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"One Solution (Linear System)" Explained

Different slopes. One intersection.

Key Idea: One Solution (Linear System)

A linear system has one solution when the two equations represent lines with different slopes, meaning they intersect at exactly one point.

General Rule:

  • Slopes are different.

  • One point of intersection exists.

  • This single point satisfies both equations simultaneously.

Why this matters for the SAT:

Identifying when a system has one solution is key to solving systems efficiently, whether by substitution, elimination, or graphing. Different slopes guarantee a unique solution.

"No Solution (Linear System)" Explained

Parallel lines. Same slope.

Key Idea: No Solution (Linear System)

A linear system has no solution when the two equations represent parallel lines.

General Rule:

  • Both lines have the same slope.

  • Lines have different y-intercepts.

  • The lines never intersect, so no point satisfies both equations.

Why this matters for the SAT:

Recognizing when a system has no solution prevents mistakes in solving and helps classify the system correctly (unique solution, no solution, or infinitely many solutions).

"Infinitely Many Solutions (Linear System)" Explained

Same line. Same slope and y-intercept.

Key Idea: Infinitely Many Solutions (Linear System)

A linear system has infinitely many solutions when the two equations represent the same line.

General Rule:

  • Both lines have the same slope.

  • Both lines have the same y-intercept.

  • Every point on one line is also on the other line.

Why this matters for the SAT:

Recognizing when a system has infinitely many solutions prevents mistakes in solving and helps classify the system correctly (unique solution, no solution, or infinite solutions).