Linear Equations in Two Variables
Linear equations in two variables form one of the most important chapters in Class 9 Mathematics under the NCERT/CBSE curriculum. A linear equation in two variables is an equation that can be written in the form ax + by + c = 0, where a, b, and c are real numbers, and both a and b are not simultaneously zero. The variables x and y each have degree one, making the equation 'linear.' Understanding linear equations in two variables is essential because they form the foundation for coordinate geometry, graphing, systems of equations, and real-world problem modelling. From calculating cost versus quantity relationships to analysing speed-distance-time scenarios, these equations appear in virtually every field of applied mathematics. In this comprehensive guide, you will learn the definition, standard form, methods of finding solutions, graphical representation, and numerous solved examples that follow the Class 9 NCERT syllabus.
What is Linear Equations in Two Variables?
A linear equation in two variables is an equation of the form:
ax + by + c = 0
where:
- a and b are the coefficients of variables x and y respectively
- c is the constant term
- a and b are real numbers such that a and b are not both zero simultaneously (a2 + b2 ≠ 0)
- x and y are the two variables, each raised to the power 1 only
This is also called the standard form or general form of a linear equation in two variables.
Key Characteristics
- The highest power (degree) of each variable is 1.
- No products of variables (like xy) appear in the equation.
- The equation produces a straight line when plotted on a Cartesian plane.
- Every point on this straight line is a solution of the equation.
Examples of Linear Equations in Two Variables
| Equation | Standard Form (ax + by + c = 0) | Values of a, b, c |
|---|---|---|
| 2x + 3y = 6 | 2x + 3y - 6 = 0 | a = 2, b = 3, c = -6 |
| x - 4y = 7 | x - 4y - 7 = 0 | a = 1, b = -4, c = -7 |
| 5x = 10 - 2y | 5x + 2y - 10 = 0 | a = 5, b = 2, c = -10 |
| y = 3x + 1 | 3x - y + 1 = 0 | a = 3, b = -1, c = 1 |
What is NOT a Linear Equation in Two Variables?
- x2 + y = 5 — x is raised to power 2 (quadratic, not linear)
- xy + 3 = 0 — contains a product of variables
- 1/x + y = 2 — x appears in the denominator (not a polynomial)
- 3x + 7 = 0 — only one variable (this is a linear equation in one variable)
Solution of a Linear Equation in Two Variables
A solution of a linear equation ax + by + c = 0 is any ordered pair (x, y) that satisfies the equation. Since the equation has two unknowns but only one equation, there are infinitely many solutions. Each solution corresponds to a point on the straight line represented by the equation.
For example, the equation x + y = 5 has solutions: (0, 5), (1, 4), (2, 3), (3, 2), (5, 0), (-1, 6), (2.5, 2.5), and infinitely many more. Every such pair lies on the line x + y = 5.
Linear Equations in Two Variables Formula
Standard Form: ax + by + c = 0 (where a, b, c are real numbers and a2 + b2 ≠ 0)
Slope-Intercept Form: y = mx + d, where m = -a/b (slope) and d = -c/b (y-intercept)
Finding Solutions: Choose any value of x, then compute y = -(ax + c) / b, or choose any value of y, then compute x = -(by + c) / a.
x-intercept: Set y = 0 and solve for x: x = -c/a
y-intercept: Set x = 0 and solve for y: y = -c/b
Derivation and Proof
Why Does a Linear Equation in Two Variables Represent a Straight Line?
Consider the equation ax + by + c = 0. We can rewrite it as:
y = (-a/b)x + (-c/b) [when b ≠ 0]
This is of the form y = mx + d, where m = -a/b is the slope and d = -c/b is the y-intercept. Since the relationship between x and y is of first degree (no squares, cubes, or higher powers), as x increases uniformly, y also changes uniformly. This constant rate of change is precisely what defines a straight line.
Graphical Proof
Take the equation 2x + y = 4. Let us compute several solutions:
| x | y = 4 - 2x | Point (x, y) |
|---|---|---|
| 0 | 4 | (0, 4) |
| 1 | 2 | (1, 2) |
| 2 | 0 | (2, 0) |
| -1 | 6 | (-1, 6) |
When these points are plotted on a Cartesian plane, they all fall on a single straight line. Moreover, every point on that line satisfies the equation 2x + y = 4. This is why a linear equation in two variables always represents a straight line, and conversely, every straight line can be expressed as a linear equation in two variables.
Special Cases
When b = 0: The equation becomes ax + c = 0, i.e., x = -c/a. This is a vertical line parallel to the y-axis.
