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Chapter – 16
System of Linear Equations
In Class 12 mathematics, the term System of Linear Equations refers to a set of two or more linear equations involving the same set of variables. A linear equation is an equation of the first degree, meaning the highest power of any variable is one.
The primary objective when dealing with a system of linear equations is to find the values of the variables that satisfy all the equations simultaneously. These solutions represent the points of intersection of the corresponding lines (or planes, in the case of three variables) in the coordinate space.
System of Linear Equations Class 12 Mathematics Solutions PDF
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System of Linear Equations
In System of Linear Equations in Class 12 mathematics involves studying methods to solve multiple linear equations simultaneously. These equations are of the form $latex a_1x + b_1y + c_1z = d_1$, $latex a_2x + b_2y + c_2z = d_2$, and so on, where $latex x$, $latex y$, and $latex z$ are variables, and $latex a_1, b_1, c_1, d_1$ are constants.
The chapter covers different methods such as substitution, elimination, and matrix approaches like Gaussian elimination, Cramer’s rule, and using determinants to find solutions. The goal is to find values of the variables that satisfy all the equations simultaneously, which is essential in various real-world applications such as economics, engineering, and physics.
Gaussian Elimination Method
One of the methods for solving a system of simultaneous linear equations is the elimination method known as Gaussian elimination method. In this process, the given system is reduced to a simpler system in which;
- the first unknown or variable x1 in the first equation will have a non-zero coefficient, and
- the first unknown or variable x1 in each of the remaining equations will have zero coefficient but the second unknown or variable x2 will have a non-zero coefficient
We then consider the subsystem of equations obtained by excluding the first equation and proceed as before to eliminate the second variable x2. This process is known as the forward elimination. Continuing the above process, we arrive at an equivalent system that looks like an inverted steps or a triangle.
The step-like form or echelon form has the leading terms with non- zero coefficients further to the right in each succeeding equation. One may make each of those non-zero coefficients equal to 1 by division. Four different cases arises. These four cases are given in the forms of examples 1, 2, 3 and 4.
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