Question

Solve the following simultaneous equations using Cramer’s rule .
\[3x - y = 7\] and \[x + 4y = 11\]
A.\[x = 3,y = 2\]
B.\[x = 4,y = 5\]
C.\[x = 5,y = 8\]
D.\[x = 4,y = 4\]

Answer 1

Hint: In the given question , the Cramer’s Rule is a method that uses determinants to solve systems of equations that have the same number of equations as variables . In this method we calculate the values of \[x\] and \[y\] using the formula \[\dfrac{{{D_x}}}{D}\] and \[\dfrac{{{D_y}}}{D}\] respectively , where \[{D_x}\] is determinant obtained using the coefficient of \[y\] and constants of equations . \[{D_y}\] is the determinant obtained using the coefficient of \[x\] and constants of equations . \[D\] is the determinant obtained using the coefficients of \[x\] and \[y\] .

Complete step-by-step answer:
Given : \[3x - y = 7\] and \[x + 4y = 11\]
First we will calculate \[D\] , which is obtained using the coefficients of \[x\] and \[y\]. Therefore ,
$\left| \begin{gathered}
3&-1\\
1&4\\
\end{gathered} \right|$
Here , the first column consists of coefficients of \[x\] and the second column consists of coefficients of \[y\] .
Now solving the determinant we get ,
\[D = 4 \times 3 - 1 \times \left( { - 1} \right)\]
On simplifying we get ,
\[D = 13\]
Now we will calculate \[{D_x}\] .
\[{D_x} = \left| \begin{gathered}
7&-1 \\
11&4 \\
\end{gathered} \right|\]
Here , the first column consists of constants from both the equations and the second column consists of coefficients of \[y\].
Now solving the determinant we get ,
\[{D_x} = 7 \times 4 - 11 \times \left( { - 1} \right)\]
On simplifying we get ,
\[{D_x} = 39\] .
Similarly for \[{D_y}\] instead of \[y\] coefficients we write constants from both the equations and coefficients of \[x\] in the first column .
\[{D_y} = \left| \begin{gathered}
3&7 \\
1&11 \\
\end{gathered} \right|\]
On solving we get
\[{D_y} = 11 \times 3 - 1 \times 7\]
On simplifying we get ,
\[{D_y} = 26\] .
Now using the formula for values of \[x\] and \[y\], we have
\[x = \dfrac{{{D_x}}}{D}\]
On putting the values we get
\[x = \dfrac{{39}}{{13}}\]
On solving we get ,
\[x = 3\]
Similarly , for \[y\] we have
\[y = \dfrac{{{D_y}}}{D}\]
On putting the values we get
\[y = \dfrac{{26}}{{13}}\]
On solving we get ,
\[y = 2\]
So, the correct answer is “Option A”.

Note: The Cramer’s rule is a short method to find the solutions for simultaneous equations as compared to other methods but use this method in the solutions when asked to do so . Also , when the number of variables are increased then the complexity of the solution also increases as you have to calculate the determinant of \[4 \times 4\] or any other figure.