When a = 0: The equation becomes by + c = 0, i.e., y = -c/b. This is a horizontal line parallel to the x-axis.
Methods
Method 1: Finding Solutions by Substitution
To find solutions of ax + by + c = 0:
- Choose a convenient value for one variable (say x = 0, 1, 2, ...)
- Substitute into the equation and solve for the other variable
- Each such pair (x, y) is one solution
Example: For 3x + 2y = 12
- Put x = 0: 2y = 12, y = 6. Solution: (0, 6)
- Put x = 2: 6 + 2y = 12, 2y = 6, y = 3. Solution: (2, 3)
- Put y = 0: 3x = 12, x = 4. Solution: (4, 0)
Method 2: Graphical Method
To draw the graph of a linear equation in two variables:
- Find at least two solutions (ordered pairs) of the equation.
- Plot these points on a Cartesian plane (x-axis and y-axis).
- Draw a straight line through the plotted points.
- Extend the line in both directions with arrows to indicate it continues infinitely.
Important: Although two points are sufficient to draw a straight line, it is recommended to find at least three points as a verification check. If all three points are collinear (lie on the same line), your solutions are correct.
Method 3: Using Intercept Form
To quickly graph a linear equation, find its x-intercept and y-intercept:
- x-intercept: Set y = 0, solve for x. Plot the point (x, 0) on the x-axis.
- y-intercept: Set x = 0, solve for y. Plot the point (0, y) on the y-axis.
- Draw a straight line through these two intercept points.
Example: For 4x + 5y = 20
- x-intercept: 4x = 20, x = 5. Point: (5, 0)
- y-intercept: 5y = 20, y = 4. Point: (0, 4)
Plot (5, 0) and (0, 4) and join them. The resulting line represents 4x + 5y = 20.
Equations of Lines Parallel to Axes
Line parallel to the x-axis: y = k (a constant). Example: y = 3 is a horizontal line passing through (0, 3).
Line parallel to the y-axis: x = k (a constant). Example: x = -2 is a vertical line passing through (-2, 0).
Solved Examples
Example 1: Express an equation in standard form
Question: Write the equation 2x = 3y - 7 in the standard form ax + by + c = 0. Identify the values of a, b, and c.
Solution:
Given equation: 2x = 3y - 7
Rearranging: 2x - 3y + 7 = 0
Comparing with ax + by + c = 0:
- a = 2
- b = -3
- c = 7
Answer: The standard form is 2x - 3y + 7 = 0, with a = 2, b = -3, c = 7.
Example 2: Finding solutions of a linear equation
Question: Find four different solutions of the equation x + 2y = 6.
Solution:
We can write: x = 6 - 2y
When y = 0: x = 6 - 0 = 6. Solution: (6, 0)
When y = 1: x = 6 - 2 = 4. Solution: (4, 1)
When y = 2: x = 6 - 4 = 2. Solution: (2, 2)
When y = 3: x = 6 - 6 = 0. Solution: (0, 3)
Answer: Four solutions are (6, 0), (4, 1), (2, 2), and (0, 3). The equation has infinitely many more solutions.
Example 3: Checking if a given pair is a solution
Question: Check whether (2, 3) is a solution of the equation 3x + 4y = 18.
Solution:
Substitute x = 2 and y = 3 in the LHS of 3x + 4y = 18:
LHS = 3(2) + 4(3) = 6 + 12 = 18
RHS = 18
Since LHS = RHS, the point (2, 3) is a solution of 3x + 4y = 18.
Answer: Yes, (2, 3) satisfies the equation.
Example 4: Finding the value of k from a given solution
Question: If (3, k) is a solution of the equation 2x - y = 5, find the value of k.
Solution:
Since (3, k) is a solution, substituting x = 3 and y = k in the equation:
2(3) - k = 5
6 - k = 5
k = 6 - 5 = 1
Answer: k = 1. So the solution is (3, 1).
Example 5: Drawing the graph of a linear equation
Question: Draw the graph of the equation 2x + 3y = 12. Find the coordinates where the line meets the axes.
Solution:
Rewrite: y = (12 - 2x) / 3
| x | y = (12 - 2x)/3 | Point |
|---|---|---|
| 0 | 4 | (0, 4) |
| 3 | 2 | (3, 2) |
| 6 | 0 | (6, 0) |
Plot these three points on a Cartesian plane and join them with a straight line.
The line meets the y-axis at (0, 4) (y-intercept) and the x-axis at (6, 0) (x-intercept).
Verification: All three points satisfy 2x + 3y = 12, confirming the graph is correct.
