- An equation ax+by=c, with a and b not both zero, represents a straight line in the Cartesian plane.
- A solution (x,y) makes the equation true when values are substituted.
- Over real numbers, one nondegenerate linear equation in two variables has infinitely many ordered-pair solutions.
- Choose values of x, calculate corresponding y, and check by substitution.
- Plot two distinct valid solutions and draw the straight line through them, extending both directions.
- y=k is horizontal and x=k is vertical. A vertical line cannot be written y=mx+b.
- When b≠0 in ax+by=c, rearrange to y=(−a/b)x+c/b; slope is −a/b.
- Set y=0 for the x-intercept and x=0 for the y-intercept, if the relevant intercept exists.
- All points on the line satisfy its equation; points off it do not.
- The x-axis has y=0; the y-axis has x=0. The origin is (0,0).
- An equation can have infinitely many real solutions while a word problem restricts solutions to nonnegative integers.
- Translate a verbal relationship into variables with units and a clear equality.
Use these notes alongside your current textbook. Check important formulas, worked answers and syllabus coverage with your teacher or official learning material.
Two Variables, One Line — NCERT Class 9 Mathematics
Edition: Ganita Manjari (2026–27), Part II, Chapter 13.
Summary: An equation ax+by=c, with a and b not both zero, represents a straight line in the Cartesian plane.
These original notes provide detailed supplementary explanations. Exact NCERT exercise numbers, activities, illustrations and all question-by-question solutions have not been checked against the complete prescribed textbook. Original practice is not represented as an official NCERT answer key.
Learning outcomes
- Use accurate definitions, formulas and theorem conditions.
- Solve and verify numerical or proof-based questions.
- Draw correctly labelled diagrams where appropriate.
- Explain reasoning, units and common errors.
Linear equations in two variables
An equation ax+by=c, with a and b not both zero, represents a straight line in the Cartesian plane.
Worked example
2x+3y=12 is linear in x and y.
Reasoning check
Explain why this method works, state any assumptions, and check the answer independently where possible.
Ordered pair solutions
A solution (x,y) makes the equation true when values are substituted.
Worked example
(3,2) satisfies 2x+3y=12.
Reasoning check
Explain why this method works, state any assumptions, and check the answer independently where possible.
Infinitely many solutions
Over real numbers, one nondegenerate linear equation in two variables has infinitely many ordered-pair solutions.
Worked example
x+y=5 has (0,5),(2,3),(5,0) and infinitely more.
Reasoning check
Explain why this method works, state any assumptions, and check the answer independently where possible.
Generating a solution table
Choose values of x, calculate corresponding y, and check by substitution.
Worked example
For y=2x+1, x=−1,0,1 gives y=−1,1,3.
Reasoning check
Explain why this method works, state any assumptions, and check the answer independently where possible.
Graphing a straight line
Plot two distinct valid solutions and draw the straight line through them, extending both directions.
Worked example
For x+y=4, plot (0,4) and (4,0).
Reasoning check
Explain why this method works, state any assumptions, and check the answer independently where possible.
Horizontal and vertical lines
y=k is horizontal and x=k is vertical. A vertical line cannot be written y=mx+b.
Worked example
y=−3 passes through (0,−3); x=2 passes through (2,0).
Reasoning check
Explain why this method works, state any assumptions, and check the answer independently where possible.
Slope-intercept form
When b≠0 in ax+by=c, rearrange to y=(−a/b)x+c/b; slope is −a/b.
Worked example
2x+y=6 gives y=−2x+6.
Reasoning check
Explain why this method works, state any assumptions, and check the answer independently where possible.
Intercepts
Set y=0 for the x-intercept and x=0 for the y-intercept, if the relevant intercept exists.
Worked example
3x+2y=12 gives (4,0) and (0,6).
Reasoning check
Explain why this method works, state any assumptions, and check the answer independently where possible.
Points on a line
All points on the line satisfy its equation; points off it do not.
Worked example
(2,1) satisfies x+2y=4; (1,1) does not.
Reasoning check
Explain why this method works, state any assumptions, and check the answer independently where possible.
Coordinate axes
The x-axis has y=0; the y-axis has x=0. The origin is (0,0).
Worked example
(0,5) lies on the y-axis.
Reasoning check
Explain why this method works, state any assumptions, and check the answer independently where possible.
Real-life constraints
An equation can have infinitely many real solutions while a word problem restricts solutions to nonnegative integers.
Worked example
If 10p+20n=100 and p,n count objects, only nonnegative integer pairs are meaningful.
