JEE Mathematics Linear Equations and Graphs Notes Set 01

Coordinates of a Point

Two perpendicular number lines, the x-axis (horizontal) and the y-axis (vertical), cross at the origin O = (0, 0) and split the plane into four quadrants. The position of a point P is given by (x, y): x (abscissa) is its signed distance from the y-axis and y (ordinate) is its signed distance from the x-axis.

A point with y = 0 lies on the x-axis and a point with x = 0 lies on the y-axis. These points belong to no quadrant. The x-axis is the graph of y = 0 and the y-axis is the graph of x = 0.

Graph of a Function

The graph of \( y=f(x) \) is the set of all points (x, y) that satisfy the equation. A point is on the graph exactly when its coordinates make the equation true: (-5, 10) is on \( y=-2x \), but (-4, 6) is not, because \( -2(-4)=8\ne6 \). The graph of a first-degree equation is a straight line, so two points are enough to draw it, with a third point as a check.

The Line y = mx + c

1. m is the slope (gradient): for two points on the line, \( m=\frac{y_2-y_1}{x_2-x_1} \). It is positive if y rises as x increases.
2. c is the intercept on the y-axis, so the line cuts the y-axis at (0, c). The x-intercept is \( -\frac cm \).
3. A line with no constant term, such as \( y=2x \), passes through the origin.
4. Lines with the same slope are parallel (for example \( y=2x \), \( y=2x+4 \), \( y=2x-3 \)). Lines with the same intercept but different slopes meet at the same point on the y-axis.
5. \( x=a \) is a vertical line, a units from the y-axis; \( y=b \) is a horizontal line, b units from the x-axis.

Using Graphs to Solve Problems

1. Equation in one variable: the solution of \( 6-1.5x=0 \) is the x-value where the graph of \( y=6-1.5x \) meets the x-axis, so x = 4.
2. Cost problems: if total cost is an initial expense c plus m per machine, then \( y=mx+c \). From 25 machines for Rs. 12,500 and 40 machines for Rs. 17,000: \( m=\frac{4500}{15}=300 \) and \( c=12500-7500=5000 \).
3. Unit conversion: \( F=32+\frac95C \) from the points (0, 32) and (100, 212), which gives \( \frac{F-32}{9}=\frac C5 \).

Simultaneous Equations by Graphs

Draw both lines; the coordinates of their common point satisfy both equations. For \( 3x+2y=12 \) (through (0, 6) and (4, 0)) and \( 2x-y=1 \) (through (0, -1) and (1, 1)) the lines cross at P(2, 3), so x = 2 and y = 3.

Inequalities on the Number Line and in the Plane

1. The solution of \( x<3 \) is every point to the left of 3 (3 itself excluded). \( -12. Multiplying or dividing by a negative number reverses the sign: \( -5x+10>0 \) gives \( x<2 \).
3. For \( ax+by+c>0 \) in the plane, draw the line \( ax+by+c=0 \) (dotted if points on it are excluded), then test a point not on it. For \( 2y-4x-5>0 \) the point (-2, 0) gives 3 > 0, so the region is above the line.
4. For a system, shade the common part of the regions. The point (1, 2) satisfies both \( 2x+y-2>0 \) and \( x-2y+2<0 \).