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The Cartesian Coordinate Plane
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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. m1 = m2

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 y-intercepts (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 “letters”and the “numbers” 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)


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