Reasoning check
Explain why this method works, state any assumptions, and check the answer independently where possible.
Forming equations
Translate a verbal relationship into variables with units and a clear equality.
Worked example
₹50 fixed charge plus ₹8 per unit gives C=50+8u.
Reasoning check
Explain why this method works, state any assumptions, and check the answer independently where possible.
Line through origin
An equation ax+by=0 passes through (0,0).
Worked example
y=3x passes through the origin.
Reasoning check
Explain why this method works, state any assumptions, and check the answer independently where possible.
Parallel lines
Two nonvertical lines with equal slopes are parallel or coincident; distinct y-intercepts make them distinct parallel lines.
Worked example
y=2x+1 and y=2x−4 are parallel.
Reasoning check
Explain why this method works, state any assumptions, and check the answer independently where possible.
Intersection of two lines
An intersection point satisfies both equations; parallel distinct lines have no common point.
Worked example
x+y=5 and x−y=1 meet at (3,2).
Reasoning check
Explain why this method works, state any assumptions, and check the answer independently where possible.
Scaling an equation
Multiplying both sides by the same nonzero constant does not change the solution set.
Worked example
2x+2y=10 and x+y=5 describe the same line.
Reasoning check
Explain why this method works, state any assumptions, and check the answer independently where possible.
Graphing errors
Use consistent scales and plot x before y; two distinct points are needed to determine a unique line.
Worked example
(2,3) and (3,2) are different points.
Reasoning check
Explain why this method works, state any assumptions, and check the answer independently where possible.
Verification
Substitute the coordinates into the original equation rather than trusting the drawing alone.
Worked example
(4,1) satisfies 2x+3y=11 because 8+3=11.
Reasoning check
Explain why this method works, state any assumptions, and check the answer independently where possible.
Reference table
| Concept | Rule | Illustration |
|---|---|---|
| Linear equations in two variables | An equation ax+by=c, with a and b not both zero, represents a straight line in the Cartesian plane. | 2x+3y=12 is linear in x and y. |
| Ordered pair solutions | A solution (x,y) makes the equation true when values are substituted. | (3,2) satisfies 2x+3y=12. |
| Infinitely many solutions | Over real numbers, one nondegenerate linear equation in two variables has infinitely many ordered-pair solutions. | x+y=5 has (0,5),(2,3),(5,0) and infinitely more. |
| Generating a solution table | Choose values of x, calculate corresponding y, and check by substitution. | For y=2x+1, x=−1,0,1 gives y=−1,1,3. |
| Graphing a straight line | Plot two distinct valid solutions and draw the straight line through them, extending both directions. | For x+y=4, plot (0,4) and (4,0). |
| Horizontal and vertical lines | y=k is horizontal and x=k is vertical. A vertical line cannot be written y=mx+b. | y=−3 passes through (0,−3); x=2 passes through (2,0). |
| Slope-intercept form | When b≠0 in ax+by=c, rearrange to y=(−a/b)x+c/b; slope is −a/b. | 2x+y=6 gives y=−2x+6. |
| Intercepts | Set y=0 for the x-intercept and x=0 for the y-intercept, if the relevant intercept exists. | 3x+2y=12 gives (4,0) and (0,6). |
| Points on a line | All points on the line satisfy its equation; points off it do not. | (2,1) satisfies x+2y=4; (1,1) does not. |
| Coordinate axes | The x-axis has y=0; the y-axis has x=0. The origin is (0,0). | (0,5) lies on the y-axis. |
| Real-life constraints | An equation can have infinitely many real solutions while a word problem restricts solutions to nonnegative integers. | If 10p+20n=100 and p,n count objects, only nonnegative integer pairs are meaningful. |
| Forming equations | Translate a verbal relationship into variables with units and a clear equality. | ₹50 fixed charge plus ₹8 per unit gives C=50+8u. |
| Line through origin | An equation ax+by=0 passes through (0,0). | y=3x passes through the origin. |
| Parallel lines | Two nonvertical lines with equal slopes are parallel or coincident; distinct y-intercepts make them distinct parallel lines. | y=2x+1 and y=2x−4 are parallel. |
| Intersection of two lines | An intersection point satisfies both equations; parallel distinct lines have no common point. | x+y=5 and x−y=1 meet at (3,2). |
| Scaling an equation | Multiplying both sides by the same nonzero constant does not change the solution set. | 2x+2y=10 and x+y=5 describe the same line. |
| Graphing errors | Use consistent scales and plot x before y; two distinct points are needed to determine a unique line. | (2,3) and (3,2) are different points. |
| Verification | Substitute the coordinates into the original equation rather than trusting the drawing alone. | (4,1) satisfies 2x+3y=11 because 8+3=11. |
Exercise A: Concept mastery
- Define linear equations in two variables, state necessary conditions and create one correct example.
