Solving Systems of Equations  Two Lines
Parallel and Coincident Equations:
→ 1. If the lines are parallel. There is no solution. The system is inconsistent.
→ If two lines are parallel, their slopes are equal. m_{1} = m_{2
}
When solving the system by the algebraic Elimination Method
if the "letters" vanish and the resulting
"sum" is not true, then
the two lines are likely parallel with no solution. Always check by
solving each equation for the form y = mx + b to see if the slopes
are the same but the yintercepts (b) are different in the two
equations
Example 1:
Multiply the first equation by − 2:
Add these equations
and note that 0 ≠ 6
Solve the equation: (1) 2x +3y = 6 and
find 


Solve the equation: (2) 4x +6y = 18 and
find 
Therefore, the system is Parallel, no solution,
inconsistent system.
→
2. If the lines are coincident. There are
infinite solutions. The system is consistent.
When solving the system by the algebraic Elimination Method
if the â€œlettersâ€and the â€œnumbersâ€ vanish, which is true,
The given equations graph on same line with infinite solutions.
Always check by solving each equation for the form
y = mx + b
to see if both yield the same equation.
Example 2:
Multiply equation (1) by 2
Add the equations to get:
Solve the equations:,
Consistent system: Infinite solutions, Coincident (same line)
