Make a conjecture about the solution of a system of equations if the result of subtracting one equation from the other is $0=0$.

A system of equations is said to be consistent if there exist a solution to the system.

When two lines intersect each other at one point then there is only one solution to system of equations then the system is said to be consistent and independent.

When two lines are equivalent and their graph overlap each other then there are infinite solutions to system of equations then the system is said to be consistent and dependent.

A system of equations is said to be inconsistent if there no solution to the system that is two lines are parallel to each other.

Consider the system of equations as.

$\begin{array}{l}x+y=6\\ 2x+2y=12\end{array}$

The algebraic method of elimination involves adding or subtracting the equations to eliminate one of the variables and forming new equation that is true. Sometimes, direct addition or subtraction of equations does not eliminate the variable then one equation requires formation of equivalent equation through multiplication so that one of the two variables has the same or opposite coefficient in both the equations. Multiplying the equation by a nonzero number, resulting new equation has same set of solutions.

To solve the equations, multiply $x+y=6$ by 2 and subtract the resulting equation and the other equation as shown below.

$\begin{array}{c}2\left(x+y\right)=2\left(6\right)\\ 2x+2y=12\end{array}$

Now, subtract $2x+2y=12$and the equation $2x+2y=12$and solve.

$\begin{array}{ccccccc}& 2x& +& 2y& =& & 12\\ & 2x& +& 2y& =& & 12\\ -& & -& & & -& \\ & 0& +& 0& =& & 0\end{array}$

This system of equations result is $0=0$ and cannot be simplified further.

The red line denotes the equation $x+y=6$ and the equation $2x+2y=12$.

The equations are equivalent and the same line denote the graph of each equation.

There are infinite solution to system of equations as all ordered pairs that lie on the line are the solution to system of equations.

Hence, the system of equations is consistent and dependent.

Thus, the conjecture about system of equations if result of subtracting one equation from other is $0=0$ then the system of equation is consistent and dependent.