\displaystyle \textbf{Ordered Pair}

\displaystyle \text{An ordered pair }(a,b)\text{ is a pair of mathematical objects written in a}
\displaystyle \text{particular order. The order in which the objects appear is significant.}

\displaystyle \text{In the ordered pair }(a,b),\ a\text{ is called the first entry or first}
\displaystyle \text{component, and }b\text{ is called the second entry or second component.}

\displaystyle \text{In general, }(a,b)\ne(b,a)\text{ unless }a=b.
\displaystyle \text{For example, }(4,1)\ne(1,4).

\displaystyle \text{Two ordered pairs }(a_1,b_1)\text{ and }(a_2,b_2)\text{ are equal if and only if}
\displaystyle a_1=a_2\text{ and }b_1=b_2.

\displaystyle \text{Thus, }(a_1,b_1)=(a_2,b_2) \iff a_1=a_2\text{ and }b_1=b_2.

\displaystyle \\

\displaystyle \textbf{Two-Dimensional Cartesian Coordinate System} Picture 1

\displaystyle \text{A Cartesian coordinate system is used to locate points uniquely in a plane.}
\displaystyle \text{It consists of two mutually perpendicular number lines called coordinate axes.}

\displaystyle \text{The horizontal axis is called the }x\text{-axis and the vertical axis is called}
\displaystyle \text{the }y\text{-axis. The point where the two axes intersect is called the origin.}

\displaystyle \text{The coordinates of the origin are }O(0,0).

\displaystyle \text{The }x\text{-axis and }y\text{-axis divide the coordinate plane into four regions}
\displaystyle \text{called quadrants, numbered I, II, III and IV in anticlockwise order.} Picture 2

\displaystyle \text{Every point }P\text{ in the coordinate plane can be represented by an ordered pair}
\displaystyle (x,y),\text{ called the coordinates of the point.}

\displaystyle \text{The first coordinate }x\text{ is called the }x\text{-coordinate or abscissa.}
\displaystyle \text{The second coordinate }y\text{ is called the }y\text{-coordinate or ordinate.}

\displaystyle \text{For example, if }P=(4,3),\text{ then the abscissa of }P\text{ is }4\text{ and its}
\displaystyle \text{ordinate is }3.

\displaystyle \text{The coordinates are signed numbers. They indicate both the distance from an axis}
\displaystyle \text{and the direction in which the point lies from the origin.}

\displaystyle \\

\displaystyle \textbf{Convention of Signs and Quadrants} Picture 3

\displaystyle \text{The }x\text{-coordinate is positive to the right of the origin and negative to}
\displaystyle \text{the left of the origin.}

\displaystyle \text{The }y\text{-coordinate is positive above the origin and negative below the origin.}

\displaystyle \begin{array}{c|c} \text{Quadrant} & \text{Signs of }(x,y)\\ \hline \text{I} & (+,+)\\ \text{II} & (-,+)\\ \text{III} & (-,-)\\ \text{IV} & (+,-) \end{array}

\displaystyle \text{A useful way to remember the signs is:}
\displaystyle \text{I}:(+,+),\quad \text{II}:(-,+),\quad \text{III}:(-,-),\quad \text{IV}:(+,-).

\displaystyle \text{A point lying on an axis does not belong to any quadrant.}
\displaystyle \text{If }P=(a,0),\text{ then }P\text{ lies on the }x\text{-axis.}
\displaystyle \text{If }P=(0,b),\text{ then }P\text{ lies on the }y\text{-axis.}

\displaystyle \\

\displaystyle \textbf{Plotting a Point on the Coordinate Plane}

\displaystyle \text{To plot points, draw the }x\text{-axis and }y\text{-axis perpendicular to each other.}
\displaystyle \text{Mark their point of intersection as the origin }O(0,0).

\displaystyle \text{Choose a suitable scale and mark equal units along both axes on either side}
\displaystyle \text{of the origin.}

\displaystyle \text{To plot a point }(a,b),\text{ first move }a\text{ units along the }x\text{-direction}
\displaystyle \text{and then move }b\text{ units parallel to the }y\text{-axis.}

\displaystyle \text{For }a>0\text{ and }b>0:

\displaystyle (a,b):\quad a\text{ units right and }b\text{ units up}\quad\text{(Quadrant I)}.

