Q31. - Algebra 2

Question

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

$2r+25s=40$

$10r+12s+6t=-2$

$36r-25s+50t=-10$

Step-by-Step Solution

Answer

The required solution is $\left(-\frac{155}{28},\frac{143}{70},\frac{673}{140}\right)$.

Step 1- Determine the value of r

Apply the Cramer’s rule for the value of r. 

$r=\frac{\left|\begin{array}{ccc}j& b& c\\ k& e& f\\ l& h& i\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}$

Use the provided equations and substitute the value of a, b, c, d, e, f, g, h, i, j, k and l into width="90" style="max-width: none; vertical-align: -68px;" $r=\frac{\left|\begin{array}{ccc}j& b& c\\ k& e& f\\ l& h& i\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}$after it perform simplification of determinants to write the value of r.

$\begin{array}{c}r=\frac{\left|\begin{array}{ccc}40& 25& 0\\ -2& 12& 6\\ -10& -25& 50\end{array}\right|}{\left|\begin{array}{ccc}2& 25& 0\\ 10& 12& 6\\ 36& -25& 50\end{array}\right|}\\ =\frac{31000}{-5600}\\ =-\frac{155}{28}\end{array}$

Step 2 – Determine the value of s

Apply the Cramer’s rule for the value of s. 

$s=\frac{\left|\begin{array}{ccc}a& j& c\\ d& k& f\\ g& l& i\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}$

Use the provided equations and substitute the value of a, b, c, d, e, f, g, h, i, j, k and l into $s=\frac{\left|\begin{array}{ccc}a& j& c\\ d& k& f\\ g& l& i\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}$after it perform simplification of determinants to write the value of s.

$\begin{array}{c}s=\frac{\left|\begin{array}{ccc}2& 40& 0\\ 10& -2& 6\\ 36& -10& 50\end{array}\right|}{\left|\begin{array}{ccc}2& 25& 0\\ 10& 12& 6\\ 36& -25& 50\end{array}\right|}\\ =\frac{-11440}{-5600}\\ =\frac{143}{70}\end{array}$

Step 3 – Determine the value of t

Apply the Cramer’s rule for the value of t. 

$t=\frac{\left|\begin{array}{ccc}a& b& j\\ d& e& k\\ g& h& l\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}$

Use the provided equations and substitute the value of a, b, c, d, e, f, g, h, i, j, k and l into $t=\frac{\left|\begin{array}{ccc}a& b& j\\ d& e& k\\ g& h& l\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}$after it perform simplification of determinants to write the value of t.

$\begin{array}{c}s=\frac{\left|\begin{array}{ccc}2& 25& 40\\ 10& 12& -2\\ 36& -25& -10\end{array}\right|}{\left|\begin{array}{ccc}2& 25& 0\\ 10& 12& 6\\ 36& -25& 50\end{array}\right|}\\ =\frac{-26920}{-5600}\\ =\frac{673}{140}\end{array}$

Step 4- Write the solution

The obtained value of r is $-\frac{155}{28}$, s is $\frac{143}{70}$ and t is $\frac{673}{140}$ so the solution is $\left(-\frac{155}{28},\frac{143}{70},\frac{673}{140}\right)$.