Q1. - Algebra 2

Question

1. Explain how you can use the methods of solving a system of two equations in two variables to solve a system of three equations in three variables.

Step-by-Step Solution

Answer

 The system of three equations in three variables can be solved using two equations in two variables.

Step-1 –Concept of solving the system of three equations in three variables

By using elimination, we obtain the system of two equations in two variables from the system of three equations in three variables.

Again, using elimination in the system of two equations in two variables we get the solution.

Step-2 –Taking an example to justify the statement

Let the system of three equations with three variables be,

$\begin{array}{l}x+2y=12\\ 3y-4z=25\\ x+6y+z=20\end{array}$

Subtracting the third equation from the first equation, we get

$\begin{array}{l}2y-6y-z=12\\ -4y-z=-8\\ 4y+z=8\end{array}$

Step-3 –Solving the system of two equations in two variables

System of equations in two variables are,

$\begin{array}{l}x+2y=12\\ 3y+16y=25+32\\ 19y=57\\ y=3\end{array}$

Substituting the value of 

   in the first equation, we get

$\begin{array}{l}x+2y=12\\ 3y+-4z=25\\ 3\left(2\right)-4z=25\\ -4z=16\\ z=-4.\end{array}$

Step-4 –Putting the value of y and z in any of the equation with all three variables

$\begin{array}{l}x+2y=12\\ x+6y=z=20\\ x+6\left(3\right)-4=20\\ x+18-4=20\\ x=20-14\\ x=6\end{array}$