Linear Equations in Two Variables
📋 Contents
- Linear Equation in One Variable
- Linear Equation in Two Variables
- Graphical Solution of a Linear Equation
- Different Forms of a Line
- Solution: One Variable
- Solution: Two Variables
- Lines Parallel to Axes
- Key Summary & Special Cases
- Let's Try — Practice Problems
- Exercise 1 (MCQs)
- Exercise 2 (Descriptive)
- Exercise 3 (Advanced)
An equation of the form ax + b = 0 where a and b are real numbers and x is a variable, is called a linear equation in one variable.
Examples: 3x + 5 = 0, 7x – 2 = 0, etc.
The solution is obtained by isolating x:
An equation of the form ax + by + c = 0 where a, b, c are real numbers, a ≠ 0, b ≠ 0, and x, y are variables, is called a linear equation in two variables.
Any pair of values of x and y which satisfies the equation ax + by + c = 0 is called a solution of the equation.
- IObtain the linear equation.
- IIIf the form is ax = b (a ≠ 0), plot the point (b/a, 0) and one more point (b/a, k) where k is any real number. If the form is ay = b, plot (0, b/a) and (k, b/a) for any real k.
- IIIJoin the plotted points to obtain the required line.
- IObtain the linear equation ax + by + c = 0.
- IIExpress y in terms of x: y = (–ax – c)/b or x in terms of y.
- IIIChoose two or three values of x (or y), and calculate the corresponding values of y (or x). Record as coordinate pairs.
- IVPlot the coordinate pairs on graph paper.
- VJoin the points. The line obtained is the graph of ax + by + c = 0.
| x | 0 | 3 | –2 |
|---|---|---|---|
| y | –3/2 | 0 | –5/2 |
| x | 0 | 2 | 4 |
|---|---|---|---|
| y | 4 | 2 | 0 |
| x | 1 | 0 | 3 |
|---|---|---|---|
| y | 0 | –2 | 4 |
If a line makes an angle θ with the positive direction of the X-axis, then tan θ is called the slope of the line, denoted by m: m = tan θ.
y = mx + cwhere m is the slope and c is the Y-intercept (the value where the line crosses the Y-axis).
y = mxWhen c = 0, the line always passes through the origin (0, 0).
x/a + y/b = 1where a is the X-intercept and b is the Y-intercept (intercepts on the positive directions of the respective axes).
For ax + b = 0 (a ≠ 0):
Multiply the equations by suitable non-zero constants to make the coefficients of one variable equal, then add or subtract the equations to eliminate that variable.
Express one variable in terms of the other from one equation, then substitute into the second equation.
A line of the form y = k (constant) is parallel to the X-axis.
🔑 Special Cases of ax + by + c = 0
- If b = 0 (a ≠ 0, c ≠ 0): Reduces to ax + c = 0, i.e., x = –c/a. Graph is a straight line parallel to the Y-axis passing through (–c/a, 0).
- If a = 0 (b ≠ 0, c ≠ 0): Reduces to by + c = 0, i.e., y = –c/b. Graph is a straight line parallel to the X-axis passing through (0, –c/b).
- If c = 0 and b = 0 (a ≠ 0): Reduces to ax = 0, i.e., x = 0. Graph is the Y-axis itself.
- If c = 0 and a = 0 (b ≠ 0): Reduces to by = 0, i.e., y = 0. Graph is the X-axis itself.
- If c = 0 (a ≠ 0, b ≠ 0): Reduces to ax + by = 0. Graph is a line passing through the origin.
🔑 Geometric Representation of ax + c = 0
- In one variable: A single point on the number line at x = –c/a.
- In two variables: A vertical line in the coordinate plane (written as a·x + 0·y = –c). The line is parallel to the Y-axis.
- A linear equation in two variables has infinitely many solutions.
- Every point on the graph of a linear equation in two variables is a solution of the equation.
| x | 0 | 2 | 1 |
|---|---|---|---|
| y | 6 | 0 | 3 |
| x (km) | 1 | 2 | 3 |
|---|---|---|---|
| y (₹) | 8 | 13 | 18 |
🔗 For more practice problems and video explanations on this topic, visit MyClass24.com.
