Graph of Linear Equation in Two Variables
A linear equation in two variables of the form ax + by + c = 0 represents a straight line when plotted on the Cartesian plane. Every solution (x, y) of the equation corresponds to a point on this line.
In Class 9 Mathematics, graphing linear equations is a core skill. The graph gives a visual representation of all solutions of the equation simultaneously.
Since a linear equation in two variables has infinitely many solutions, its graph is a continuous straight line extending in both directions. Any two distinct solutions are sufficient to draw the line, although a third point is used for verification.
What is Graph of Linear Equation in Two Variables?
Definition: The graph of a linear equation in two variables ax + by + c = 0 (where a and b are not both zero) is a straight line on the Cartesian plane.
Every point on the line is a solution of the equation, and every solution of the equation is a point on the line.
Key terms:
- x-intercept: The point where the line crosses the x-axis (set y = 0 and solve for x).
- y-intercept: The point where the line crosses the y-axis (set x = 0 and solve for y).
- Slope (gradient): The steepness of the line, equal to −a/b for the equation ax + by + c = 0.
Important:
- A linear equation in one variable (e.g., x = 3) also has a graph — it is a line parallel to the y-axis.
- The equation y = k gives a line parallel to the x-axis.
- The equation y = mx passes through the origin.
Graph of Linear Equation in Two Variables Formula
Key Formulas and Forms:
1. Standard Form:
ax + by + c = 0
where a, b, c are real numbers and a, b are not both zero.
2. Slope-Intercept Form:
y = mx + c
Where:
- m = slope of the line = −a/b
- c = y-intercept (value of y when x = 0)
3. Intercept Form:
x/p + y/q = 1
Where:
- p = x-intercept
- q = y-intercept
4. Finding the x-intercept: Put y = 0 in the equation and solve for x.
5. Finding the y-intercept: Put x = 0 in the equation and solve for y.
Derivation and Proof
Steps to Plot the Graph of a Linear Equation
Step 1: Write the equation in the form y = mx + c
- Rearrange the given equation to express y in terms of x.
- This makes it easy to substitute values of x and find corresponding y values.
Step 2: Find at least two solutions
- Choose convenient values of x (such as 0, 1, 2 or any value that simplifies calculation).
- Substitute each value of x into y = mx + c to get y.
- Record the ordered pairs (x, y) in a table.
Step 3: Choose a third point for verification
- Select one more value of x.
- If the third point lies on the same line as the first two, the graph is correct.
Step 4: Plot the points on the Cartesian plane
- Draw the x-axis (horizontal) and y-axis (vertical) with proper scale.
- Mark each ordered pair as a point.
- Join the points with a straight line.
- Extend the line on both sides with arrows to show it continues infinitely.
Step 5: Label the line
- Write the equation of the line near it.
- Mark the intercepts if required.
Types and Properties
Types of Graphs Based on the Equation:
1. Lines with positive slope (m > 0)
- The line rises from left to right.
- As x increases, y also increases.
- Example: y = 2x + 1
2. Lines with negative slope (m < 0)
- The line falls from left to right.
- As x increases, y decreases.
- Example: y = −3x + 5
3. Lines with zero slope (m = 0)
- The line is horizontal, parallel to the x-axis.
- The equation is of the form y = c (constant).
- Example: y = 4
4. Lines with undefined slope (vertical lines)
- The line is vertical, parallel to the y-axis.
- The equation is of the form x = k (constant).
- Example: x = −2
5. Lines through the origin
- The equation has no constant term: y = mx.
- Both x-intercept and y-intercept are zero.
- Example: y = 3x
Solved Examples
Example 1: Example 1: Graph of y = 2x + 3
Problem: Draw the graph of y = 2x + 3.
Solution:
Step 1: Find solutions by substituting values of x:
- When x = 0: y = 2(0) + 3 = 3 → Point (0, 3)
- When x = 1: y = 2(1) + 3 = 5 → Point (1, 5)
- When x = −1: y = 2(−1) + 3 = 1 → Point (−1, 1)
- When x = −2: y = 2(−2) + 3 = −1 → Point (−2, −1)
Step 2: Plot the points (0, 3), (1, 5), (−1, 1), (−2, −1) on the Cartesian plane.
Step 3: Join the points with a straight line.
Observations:
- The y-intercept is 3 (the line crosses the y-axis at (0, 3)).
- The x-intercept is −3/2 (put y = 0: 0 = 2x + 3, so x = −3/2).
- The slope is 2 (the line rises steeply from left to right).
Example 2: Example 2: Graph of x + y = 7
Problem: Draw the graph of x + y = 7.
Solution:
Rewrite as: y = 7 − x
Find solutions:
- When x = 0: y = 7 → Point (0, 7)
- When x = 7: y = 0 → Point (7, 0)
- When x = 3: y = 4 → Point (3, 4)
Plot the points (0, 7), (7, 0), (3, 4) and join with a straight line.