Example 6: Writing a linear equation from a word problem
Question: A pen costs Rs x and a notebook costs Rs y. Rahul buys 5 pens and 3 notebooks for Rs 190. Write the linear equation representing this situation and find three solutions.
Solution:
The equation is: 5x + 3y = 190
Standard form: 5x + 3y - 190 = 0
Finding solutions (y = (190 - 5x) / 3):
When x = 10: y = (190 - 50)/3 = 140/3 = 46.67 (not a whole number)
When x = 20: y = (190 - 100)/3 = 90/3 = 30. Solution: (20, 30)
When x = 8: y = (190 - 40)/3 = 150/3 = 50. Solution: (8, 50)
When x = 32: y = (190 - 160)/3 = 30/3 = 10. Solution: (32, 10)
Answer: The equation is 5x + 3y = 190. Three solutions are (20, 30), (8, 50), and (32, 10).
Example 7: Equation of a line parallel to the y-axis
Question: Give the equation of a line which is parallel to the y-axis and passes through the point (-3, 5). Express it in the form ax + by + c = 0.
Solution:
A line parallel to the y-axis has the equation x = constant.
Since the line passes through (-3, 5), the x-coordinate is -3 for all points on this line.
Equation: x = -3
In standard form: x + 0y + 3 = 0, or simply x + 3 = 0
Here a = 1, b = 0, c = 3.
Answer: The equation is x + 3 = 0 (or equivalently x = -3).
Example 8: Verifying collinearity of three points
Question: Verify whether the points (1, 5), (3, 1), and (0, 7) lie on the line represented by 2x + y = 7.
Solution:
Point (1, 5): LHS = 2(1) + 5 = 2 + 5 = 7 = RHS. Lies on the line.
Point (3, 1): LHS = 2(3) + 1 = 6 + 1 = 7 = RHS. Lies on the line.
Point (0, 7): LHS = 2(0) + 7 = 0 + 7 = 7 = RHS. Lies on the line.
All three points satisfy the equation 2x + y = 7, so they are collinear (they all lie on the same straight line).
Answer: Yes, all three points lie on the line 2x + y = 7.
Example 9: Expressing a real-life relationship as a linear equation
Question: The temperature in Fahrenheit (F) and Celsius (C) are related by the formula F = (9/5)C + 32. Express this as a linear equation in two variables in standard form. Find the temperature in Fahrenheit when C = 35.
Solution:
Given: F = (9/5)C + 32
Multiply both sides by 5: 5F = 9C + 160
Rearrange to standard form: 9C - 5F + 160 = 0
Here, the two variables are C and F, with a = 9, b = -5, c = 160.
When C = 35:
F = (9/5)(35) + 32 = 63 + 32 = 95
Answer: The standard form is 9C - 5F + 160 = 0. When C = 35, F = 95 degrees Fahrenheit.
Example 10: Finding solutions with fractional values
Question: Find two solutions of the equation 4x + 3y = 7, where x and y need not be whole numbers.
Solution:
From the equation: y = (7 - 4x) / 3
When x = 1: y = (7 - 4)/3 = 3/3 = 1. Solution: (1, 1)
When x = 1/2: y = (7 - 2)/3 = 5/3. Solution: (1/2, 5/3)
Verification of (1/2, 5/3):
LHS = 4(1/2) + 3(5/3) = 2 + 5 = 7 = RHS. Correct!
Answer: Two solutions are (1, 1) and (1/2, 5/3).
Real-World Applications
Linear equations in two variables are widely used in real-world scenarios:
- Cost and Revenue Analysis: If a factory produces x items at a cost of Rs 50 each with a fixed overhead of Rs 2000, the total cost C = 50x + 2000. Plotting C against x gives a straight line.
- Distance-Speed-Time Problems: If a car travels at a uniform speed, the relationship between distance (d) and time (t) is d = vt, which is a linear equation in two variables d and t.
- Temperature Conversion: The relationship F = (9/5)C + 32 between Fahrenheit (F) and Celsius (C) is a linear equation in two variables.
- Profit and Sales: Profit = Revenue - Cost. If revenue depends linearly on units sold, the profit equation is linear in profit and units.
- Mixture Problems: When mixing two solutions of different concentrations, the resulting concentration can be expressed as a linear equation.
- Age Problems: Relationships between ages of two people at different times can be modelled as linear equations.
Key Points to Remember
- A linear equation in two variables has the general form ax + by + c = 0, where a and b are not both zero.
- The graph of a linear equation in two variables is always a straight line.
