Carmer's Rule.

3. It is based on determinants.

4. The constant term of given linear equation is transferred to right hand side. Therefore, the general form of Carmer's rule for simultaneous equations is

ax + by = c

Where, a, b, c are real numbers and

a ≠ 0, b ≠ 0.

5. First, determinant D is calculated and then $\sf\:{D}_{x}\:and\:{D}_{y}$ are calculated step-by-step.

6. By using Carmer's rule values of x and y ( the variables used in the given equations ) are calculated.

7. Carmer's rule is as follows:

$\boxed{\red{\sf\:x =\frac{D_x}{D}}}\:\:\:\&\:\:\:\boxed{\red{\sf\:y =\frac{D_y}{D}}}$