Question

3: If a system of a three equation in three variables has one solution, the graph of the equation intersects in a (point, plane).

Short answer

If a system of a three equation in three variables has one solution, the graph of the equation intersects in a point.

Step-by-step solution

Step-1 –Taking an example to justify the statement

Let the system of three equations with three variables be,

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

Subtracting the third equation from the first equation, we get

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

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

System of equations in two variables are,

$\begin{array}{c}3y-4z=25\\ 4y+z=8\end{array}$

Multiplying the second equation by $4$and adding to the first equation, we get

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

Substituting the value of $y$in the first equation, we get

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

Step-3 –Putting the value of yand z in any of the equation with all three variables

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