Question

Use Cramer’s rule to solve each system of equations.

10.

$\begin{array}{l}\mathbf{3}\mathbf{a}-\mathbf{5}\mathbf{b}+\mathbf{2}\mathbf{c}=-\mathbf{5}\\ \mathbf{4}\mathbf{a}+\mathbf{b}+\mathbf{3}\mathbf{c}=\mathbf{9}\\ \mathbf{2}\mathbf{a}-\mathbf{c}=\mathbf{1}\end{array}$

Short answer

The solution of the given system of equations is $\left(\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{1}\right)$.

Step-by-step solution

­- Description of step.

The solution of the system of linear equations in three variables:

$\begin{array}{l}{a}_{1}a+b_{1}b+c_{1}c=d_1\\ a_{2}a+b_{2}b+c_{2}c=d_2\\ a_{3}a+b_{3}b+c_{3}c=d_3\end{array}$

by Cramer’s rule is given by $\left(a,b,c\right)$where

$a=\frac{\left|\begin{array}{ccc}{d}_1& b_1& c_1\\ d_2& b_2& c_2\\ d_3& b_3& c_3\end{array}\right|}{\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|}$$b=\frac{\left|\begin{array}{ccc}{a}_1& d_1& c_1\\ a_2& d_2& c_2\\ a_3& d_3& c_3\end{array}\right|}{\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|}$$c=\frac{\left|\begin{array}{ccc}{a}_1& b_1& d_1\\ a_2& b_2& d_2\\ a_3& b_3& d_3\end{array}\right|}{\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|}$and$\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|\ne 0$

­- Description of step.

The given system of linear equations in three variables is:

$\begin{array}{c}3a-5b+2c=-5\\ 4a+b+3c=9\\ 2a-c=1\end{array}$

Therefore, by comparing the given system of linear equations with the

system of linear equations$\begin{array}{l}{a}_{1}a+b_{1}b+c_{1}c=d_1\\ a_{2}a+b_{2}b+c_{2}c=d_2\\ a_{3}a+b_{3}b+c_{3}c=d_3\end{array}$it can be obtained that:

$a_1=3$, $b_1=-5$, $c_1=2$, $d_1=-5$, $a_2=4$, $b_2=1$,$c_2=3$, $d_2=9$, $a_3=2$,$b_3=0$, $c_3=-1$and $d_3=1$.

­- Find the values of a, b and c.

The value of $a$is given by:

$\begin{array}{c}a=\frac{\left|\begin{array}{ccc}-5& -5& 2\\ 9& 1& 3\\ 1& 0& -1\end{array}\right|}{\left|\begin{array}{ccc}3& -5& 2\\ 4& 1& 3\\ 2& 0& -1\end{array}\right|}\\ =\frac{-5\left|\begin{array}{cc}1& 3\\ 0& -1\end{array}\right|-\left(-5\right)\left|\begin{array}{cc}9& 3\\ 1& -1\end{array}\right|+2\left|\begin{array}{cc}9& 1\\ 1& 0\end{array}\right|}{3\left|\begin{array}{cc}1& 3\\ 0& -1\end{array}\right|-\left(-5\right)\left|\begin{array}{cc}4& 3\\ 2& -1\end{array}\right|+2\left|\begin{array}{cc}4& 1\\ 2& 0\end{array}\right|}\\ =\frac{-5\left(-1-0\right)+5\left(-9-3\right)+2\left(0-1\right)}{3\left(-1-0\right)+5\left(-4-6\right)+2\left(0-2\right)}\\ =\frac{5-60-2}{-3-50-4}\\ =\frac{-57}{-57}\\ =1\end{array}$

The value of $b$is given by:

$\begin{array}{c}b=\frac{\left|\begin{array}{ccc}3& -5& 2\\ 4& 9& 3\\ 2& 1& -1\end{array}\right|}{\left|\begin{array}{ccc}3& -5& 2\\ 4& 1& 3\\ 2& 0& -1\end{array}\right|}\\ =\frac{3\left|\begin{array}{cc}9& 3\\ 1& -1\end{array}\right|-\left(-5\right)\left|\begin{array}{cc}4& 3\\ 2& -1\end{array}\right|+2\left|\begin{array}{cc}4& 9\\ 2& 1\end{array}\right|}{3\left|\begin{array}{cc}1& 3\\ 0& -1\end{array}\right|-\left(-5\right)\left|\begin{array}{cc}4& 3\\ 2& -1\end{array}\right|+2\left|\begin{array}{cc}4& 1\\ 2& 0\end{array}\right|}\\ =\frac{3\left(-9-3\right)+5\left(-4-6\right)+2\left(4-18\right)}{3\left(-1-0\right)+5\left(-4-6\right)+2\left(0-2\right)}\\ =\frac{-36-50-28}{-3-50-4}\\ =\frac{-114}{-57}\\ =2\end{array}$

The value of $c$is given by:

$\begin{array}{c}c=\frac{\left|\begin{array}{ccc}3& -5& -5\\ 4& 1& 9\\ 2& 0& 1\end{array}\right|}{\left|\begin{array}{ccc}3& -5& 2\\ 4& 1& 3\\ 2& 0& -1\end{array}\right|}\\ =\frac{3\left|\begin{array}{cc}1& 9\\ 0& 1\end{array}\right|-\left(-5\right)\left|\begin{array}{cc}4& 9\\ 2& 1\end{array}\right|+\left(-5\right)\left|\begin{array}{cc}4& 1\\ 2& 0\end{array}\right|}{3\left|\begin{array}{cc}1& 3\\ 0& -1\end{array}\right|-\left(-5\right)\left|\begin{array}{cc}4& 3\\ 2& -1\end{array}\right|+2\left|\begin{array}{cc}4& 1\\ 2& 0\end{array}\right|}\\ =\frac{3\left(1-0\right)+5\left(4-18\right)-5\left(0-2\right)}{3\left(-1-0\right)+5\left(-4-6\right)+2\left(0-2\right)}\\ =\frac{3-70+10}{-3-50-4}\\ =\frac{-57}{-57}\\ =1\end{array}$

The values of $a$, $b$ and $c$ are 1, 2 and 1 respectively.

­- Description of step.

The solution of the given system of equations is $\left(1,2,1\right)$.