Observations:
- x-intercept = 7, y-intercept = 7.
- Slope = −1 (the line falls from left to right at 45°).
- The line passes through the first quadrant and intersects both axes at equal distances from the origin.
Example 3: Example 3: Graph of 2x + 3y = 12
Problem: Draw the graph of 2x + 3y = 12 and find the intercepts.
Solution:
Rewrite as: y = (12 − 2x) / 3
Find solutions:
- When x = 0: y = 12/3 = 4 → Point (0, 4)
- When x = 6: y = (12 − 12)/3 = 0 → Point (6, 0)
- When x = 3: y = (12 − 6)/3 = 2 → Point (3, 2)
Plot the points and join with a straight line.
Intercepts:
- x-intercept = 6 (put y = 0)
- y-intercept = 4 (put x = 0)
Answer: The line passes through (0, 4) and (6, 0) with x-intercept 6 and y-intercept 4.
Example 4: Example 4: Graph of y = −x + 5
Problem: Draw the graph of y = −x + 5 and state the quadrants it passes through.
Solution:
Find solutions:
- When x = 0: y = 5 → Point (0, 5)
- When x = 5: y = 0 → Point (5, 0)
- When x = −2: y = 7 → Point (−2, 7)
- When x = 8: y = −3 → Point (8, −3)
Plot the points and draw the line.
Quadrants:
- The line passes through Quadrants I, II, and IV.
- It does NOT pass through Quadrant III because both intercepts are positive.
Answer: The line y = −x + 5 passes through Quadrants I, II, and IV.
Example 5: Example 5: Graph of y = 3x (line through origin)
Problem: Draw the graph of y = 3x.
Solution:
Find solutions:
- When x = 0: y = 0 → Point (0, 0)
- When x = 1: y = 3 → Point (1, 3)
- When x = −1: y = −3 → Point (−1, −3)
- When x = 2: y = 6 → Point (2, 6)
Plot the points and join with a straight line.
Observations:
- The line passes through the origin (0, 0).
- Both intercepts are 0.
- Slope = 3 (steep rise from left to right).
- The line passes through Quadrants I and III.
Example 6: Example 6: Graph of x = 4 (vertical line)
Problem: Draw the graph of x = 4.
Solution:
The equation x = 4 means: for every point on the line, the x-coordinate is 4, regardless of the value of y.
Points on the line:
- (4, 0), (4, 1), (4, 2), (4, −1), (4, −3)
Plot these points. They all lie on a vertical line passing through x = 4.
Observations:
- The line is parallel to the y-axis.
- It has no y-intercept (it never crosses the y-axis).
- The x-intercept is 4.
- The slope is undefined (vertical line).
Example 7: Example 7: Graph of y = −2 (horizontal line)
Problem: Draw the graph of y = −2.
Solution:
The equation y = −2 means: for every point on the line, the y-coordinate is −2, regardless of x.
Points on the line:
- (0, −2), (1, −2), (−1, −2), (3, −2), (−4, −2)
Plot these points. They form a horizontal line at y = −2.
Observations:
- The line is parallel to the x-axis.
- y-intercept = −2.
- It has no x-intercept (never crosses the x-axis).
- Slope = 0.
Example 8: Example 8: Finding coordinates from a graph
Problem: The graph of a linear equation passes through the points (2, 5) and (4, 9). Find the equation and draw the graph.
Solution:
Step 1: Find the slope
- m = (y&sub2; − y&sub1;) / (x&sub2; − x&sub1;) = (9 − 5) / (4 − 2) = 4/2 = 2
Step 2: Use point-slope form
- y − 5 = 2(x − 2)
- y − 5 = 2x − 4
- y = 2x + 1
Verification:
- At x = 2: y = 2(2) + 1 = 5 ✔
- At x = 4: y = 2(4) + 1 = 9 ✔
Answer: The equation is y = 2x + 1.
Example 9: Example 9: Reading intercepts from the equation
Problem: Find the x-intercept and y-intercept of the equation 3x − 4y = 24 without drawing the graph.
Solution:
Finding x-intercept (put y = 0):
- 3x − 4(0) = 24
- 3x = 24
- x = 8
- x-intercept = 8, i.e., point (8, 0)
Finding y-intercept (put x = 0):
- 3(0) − 4y = 24
- −4y = 24
- y = −6
- y-intercept = −6, i.e., point (0, −6)
Answer: x-intercept = 8; y-intercept = −6.
Example 10: Example 10: Real-life graphing problem
Problem: The fare of a bus ride is given by F = 5d + 10, where F is the fare in rupees and d is the distance in km. Draw the graph and find the fare for 8 km.
