\r\n A system means a set, a group or a family of two or more objects.\r\n A linear equation is an equation that is linear in its constituent variable/s — in simple words, it means that the highest order of any of the constituent variables cannot be greater than unity. For example: \\(f(x) = 3 x + 4\\) is an example of a linear equation in \\(x\\) - compare this to \\(g(x) = 4 x^2 + 3 x + 9\\) which is a quadratic function in \\(x\\).\r\n Thus, a system of linear equations simply means a set of equations that are linear in the constituent variables. For example, the general form of a system of linear equations in two variables x and y is given as:\r\n $$ m_1 x + n_1 y = l_1$$\r\n $$ m_2 x + n_2 y = l_2$$\r\n where \\(m_1 \\), \\(n_1\\), \\(m_2\\), \\(n_2\\), \\(l_1\\) and \\(l_2\\) all are real numbers.
\r\n Graphically, each linear equation in two variables (say \\(x\\) and \\(y\\)) represents a straight line in the \\(x-y\\) plane.\r\n Thus, the solution of a system of linear equations in two variables essentially represents the point of intersection of the two lines.\r\n
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\n The necessary condition for existence of a solution is that the number of equations must be same as the number of variables. Note that this is just a necessary condition - meaning it must be met whenever a solution exists but it alone is not sufficient to guarantee the existence of a solution. A system of linear equations can have one, infinitely many or no solution. A system of equations with no solution is called \"inconsistent\" while a system with one or infinitely many solution is called \"consistent\".\r\n
There are several different ways of solving a system of linear equations such as substitution method, elimination method, graphical method. These methods are described in below.
\r\n\r\n Let's consider the following system of linear equations in two variables (\\(x\\) and \\(y\\)):\r\n $$ 3 x + 4 y = 5 \\tag{1}$$\r\n $$ 4 x + 8 y = 8 \\tag{2}$$\r\n First, we solve one of the linear equations in the system (equation # 1) to obtain the expression for one variable (say \\(y\\)) in terms of the other variable (say \\(x\\)).\r\n $$ y = \\frac{5}{4} - \\frac{3}{4} x \\tag{3}$$\r\n Then, we substitute for \\(y\\) in terms of \\(x\\) in equation #1 using the above expression. Thus, we obtain an equation which is entirely in \\(x\\).\r\n $$ 4 x + 8 \\left(\\frac{5}{4} - \\frac{3}{4} x \\right) = 8 $$ $$ \\Rightarrow 4 x + 10 - 6 x = 8$$\r\n $$\\Rightarrow 2 x = 2$$\r\n Solve the resulting equation (equation # 3) for \\(x\\):\r\n $$ x =1$$\r\n Substitute the value of \\(x\\) in the expression for \\(y\\) (equation # 3) to obtain the value \\(y\\)\r\n $$y = \\frac{5}{4} - \\frac{3}{4} = \\frac{1}{2}$$\r\n
\r\nGraphical method involves plotting the lines described by each of the equations on a graph. Then, the solution corresponds to the point at which the two lines intersect.
\r\nThis MagicGraph offers an interactive way to learn solving a system of linear equations using the graphical method.
\r\n\r\n Start by entering the coefficients of two equations of the system. The MagicGraph automatically draws the lines corresponding to the\r\n two equations and finds the point of intersection. This intersection represents the solution of the system of equations.\n
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