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Mathematics · Class 9 · Chapter 7: Linear Graphs and Their Application

8 min

Key points

Linear equation in two variables has the general form ax + by + c = 0 where a, b are not both zero
Graph of linear equation in two variables is always a straight line
To draw a graph, find at least two points by substituting values of x and y
x-intercept is the point where line crosses x-axis (y = 0)
y-intercept is the point where line crosses y-axis (x = 0)
Slope of a line is the ratio of vertical change to horizontal change
Two lines are parallel if their slopes are equal
Two lines are perpendicular if product of their slopes is -1
Linear graphs are used in real-life problems like cost analysis, profit calculation, and distance-time relations

Definitions

Linear Equation in Two Variables: An equation of the form ax + by + c = 0, where a and b are real numbers and both a and b are not zero simultaneously, is called a linear equation in two variables x and y.
Graph of Linear Equation: The graph of a linear equation in two variables is the set of all points whose coordinates satisfy the equation and it represents a straight line.
Slope of a Line: The slope of a line is the ratio of the vertical change to the horizontal change between any two points on the line. It is denoted by m and calculated as m = (y₂ - y₁)/(x₂ - x₁).
x-intercept: The x-intercept of a line is the x-coordinate of the point where the line crosses the x-axis. At this point, y = 0.
y-intercept: The y-intercept of a line is the y-coordinate of the point where the line crosses the y-axis. At this point, x = 0.

Worked example

Draw a linear graph for the equation y = 2x + 3 using values of x from -2 to +2. Find the slope and y-intercept. (7 marks)

The linear graph is a straight line passing through the plotted points. Slope = 2, Y-intercept = 3

Common mistakes

Students forget to write 'where a and b are not both zero' in definition of linear equation
Mixing up x-intercept and y-intercept calculations
Not plotting points accurately on graph paper
Forgetting to extend the line beyond plotted points with arrows
Calculating slope incorrectly by reversing the formula

Quick recap

Linear graphs are straight line representations of linear equations in the form y = mx + c. The slope (m) shows steepness and direction, while y-intercept (c) is where the line crosses y-axis. Always plot minimum 3 points, use ruler for straight lines, and label axes properly. Remember that positive slope goes upward, negative slope goes downward.

Generated by AI from Sindh Textbook Board material; review state: ai_unreviewed. MCQs, past-paper references and examiner notes are held back until a teacher has checked them.