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 ${D_x}$ .

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.