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System of Linear Equations

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In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same set of variables. For example, a system with two variables has the following general form:

\begin{align} \left\{\begin{matrix} a_{1}x+b_{1}y&=c_{1},\\ a_{2}x+b_{2}y&=c_{2}. \end{matrix}\right. \end{align}


where,

x,y— variables, unknowns;
a1, a2, b1, b2, c1, c2 — coefficients of the system, constants.

For solving systems of linear equations, the equations must be converted to a general form and then solved by an appropriate technique.

Elimination of variables

The simplest method for solving a system of linear equations with two variables can be described as follows:

  • In the first equation, solve for one of the variables in terms of the others;
  • Substitute this expression into the remaining equation;
  • Solve the single linear equation;
  • Back-substitute for solving the remaining variable.

Ex 1.

\begin{align} \left\{\begin{matrix} 3x-7y =16,\\ x-5y =0. \end{matrix}\right. \\ \\ 3\times 5y-7y = 16 \Rightarrow y &= 2 \\ x-5 \times 2 = 0 \Rightarrow x &= 10 \end{align}


Addition method

Solving system of linear equations with the addition method:

  • If one of the variables in the top equation has the opposite coefficient of the same variable in the bottom equation, add the equations together, eliminating one variable. If not, use multiplication by a nonzero number so that one of the variables in the top equation has the opposite coefficient of the same variable in the bottom equation, then add the equations to eliminate the variable;
  • Solve the resulting equation for the remaining variable;
  • Substitute that value into one of the original equations and solve for the second variable.

Ex 2.

\begin{align} \left\{\begin{matrix} 3x-7y =16\\ x-5y =0 \end{matrix}\right| \begin{matrix} \\ \times (-3) \end{matrix} \\ \\ 3x-3x-7y+15y = 16+0 \Rightarrow y &= 2 \\ x-5 \times 2 = 0 \Rightarrow x &= 10 \end{align}


See also:



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