- Define ordered pair solutions, state necessary conditions and create one correct example.
- Define infinitely many solutions, state necessary conditions and create one correct example.
- Define generating a solution table, state necessary conditions and create one correct example.
- Define graphing a straight line, state necessary conditions and create one correct example.
- Define horizontal and vertical lines, state necessary conditions and create one correct example.
- Define slope-intercept form, state necessary conditions and create one correct example.
- Define intercepts, state necessary conditions and create one correct example.
- Define points on a line, state necessary conditions and create one correct example.
- Define coordinate axes, state necessary conditions and create one correct example.
- Define real-life constraints, state necessary conditions and create one correct example.
- Define forming equations, state necessary conditions and create one correct example.
Exercise B: Applications and worked problems
- Verify this example: 2x+3y=12 is linear in x and y. Change valid numerical values and solve a related problem with complete steps.
- Verify this example: (3,2) satisfies 2x+3y=12. Change valid numerical values and solve a related problem with complete steps.
- Verify this example: x+y=5 has (0,5),(2,3),(5,0) and infinitely more. Change valid numerical values and solve a related problem with complete steps.
- Verify this example: For y=2x+1, x=−1,0,1 gives y=−1,1,3. Change valid numerical values and solve a related problem with complete steps.
- Verify this example: For x+y=4, plot (0,4) and (4,0). Change valid numerical values and solve a related problem with complete steps.
- Verify this example: y=−3 passes through (0,−3); x=2 passes through (2,0). Change valid numerical values and solve a related problem with complete steps.
- Verify this example: 2x+y=6 gives y=−2x+6. Change valid numerical values and solve a related problem with complete steps.
- Verify this example: 3x+2y=12 gives (4,0) and (0,6). Change valid numerical values and solve a related problem with complete steps.
- Verify this example: (2,1) satisfies x+2y=4; (1,1) does not. Change valid numerical values and solve a related problem with complete steps.
- Verify this example: (0,5) lies on the y-axis. Change valid numerical values and solve a related problem with complete steps.
- Verify this example: If 10p+20n=100 and p,n count objects, only nonnegative integer pairs are meaningful. Change valid numerical values and solve a related problem with complete steps.
- Verify this example: ₹50 fixed charge plus ₹8 per unit gives C=50+8u. Change valid numerical values and solve a related problem with complete steps.
Exercise C: Challenge and reasoning
- Prove or derive a central theorem or formula.
- Provide a counterexample to a false generalisation.
- Compare two similar concepts and explain the difference.
- Write a practical application with realistic units and constraints.
- Find and correct an error in a proposed solution.
- Check an answer using a second valid method.
- Create and solve a multi-step original problem.
- Explain a concept using a labelled diagram or table.
FAQs with answers
Explain linear equations in two variables with an example.
An equation ax+by=c, with a and b not both zero, represents a straight line in the Cartesian plane. Example: 2x+3y=12 is linear in x and y.
Explain ordered pair solutions with an example.
A solution (x,y) makes the equation true when values are substituted. Example: (3,2) satisfies 2x+3y=12.
Explain infinitely many solutions with an example.
Over real numbers, one nondegenerate linear equation in two variables has infinitely many ordered-pair solutions. Example: x+y=5 has (0,5),(2,3),(5,0) and infinitely more.
Explain generating a solution table with an example.
Choose values of x, calculate corresponding y, and check by substitution. Example: For y=2x+1, x=−1,0,1 gives y=−1,1,3.
Explain graphing a straight line with an example.
Plot two distinct valid solutions and draw the straight line through them, extending both directions. Example: For x+y=4, plot (0,4) and (4,0).
Explain horizontal and vertical lines with an example.
y=k is horizontal and x=k is vertical. A vertical line cannot be written y=mx+b. Example: y=−3 passes through (0,−3); x=2 passes through (2,0).
Explain slope-intercept form with an example.
When b≠0 in ax+by=c, rearrange to y=(−a/b)x+c/b; slope is −a/b. Example: 2x+y=6 gives y=−2x+6.
Explain intercepts with an example.
Set y=0 for the x-intercept and x=0 for the y-intercept, if the relevant intercept exists. Example: 3x+2y=12 gives (4,0) and (0,6).
Explain points on a line with an example.
All points on the line satisfy its equation; points off it do not. Example: (2,1) satisfies x+2y=4; (1,1) does not.