- If x is the number of hours a labourer works and y is his wages in rupees, then y = 5x + 7. Draw the work–wages graph. From the graph, find the wages for 6 hours of work.
- If y = 100x, find the value of y when x = 6 (i.e., x = 6).
- Verify which of the following are solutions of x – 2y + 4 = 0 and which are not: (0, 2), (2, 0), (4, 0), (6, 1), (–2, 1).
- Find the value of b if (3, 4) is a solution of the equation 5x + by = 13.
- Express y in terms of x in 2x + 3y = 6. Find the point where the line cuts the Y-axis.
- Express x in terms of y in 3x – 2y = 12. Find the point where the line cuts the X-axis.
- Give the geometric representation of the equation 3x + 5 = 0 as an equation in (i) one variable and (ii) two variables.
- Give the geometric representation of 3y – 9 = 0 as an equation in (i) one variable and (ii) two variables.
- A man drives at a uniform speed of 90 km/h. Draw the time–distance graph. From the graph find the distance in (i) 1/2 hour and (ii) 2.5 hours.
- Nidhi and Nisha together contributed ₹300 towards the PM's Relief Fund. Express this as a linear equation in two variables and draw the graph.
- If x + 1/x = 3, find the value of x³ + x² + x + 1.
- Determine whether x = 5, y = 4 is a solution of x – 2y = –3.
- Solve: 8x – 5y = 34 and 3x – 2y = 13.
- Solve: 20x + 3y = 7 and 8y – 15x = 5.
- Solve: 2x – 3y – 3 = 0 and (x/3) + 4y + (1/2) = 0.
📹 Watch step-by-step video solutions for Exercise 2 on MyClass24.com.
- Draw the graph of 2x + 3y = 6 and use it to find the area of the triangle formed by the line and the coordinate axes.
- Draw the graph of 4x – y = 5 and 5y – 4x = 7 on the same graph paper and find the coordinates of their point of intersection.
- Find two numbers such that five times the greater exceeds four times the lesser by 22, and three times the greater together with seven times the lesser is 32.
- Draw the graph of x – y + 1 = 0 and 3x + 2y – 12 = 0 on the same graph. Calculate the area bounded by these lines and the X-axis.
- If p = 3x + 1 and q = 9x + 13 and p : q = 6 : 5, find the value of x.
FAQs on CBSE Class 9 Maths Notes – Linear Equations in Two Variables
Linear Equations in Two Variables are algebraic equations that contain two variables, usually represented by x and y, where the highest power of each variable is one. The standard form of a linear equation is ax + by + c = 0, where a, b, and c are constants. This chapter is one of the most important topics in Class 9 Mathematics because it introduces students to the relationship between algebra and graphs. Frequently searched keywords include linear equations in two variables, standard form of linear equation, algebraic equations, graph of linear equations, and NCERT solutions. Students learn how different pairs of values satisfy an equation and how these values can be represented graphically. Understanding these concepts helps build a strong foundation for coordinate geometry, graphing techniques, and advanced algebra topics taught in higher classes.
To score well in this chapter, students should first understand the concept of variables, constants, coefficients, and the standard form of a linear equation. Regular practice of finding solutions and plotting graphs is essential. Students often search for linear equations formulas, important questions, solved examples, NCERT exercise solutions, graph questions, and revision notes while preparing for examinations. Creating tables of values and drawing accurate graphs improves understanding and reduces mistakes. Students should also revise key concepts such as ordered pairs, Cartesian plane, x-axis, y-axis, and graphical representation of equations. Solving sample papers and previous examination questions helps strengthen problem-solving skills. Consistent revision and practice with graph-based exercises enable students to develop confidence and achieve high marks in the Linear Equations in Two Variables chapter.
The solution of a linear equation in two variables is any ordered pair of values that satisfies the equation. To graph a linear equation, students first create a table of values by assigning values to one variable and finding the corresponding values of the other variable. These ordered pairs are then plotted on a Cartesian plane. High-search keywords for this topic include graph of linear equations, plotting points on a graph, solution of linear equations, coordinate geometry graphs, and graphing linear equations examples. After plotting at least two points, a straight line is drawn through them. Every point on this line represents a solution to the equation. Practicing graph-based questions helps students understand the relationship between algebraic expressions and graphical representations, which is a key objective of this chapter.