- A linear equation in two variables has infinitely many solutions, each represented as an ordered pair (x, y).
- Every solution of the equation corresponds to a point on the line, and every point on the line is a solution.
- To draw the graph, find at least two solutions, plot them, and join with a straight line.
- The equation y = k (constant) represents a line parallel to the x-axis.
- The equation x = k (constant) represents a line parallel to the y-axis.
- The x-axis is represented by y = 0 and the y-axis by x = 0.
- An equation like 2x + 3 = 0 can be treated as a linear equation in two variables by writing it as 2x + 0y + 3 = 0.
- Two distinct points uniquely determine a straight line, but always verify with a third point.
Practice Problems
- Write each of the following equations in the form ax + by + c = 0 and identify a, b, and c: (i) 3x - 5y = 15, (ii) 7x = 2y + 4, (iii) y = (3/2)x - 8.
- Find five solutions of the equation 2x - 3y = 6 and verify that each solution lies on the graph of the equation.
- Check whether (-2, 4) is a solution of the equation x + 2y = 6. If not, find the correct value of y when x = -2.
- If (m, 3) is a solution of the equation 5x - 2y = 4, find the value of m.
- Draw the graph of the equation x - y = 3. From the graph, find: (i) the x-intercept, (ii) the y-intercept, (iii) the point where x = 5.
- A taxi charges a fixed amount of Rs 30 plus Rs 12 per kilometre. Write a linear equation for the total fare (y) in terms of the distance travelled (x). Use this equation to find the fare for a 15 km ride.
- Give the equations of two lines passing through the point (4, 2). How many such lines are possible?
- Draw the graph of x = 5 and y = -3 on the same Cartesian plane. What is the point of intersection of these two lines?
Frequently Asked Questions
Q1. What is a linear equation in two variables?
A linear equation in two variables is an equation of the form ax + by + c = 0, where a, b, and c are real numbers, and a and b are not both zero. The variables x and y each have degree 1. The graph of such an equation is always a straight line on the Cartesian plane.
Q2. How many solutions does a linear equation in two variables have?
A single linear equation in two variables has infinitely many solutions. This is because for every value chosen for one variable, we can compute a corresponding value for the other variable. Each solution is an ordered pair (x, y) representing a point on the straight line.
Q3. What is the difference between a linear equation in one variable and two variables?
A linear equation in one variable (like 3x + 5 = 0) has exactly one solution and can be represented as a point on a number line. A linear equation in two variables (like 3x + 2y = 6) has infinitely many solutions and is represented by a straight line on the Cartesian plane.
Q4. Can a linear equation in two variables have no solution?
A single linear equation in two variables always has infinitely many solutions. However, a system of two linear equations in two variables can have no solution when the two lines are parallel (same slope but different intercepts). Such a system is called inconsistent.
Q5. How do you graph a linear equation in two variables?
To graph a linear equation in two variables: (1) Find at least two solutions (ordered pairs) by substituting values, (2) Plot these points on the Cartesian plane, (3) Draw a straight line through these points. It is good practice to find three points and check that they are collinear.
Q6. What does the graph of x = 5 look like?
The equation x = 5 (or x - 5 = 0) represents a vertical straight line parallel to the y-axis, passing through the point (5, 0) on the x-axis. Every point on this line has its x-coordinate equal to 5 regardless of the y-coordinate.
Q7. What does the graph of y = 0 represent?
The equation y = 0 represents the x-axis itself. Every point on the x-axis has y-coordinate equal to 0. Similarly, x = 0 represents the y-axis.
Q8. Is the equation y = 7 a linear equation in two variables?
Yes. The equation y = 7 can be written as 0x + 1y - 7 = 0 (or 0x + y = 7), which is of the form ax + by + c = 0 with a = 0, b = 1, c = -7. It represents a horizontal line parallel to the x-axis, passing through (0, 7).
Q9. What are the x-intercept and y-intercept of a linear equation?
The x-intercept is the point where the line crosses the x-axis (y = 0), found by substituting y = 0 in the equation. The y-intercept is the point where the line crosses the y-axis (x = 0), found by substituting x = 0. For example, in 3x + 4y = 12, the x-intercept is (4, 0) and the y-intercept is (0, 3).
Q10. How are linear equations in two variables used in real life?
Linear equations in two variables model many real-life situations: calculating total cost based on quantity (C = price per unit x quantity + fixed cost), converting temperatures (F = 9C/5 + 32), computing distances from speed and time (d = vt), budgeting expenses, and planning resource allocation. Whenever two quantities have a constant rate of change relationship, a linear equation describes it.
