Linear algebraic equations: Are they are the same?

Abstract

Introduction to the solution of linear algebraic equations often begins with equations in one unknown of the form ax±b=c, where x is the unknown. More complex linear equations are introduced culminating in the solution of rational equations. Solution of linear equations requires the construction of a system of equivalent equations. Such construction requires the understanding of the properties of equivalence, in particular that of symmetric property where if a=b, then b=a. When solving linear equations of the form c=ax±b, many students do not exploit this reversibility. Instead these students often transform the linear equations of the form c=ax±b to ax±b=c or they will transpose the unknown from the right side of the equal sign to the left side of the equal sign. This is because the answer d=x does not seem right. Unfortunately the transposition process is not straightforward. Often students make errors involving the change of signs as solution of linear equations requires sound knowledge of symmetric equivalence and the properties of the four basic operations. The study reported in this paper, examined the types of linear equations students find the least and the most challenging. We then examine the different types of errors made by the students when they solved each type of linear equations. We used the extant literature examining the difficulties encountered by students in solving linear equations to construct an 18-item instrument. This instrument was administered to 38 Secondary Two Normal Academic stream students with 21 boys and 17 girls. This study found that students were most challenged by (i) rational equations where the unknown had negative coefficients, (ii) unknown is on the right of the equal sign, and (iii) equations with unknown with negative coefficients, in this order. Students made the most errors during transposing of terms. This suggests that greater attention should (i) focus on helping students understand the notion of symmetric equivalence and (ii) improve their facility with the negative sign.