Solution:
Find solutions:
- When d = 0: F = 5(0) + 10 = 10 → Point (0, 10)
- When d = 2: F = 5(2) + 10 = 20 → Point (2, 20)
- When d = 4: F = 5(4) + 10 = 30 → Point (4, 30)
- When d = 6: F = 5(6) + 10 = 40 → Point (6, 40)
Plot the points with d on x-axis and F on y-axis. Join with a straight line.
From the graph or formula:
- When d = 8: F = 5(8) + 10 = 50
Answer: The fare for 8 km is Rs 50.
Real-World Applications
Applications of Graphing Linear Equations:
- Distance-Time Graphs: In physics, the relationship between distance and time at constant speed is linear. The graph gives the speed as the slope.
- Cost and Revenue Analysis: Businesses use linear graphs to visualise costs (y = mx + c where m is the variable cost per unit and c is the fixed cost).
- Temperature Conversion: The relationship F = (9/5)C + 32 between Fahrenheit and Celsius is linear. Its graph allows quick conversions.
- Currency Exchange: If 1 USD = 83 INR, the equation R = 83D is a line through the origin. The graph gives the exchange amount for any value.
- Simple Interest: The formula SI = PRT/100 is linear in T (for fixed P and R). Plotting SI versus T gives a straight line.
- Science Experiments: In chemistry and physics, many relationships (Ohm’s law V = IR, Hooke’s law F = kx) are linear and verified by graphing experimental data.
Key Points to Remember
- The graph of every linear equation in two variables (ax + by + c = 0) is a straight line.
- Every point on the line is a solution of the equation.
- To draw the graph, find at least two solutions and plot them; use a third point for verification.
- The x-intercept is found by putting y = 0; the y-intercept by putting x = 0.
- Equations of the form y = c give horizontal lines parallel to the x-axis.
- Equations of the form x = k give vertical lines parallel to the y-axis.
- Equations of the form y = mx (no constant term) pass through the origin.
- A positive slope means the line rises from left to right; a negative slope means it falls.
- Two lines are parallel if and only if they have the same slope.
- The graph of a linear equation extends infinitely in both directions.
Practice Problems
- Draw the graph of 2x + y = 6. Find the x-intercept and y-intercept from the graph.
- Plot the graph of y = 3x − 2. State which quadrants the line passes through.
- Draw the graph of x − y = 0. What do you observe about this line?
- Plot the graphs of y = 2x + 1 and y = 2x − 3 on the same Cartesian plane. What is the relationship between the two lines?
- The cost C (in rupees) of producing x items is given by C = 15x + 200. Draw the graph and find the cost of producing 20 items.
- Draw the graph of 3x − 2y = 6 and find the area of the triangle formed by the line with the coordinate axes.
- Plot the graph of y = −(1/2)x + 4. Find the coordinates of the point where the line crosses the x-axis.
- The temperature T (in °C) of a cooling liquid after t minutes is given by T = −3t + 90. Draw the graph for t = 0 to t = 30 and find when the temperature reaches 0°C.
Frequently Asked Questions
Q1. What is the graph of a linear equation in two variables?
The graph of a linear equation ax + by + c = 0 is a straight line on the Cartesian plane. Every point on this line satisfies the equation.
Q2. How many points are needed to draw the graph of a linear equation?
A minimum of two distinct points are needed since two points determine a unique straight line. A third point should be plotted for verification.
Q3. What is the x-intercept of a line?
The x-intercept is the x-coordinate of the point where the line crosses the x-axis. To find it, substitute y = 0 in the equation and solve for x.
Q4. What is the y-intercept of a line?
The y-intercept is the y-coordinate of the point where the line crosses the y-axis. To find it, substitute x = 0 in the equation and solve for y.
Q5. What does the slope of a line indicate?
The slope indicates how steeply the line is inclined. A positive slope means the line rises from left to right; a negative slope means it falls. Zero slope gives a horizontal line.
Q6. Can the graph of a linear equation be a curve?
No. The graph of a linear equation in two variables is always a straight line. If the graph is a curve, the equation is not linear.
Q7. What does x = 5 look like on a graph?
The equation x = 5 is a vertical line passing through the point (5, 0), parallel to the y-axis. Every point on this line has x-coordinate equal to 5.
Q8. Why does y = mx pass through the origin?
In y = mx, when x = 0, y = m(0) = 0. So the point (0, 0) always satisfies the equation. There is no constant term to shift the line away from the origin.
Q9. How do you verify if a point lies on the line?
Substitute the x and y coordinates of the point into the equation. If the left-hand side equals the right-hand side, the point lies on the line.
Q10. Is the graph of a linear equation in two variables always a line in NCERT Class 9?
Yes. NCERT Class 9 Chapter 4 establishes that every equation of the form ax + by + c = 0 (a and b not both zero) has a graph that is a straight line, and conversely every straight line in the plane is the graph of some linear equation.