Explain coordinate axes with an example.
The x-axis has y=0; the y-axis has x=0. The origin is (0,0). Example: (0,5) lies on the y-axis.
Explain real-life constraints with an example.
An equation can have infinitely many real solutions while a word problem restricts solutions to nonnegative integers. Example: If 10p+20n=100 and p,n count objects, only nonnegative integer pairs are meaningful.
Explain forming equations with an example.
Translate a verbal relationship into variables with units and a clear equality. Example: ₹50 fixed charge plus ₹8 per unit gives C=50+8u.
Explain line through origin with an example.
An equation ax+by=0 passes through (0,0). Example: y=3x passes through the origin.
Explain parallel lines with an example.
Two nonvertical lines with equal slopes are parallel or coincident; distinct y-intercepts make them distinct parallel lines. Example: y=2x+1 and y=2x−4 are parallel.
Explain intersection of two lines with an example.
An intersection point satisfies both equations; parallel distinct lines have no common point. Example: x+y=5 and x−y=1 meet at (3,2).
Revision checklist
- I can explain and apply linear equations in two variables.
- I can explain and apply ordered pair solutions.
- I can explain and apply infinitely many solutions.
- I can explain and apply generating a solution table.
- I can explain and apply graphing a straight line.
- I can explain and apply horizontal and vertical lines.
- I can explain and apply slope-intercept form.
- I can explain and apply intercepts.
- I can explain and apply points on a line.
- I can explain and apply coordinate axes.
- I can explain and apply real-life constraints.
- I can explain and apply forming equations.
- I can explain and apply line through origin.
- I can explain and apply parallel lines.
- I can explain and apply intersection of two lines.
- I can explain and apply scaling an equation.
- I can explain and apply graphing errors.
- I can explain and apply verification.
Official textbook reference
NCERT official textbook portal. These are original supplementary notes and practice questions, not verified official numbered exercise solutions.
Frequently asked questions
Explain linear equations in two variables with an example.
An equation ax+by=c, with a and b not both zero, represents a straight line in the Cartesian plane. Example: 2x+3y=12 is linear in x and y.
Explain ordered pair solutions with an example.
A solution (x,y) makes the equation true when values are substituted. Example: (3,2) satisfies 2x+3y=12.
Explain infinitely many solutions with an example.
Over real numbers, one nondegenerate linear equation in two variables has infinitely many ordered-pair solutions. Example: x+y=5 has (0,5),(2,3),(5,0) and infinitely more.
Explain generating a solution table with an example.
Choose values of x, calculate corresponding y, and check by substitution. Example: For y=2x+1, x=−1,0,1 gives y=−1,1,3.
Explain graphing a straight line with an example.
Plot two distinct valid solutions and draw the straight line through them, extending both directions. Example: For x+y=4, plot (0,4) and (4,0).
Explain horizontal and vertical lines with an example.
y=k is horizontal and x=k is vertical. A vertical line cannot be written y=mx+b. Example: y=−3 passes through (0,−3); x=2 passes through (2,0).
Explain slope-intercept form with an example.
When b≠0 in ax+by=c, rearrange to y=(−a/b)x+c/b; slope is −a/b. Example: 2x+y=6 gives y=−2x+6.
Explain intercepts with an example.
Set y=0 for the x-intercept and x=0 for the y-intercept, if the relevant intercept exists. Example: 3x+2y=12 gives (4,0) and (0,6).
Explain points on a line with an example.
All points on the line satisfy its equation; points off it do not. Example: (2,1) satisfies x+2y=4; (1,1) does not.
Explain coordinate axes with an example.
The x-axis has y=0; the y-axis has x=0. The origin is (0,0). Example: (0,5) lies on the y-axis.
Explain real-life constraints with an example.
An equation can have infinitely many real solutions while a word problem restricts solutions to nonnegative integers. Example: If 10p+20n=100 and p,n count objects, only nonnegative integer pairs are meaningful.
Explain forming equations with an example.
Translate a verbal relationship into variables with units and a clear equality. Example: ₹50 fixed charge plus ₹8 per unit gives C=50+8u.
Explain line through origin with an example.
An equation ax+by=0 passes through (0,0). Example: y=3x passes through the origin.
Explain parallel lines with an example.
Two nonvertical lines with equal slopes are parallel or coincident; distinct y-intercepts make them distinct parallel lines. Example: y=2x+1 and y=2x−4 are parallel.
Explain intersection of two lines with an example.
An intersection point satisfies both equations; parallel distinct lines have no common point. Example: x+y=5 and x−y=1 meet at (3,2).
