Q3.

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

Verified

Answer

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

1Step-1 –Taking an example to justify the statement

Let the system of three equations with three variables be,

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

Subtracting the third equation from the first equation, we get

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

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

System of equations in two variables are,

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

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

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

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

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

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

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