\displaystyle (-a,b):\quad a\text{ units left and }b\text{ units up}\quad\text{(Quadrant II)}.

\displaystyle (-a,-b):\quad a\text{ units left and }b\text{ units down}\quad\text{(Quadrant III)}.

\displaystyle (a,-b):\quad a\text{ units right and }b\text{ units down}\quad\text{(Quadrant IV)}.

\displaystyle \text{Always read the coordinates in the order }x\text{ first and }y\text{ second.}

\displaystyle \\

\displaystyle \textbf{Graph of a Linear Equation in Two Variables} Picture 4

\displaystyle \text{A linear equation in two variables represents a straight line on the Cartesian}
\displaystyle \text{plane. Every point on the line satisfies the given linear equation.}

\displaystyle \text{Consider the linear equation }y=mx+c.

\displaystyle \text{To find its }y\text{-intercept, put }x=0.
\displaystyle y=m(0)+c=c.
\displaystyle \therefore \text{ the line intersects the }y\text{-axis at }(0,c).

\displaystyle \text{To find its }x\text{-intercept, put }y=0.
\displaystyle 0=mx+c
\displaystyle \Rightarrow mx=-c
\displaystyle \Rightarrow x=-\frac{c}{m},\qquad m\ne0.
\displaystyle \therefore \text{ the line intersects the }x\text{-axis at } \left(-\frac{c}{m},0\right).

\displaystyle \text{Plot these two points and draw the straight line passing through them.}

\displaystyle \text{More generally, any two distinct points satisfying a linear equation are}
\displaystyle \text{sufficient to draw its graph. A third point may be used as a check.}

\displaystyle \\

\displaystyle \textbf{Example 1: }\quad y=3x+6

\displaystyle \text{Putting }x=0,\quad y=6.
\displaystyle \therefore \text{ one point is }(0,6).

\displaystyle \text{Putting }y=0,\quad 0=3x+6.
\displaystyle \Rightarrow 3x=-6
\displaystyle \Rightarrow x=-2.
\displaystyle \therefore \text{ another point is }(-2,0).

\displaystyle \text{Plot }(0,6)\text{ and }(-2,0)\text{ and join them to obtain the straight line.}

\displaystyle \\

\displaystyle \textbf{Example 2: }\quad 3y=2x-12

\displaystyle \text{Putting }x=0,\quad 3y=-12.
\displaystyle \Rightarrow y=-4.
\displaystyle \therefore \text{ one point is }(0,-4).

\displaystyle \text{Putting }y=0,\quad 0=2x-12.
\displaystyle \Rightarrow x=6.
\displaystyle \therefore \text{ another point is }(6,0).

\displaystyle \text{Plot }(0,-4)\text{ and }(6,0)\text{ and join them to obtain the straight line.}

\displaystyle \\

\displaystyle \textbf{Graphical Solution of Simultaneous Linear Equations}

\displaystyle \text{Each linear equation in two variables represents a straight line. Therefore,}
\displaystyle \text{a pair of simultaneous linear equations can be solved graphically by drawing}
\displaystyle \text{the two corresponding straight lines on the same coordinate plane.}

\displaystyle \text{The coordinates of a point common to both lines satisfy both equations.}

\displaystyle \text{There are three possible cases:}

\displaystyle \text{(i) If the two lines intersect at one point, the equations have one unique solution.}

\displaystyle \text{If the lines intersect at }(a,b),\text{ then the solution is }x=a,\ y=b.

\displaystyle \text{(ii) If the two lines are distinct and parallel, they have no common point.}
\displaystyle \therefore \text{ the simultaneous equations have no solution.}

\displaystyle \text{(iii) If the two lines coincide, every point on one line also lies on the other.}
\displaystyle \therefore \text{ the simultaneous equations have infinitely many solutions.}

\displaystyle \\

Discover more from ICSE / ISC / CBSE Mathematics Portal for K12 Students

Subscribe to get the latest posts sent to your email.