\displaystyle \textbf{Linear Equation in Two Variables}

\displaystyle \text{An equation of the form }ax+by+c=0,\text{ where }a,\ b,\ c \text{ are real numbers}
\displaystyle \text{and }a\text{ and }b\text{ are not both zero, is called a linear equation}
\displaystyle \text{in two variables }x\text{ and }y.
\displaystyle \text{The highest degree of the variables in such an equation is }1.

\displaystyle \text{Examples:}

\displaystyle 2x+5y=10

\displaystyle -2x+3y-5=0

\displaystyle 78x+\frac{1}{2}y-23=0

\displaystyle \text{The variables need not always be represented by }x\text{ and }y.
\displaystyle \text{Other letters such as }a,\ b\text{ or }m,\ n\text{ may also be used.}
\displaystyle \text{In our discussion, we will generally use }x\text{ and }y,\text{ as these}
\displaystyle \text{are commonly used to represent variables.}

\displaystyle \text{A linear equation in two variables has infinitely many solutions.}
\displaystyle \text{Each solution is an ordered pair }(x,y)\text{ that satisfies the equation.}
\displaystyle \text{For example, for }x+y=5,\text{ some solutions are }(1,4),(2,3),(3,2).

\displaystyle \text{Geometrically, a linear equation in two variables represents a}
\displaystyle \text{straight line in the Cartesian plane.}
\displaystyle \\

\displaystyle \textbf{Simultaneous Linear Equations in Two Variables}

\displaystyle \text{Two linear equations involving the same two variables are called}
\displaystyle \text{simultaneous linear equations, or a pair of linear equations in}
\displaystyle \text{two variables. They are considered together to find values of the}
\displaystyle \text{variables that satisfy both equations simultaneously.}

\displaystyle \text{A solution of a pair of simultaneous linear equations is an ordered}
\displaystyle \text{pair }(x,y)\text{ that satisfies both equations.}

\displaystyle \text{Examples:}

\displaystyle 5x+3y=12\qquad\text{and}\qquad3x+5y=15

\displaystyle 4x-5y=20\qquad\text{and}\qquad6x+5y=-2

\displaystyle \text{Geometrically, each equation represents a straight line. If the two}
\displaystyle \text{lines intersect at one point, the coordinates of that point give the}
\displaystyle \text{unique solution of the simultaneous equations.}

\displaystyle \text{A pair of linear equations may have:}

\displaystyle \text{(i) one solution, if the two lines intersect at one point;}

\displaystyle \text{(ii) no solution, if the two lines are parallel;}

\displaystyle \text{(iii) infinitely many solutions, if the two lines coincide.}
\displaystyle \\

\displaystyle \textbf{Methods of Solving Simultaneous Linear Equations}

\displaystyle \text{Two commonly used algebraic methods for solving simultaneous linear}
\displaystyle \text{equations in two variables are:}

\displaystyle \text{(i) Substitution Method}

\displaystyle \text{(ii) Elimination Method}
\displaystyle \\

\displaystyle \textbf{Substitution Method}

\displaystyle \text{Suppose we are given two linear equations in }x\text{ and }y.
\displaystyle \text{They can be solved by the substitution method as follows:}

\displaystyle \text{1. Express one variable, say }y,\text{ in terms of }x\text{ from either}
\displaystyle \text{of the given equations.}

\displaystyle \text{2. Substitute this expression for }y\text{ in the other equation.}
\displaystyle \text{This gives an equation containing only }x.

\displaystyle \text{3. Solve the resulting equation to find the value of }x.

\displaystyle \text{4. Substitute the value of }x\text{ in either of the original equations}
\displaystyle \text{or in the expression obtained in Step 1 to find }y.

\displaystyle \text{Note: The roles of }x\text{ and }y\text{ may be interchanged. It is often}
\displaystyle \text{convenient to isolate the variable whose coefficient is }1\text{ or }-1.
\displaystyle \\

\displaystyle \textbf{Example: Substitution Method}

\displaystyle \text{Solve the following simultaneous equations:}

\displaystyle 2x+y=10\qquad\text{...(i)}

\displaystyle x+2y=8\qquad\text{...(ii)}

\displaystyle \text{From (i),}

\displaystyle y=10-2x

\displaystyle \text{Substituting }y=10-2x\text{ in (ii),}

\displaystyle x+2(10-2x)=8

\displaystyle \Rightarrow x+20-4x=8

\displaystyle \Rightarrow 20-3x=8

\displaystyle \Rightarrow -3x=-12

\displaystyle \Rightarrow x=4

\displaystyle \text{Substituting }x=4\text{ in }y=10-2x,

\displaystyle y=10-2(4)=2

\displaystyle \therefore x=4,\qquad y=2.

\displaystyle \text{Hence, the solution is the ordered pair }(4,2).

\displaystyle \text{Check:}

\displaystyle 2(4)+2=10

\displaystyle 4+2(2)=8

\displaystyle \text{Thus, }(4,2)\text{ satisfies both equations.}

\displaystyle \text{Geometrically, the two lines intersect at the point }(4,2).
\displaystyle \\

\displaystyle \textbf{Elimination Method}

\displaystyle \text{In the elimination method, one variable is eliminated by adding or}
\displaystyle \text{subtracting suitably chosen multiples of the two equations.}

\displaystyle \text{The steps are:}

\displaystyle \text{1. If necessary, multiply one or both equations by suitable non-zero}
\displaystyle \text{numbers so that the coefficients of one variable become equal in}
\displaystyle \text{magnitude.}

\displaystyle \text{2. If these coefficients have opposite signs, add the equations. If they}
\displaystyle \text{have the same sign, subtract one equation from the other.}

\displaystyle \text{3. Solve the resulting equation to find the value of one variable.}

\displaystyle \text{4. Substitute this value in either of the original equations to find}
\displaystyle \text{the value of the other variable.}
\displaystyle \\

\displaystyle \textbf{Example: Elimination Method}

\displaystyle \text{Solve the following simultaneous equations:}

\displaystyle 2x+y=10\qquad\text{...(i)}

\displaystyle x+2y=8\qquad\text{...(ii)}

\displaystyle \text{Multiplying equation (ii) by }2,\text{ we get}

\displaystyle 2x+4y=16\qquad\text{...(iii)}

\displaystyle \text{Subtracting equation (i) from equation (iii),}

\displaystyle (2x+4y)-(2x+y)=16-10

\displaystyle \Rightarrow 3y=6

\displaystyle \Rightarrow y=2

\displaystyle \text{Substituting }y=2\text{ in equation (i),}

\displaystyle 2x+2=10

\displaystyle \Rightarrow 2x=8

\displaystyle \Rightarrow x=4

\displaystyle \therefore x=4,\qquad y=2.

\displaystyle \text{Hence, the solution is the ordered pair }(4,2).

\displaystyle \text{Geometrically, the two lines intersect at the point }(4,2).
\displaystyle \\

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