Question

Solve each system of equations.

$\begin{array}{l}2r+3s-4t=20\\ 4r-s+5t=13\\ 3r+2s+4t=15\end{array}$

Short answer

The solution is$r=5,s=-2,t=-1.$

Step-by-step solution

Step-1 –Using elimination to obtain two equations with two variables

Given system of equations are

$\begin{array}{c}2r+3s-4t=20\\ 4r-s+5t=13\\ 3r+2s+4t=15\end{array}$

Multiplying second equation by 3 and adding to the first equation, we get

$\begin{array}{c}2r+12r-4t+15t=20+39\\ 14r+11t=59\end{array}$

Again multiplying second equation by 2 and adding to the third equation, we get

$\begin{array}{c}8r+3r+10t+4t=26+15\\ 11r+14t=41\end{array}$

Step-2 –Solving the system of two equations with two variables

System of two equations with two variables are

$\begin{array}{c}14r+11t=59\\ 11r+14t=41\end{array}$

Multiplying the first equation by 11 and second by 14 and subtracting them we get

$\begin{array}{c}121t-196t=649-574\\ -75t=75\\ t=-1\end{array}$

Putting the value of in equation we get

$\begin{array}{c}14r-11=59\\ 14r=59+11\\ 14r=70\\ r=5\end{array}$

Step-3 –Finding the value of using the value o f   r  and t

Putting the value of r and t in the equation$4r-s+5t=13$we get,

$\begin{array}{c}4\left(5\right)-s+5(-1)=13\\ 20-s-5=13\\ -s=13-15\\ -s=-2\\ s=2\end{array}$