Efficient solution of a two-variable linear system depends on its structure: substitution is often suitable when a variable is isolated, elimination when coefficients align or can be scaled easily, and graphing when intersections or approximate solutions are the focus. The learner understands that equivalent algebraic transformations preserve the common solution, verifies an ordered pair in both equations, and interprets intersecting, parallel, or coincident lines as one solution, no solution, or infinitely many solutions. Matrix methods, nonlinear systems, and systems with more than two variables are outside this scope.
Before solving, look at the structure of the equations:
Solve the system:
The first equation already has isolated, so substitution is the most efficient method.
The expression equals . Replace in the second equation with :
Combine like terms:
Subtract from both sides:
Divide by :
Substitute into the first equation:
So the solution is:
Check in both original equations:
This is true. Also,
This is true.
Therefore, the lines intersect at , so the system has one solution. Equivalent algebraic steps do not change the common solution.
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