<![CDATA[Mathematical thinking ability is an essential component of physics learning because many physical phenomena are represented through mathematical models and systems of equations. However, students often have difficulty solving fixed-pulley system problems because they must simultaneously interpret physical concepts and solve systems of linear equations with multiple variables. This study aimed to analyze the application of the Gauss-Jordan elimination method in learning fixed pulley systems and to examine its effect on mathematical thinking ability, learning outcomes, and student responses. The study employed a quantitative approach with a posttest-only control-group quasi-experimental design. The participants comprised 130 respondents across two educational levels: 61 university students from the Physics Education Study Program at the University of Jember and 69 eleventh-grade students from SMAN 3 Jember. The experimental class was taught using the Gauss-Jordan elimination method, while the control class was taught using the substitution-elimination method. Data were collected through essay tests, learning outcome tests, and response questionnaires, and analyzed using the Independent Samples t-test and the Mann-Whitney U test, depending on the data distribution. The results showed that among school students, significant differences were found in the indicators of specializing (0.016), generalizing (0.001), and conjecturing (0.031), but not in convincing (0.468). Among university students, significant differences were found across all mathematical thinking indicators: specializing (0.007), generalizing (0.000), conjecturing (0.018), and convincing (0.004). Learning outcomes also differed significantly for both university students (0.000) and school students (0.010). In addition, student responses to the method's implementation were positive, averaging 71%. The novelty of this study lies in the integration of the Gauss-Jordan elimination method as a structured mathematical procedure in solving fixed pulley system problems in physics learning across two educational levels. In conclusion, the Gauss-Jordan elimination method can serve as an alternative instructional strategy that supports mathematical thinking, improves learning outcomes, and strengthens the connection between physical concepts and mathematical representations, thereby contributing to more systematic and meaningful physics education.]]>
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