Question

Solve the following equations by cramer's method.
4m-2n=-4; 4m+3n=16.

Answer 1

Answer:

The solution of the given simultaneous equations is ( m, n ) = ( 1, 4 ).

Step-by-step-explanation:

The given simultaneous equations are

$\sf\:4m\:-\:2n\:=\:-\:4\:\:\:\&\\\\\sf\:4m\:+\:3n\:=\:16$

$\sf\:4m\:-\:2n\:=\:-\:4\\\\\\\bullet\sf\:a_1\:=\:4\\\\\\\bullet\sf\:b_1\:=\:-\:2\\\\\\\bullet\sf\:c_1\:=\:-\:4\\\\\\\sf\:4m\:+\:3n\:=\:16\\\\\\\bullet\sf\:a_2\:=\:4\\\\\\\bullet\sf\:b_2\:=\:3\\\\\\\bullet\sf\:c_2\:=\:16$

Now, we know that,

$\sf\:D\:=\:\left|\begin{array}{cc}\sf\:a_1 & \sf\:b_1\\\\\sf\:a_2 & \sf\:b_2\end{array}\right|\\\\\\\implies\sf\:D\:=\:\left|\begin{array}{cc}\sf\:4 & \sf\:-\:2\\\\\sf\:4 & \sf\:3\end{array}\right|\\\\\\\implies\sf\:D\:=\:4\:\times\:3\:-\:(\:-\:2\:)\:\times\:4\\\\\\\implies\sf\:D\:=\:12\:+\:8\\\\\\\implies\pink{\sf\:D\:=\:20}$

Now,

$\sf\:D_x\:=\:\left|\begin{array}{cc}\sf\:c_1 & \sf\:b_1\\\\\sf\:c_2 & \sf\:b_2\end{array}\right|\\\\\\\implies\sf\:D_x\:=\:\left|\begin{array}{cc}\sf\:-\:4 & \sf\:-\:2\\\\\sf\:16 & \sf\:3\end{array}\right|\\\\\\\implies\sf\:D_x\:=\:(\:-\:4\:)\:\times\:3\:-\:(\:-\:2\:)\:\times\:16\\\\\\\implies\sf\:D_x\:=\:-\:12\:+\:32\\\\\\\implies\pink{\sf\:D_x\:=\:20}$

Now,

$\sf\:D_y\:=\:\left|\begin{array}{cc}\sf\:a_1 & \sf\:c_1\\\\\sf\:a_2 & \sf\:c_2\end{array}\right|\\\\\\\implies\sf\:D_y\:=\:\left|\begin{array}{cc}\sf\:4 & \sf\:-\:4\\\\\sf\:4 & \sf\:16\end{array}\right|\\\\\\\implies\sf\:D_y\:=\:4\:\times\:16\:-\:(\:-\:4\:)\:\times\:4\\\\\\\implies\sf\:D_y\:=\:64\:+\:16\\\\\\\implies\pink{\sf\:D_y\:=\:80}$

Now, by Carmer's rule,

$\sf\:m\:=\:\dfrac{D_x}{D}\\\\\\\implies\sf\:m\:=\:\cancel{\frac{20}{20}}\\\\\\\implies\boxed{\red{\sf\:m\:=\:1}}\\\\\\\sf\:n\:=\:\dfrac{D_y}{D}\\\\\\\implies\sf\:n\:=\:\cancel{\frac{80}{20}}\\\\\\\implies\boxed{\red{\sf\:n\:=\:4}}$

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:

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