Complete Solutions and Summary of Linear Equations in Two Variables – NCERT Class 9, Mathematics, Chapter 4 – Summary, Questions, Answers, Extra Questions
Detailed summary and explanation of Chapter 4 'Linear Equations in Two Variables' with all question answers, extra questions, and solutions from NCERT Class IX, Mathematics.
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Categories: NCERT, Class IX, Mathematics, Summary, Extra Questions, Linear Equations in Two Variables, Chapter 4
Tags: Linear Equations, Two Variables, Solutions, Cartesian Plane, Graphs, Unique Solution, Infinite Solutions, Ordered Pair, Equation Format, ax+by+c=0, Substitution Method, Elimination Method, Graphical Method, NCERT, Mathematics, Class 9, Chapter 4, Answers, Extra Questions
Linear Equations in Two Variables
Chapter 4: Mathematics - Complete Study Guide
Chapter Overview
Linear
ax+by+c=0
Solutions
Infinite
Graph
Straight Line
Parallel
Axes Lines
What You'll Learn
Equation Form
ax + by + c = 0.
Solutions
Infinite solutions, graph line.
Graphing
Plot equations on plane.
Special Cases
Parallel to axes.
Key Highlights
Linear equations in two variables have form ax + by + c = 0, infinite solutions forming straight line graph. Solutions satisfy equation. Graph by finding points.
Comprehensive Chapter Summary
1. Introduction
- Recall linear equations in one variable: x + 1 = 0, solution unique.
- Extend to two variables: Operations, factorisation from earlier.
- Questions: Has solution? Unique? On plane?
- Identities: (x+y)^2 etc. for factorisation.
- Build on Ch 3 concepts.
- Applications: Real-life like scores.
- Equation solving principles: Add/subtract same, multiply/divide non-zero.
- Examples: Cricket scores x+y=176.
- General form: ax+by+c=0.
- Examples: Rewrite to standard.
- a,b not both zero.
- More examples, values of a,b,c.
- Transition to solutions.
- Graph representation.
- Importance in math.
- Edmund Halley quote on equations.
Cricket Scores
x + y = 176 equation.
2. Linear Equations
- Form ax + by + c = 0, a,b,c real, a,b not both zero.
- Examples: 2x + 3y = 4.37 as 2x + 3y -4.37=0.
- x-4=\sqrt{3} y as x - \sqrt{3} y -4=0.
- 4=5x-3y as 5x -3y -4=0.
- 2x=y as 2x - y =0.
- One variable recall: Unique solution.
- Two variables: Infinite solutions.
- Solution: Pair (x,y) satisfying equation.
- Examples: For x+y=4, (0,4),(1,3) etc.
- Any real x, y=4-x solution.
- Graph: Straight line.
- Points on line solutions.
- More examples: 2x=3, y=5.
- General: Infinite on line.
- Notation: Ordered pair.
- Verification: Substitute.
Standard Form
ax + by + c = 0.
Solutions
Infinite pairs.
Example: Solutions
For x+y=4: (0,4),(1,3),(2,2),(3,1).
3. Solution of a Linear Equation
- Infinite solutions: Any on line.
- One variable: One solution.
- Two variables: Infinite.
- Examples: Find four solutions for equations.
- Method: Choose x, find y.
- Graphically: Line intersection with values.
- Applications: Real problems.
- More solutions possible.
- Non-integer solutions if needed.
- Positive solutions in context.
- Verification important.
- Extension to systems later.
4. Graph of a Linear Equation in Two Variables
- Plot points: Find solutions, plot.
- Straight line: Join points.
- Examples: Graph x+y=4.
- Table: x,y pairs.
- Any point on line solution.
- Check points.
- More examples: 3x+2y=6.
- Infinite points on line.
- Cartesian plane use.
- Scale on axes.
- Applications: Trends, predictions.
- Intercepts: x-intercept, y-intercept.
Example: Graph
x+y=4 line.
5. Equations of Lines Parallel to the x-axis and y-axis
- Parallel to x-axis: y=constant.
- Horizontal line.
- Parallel to y-axis: x=constant.
- Vertical line.
- Examples: y=4, x=-2.
- Graph: Horizontal/vertical.
- Solutions: All x for y=constant.
- Infinite again.
- Special cases of linear.
- a=0 or b=0.
- Applications: Boundaries.
- Summary: All linear graph straight lines.
Key Concepts and Definitions
Linear Equation
ax + by + c = 0.
Solution
(x,y) satisfying equation.
Graph
Straight line.
Horizontal
y = k.
Vertical
x = k.
Ordered Pair
(x,y).
Infinite Solutions
All on line.
Important Facts
One Var
Unique Sol
Two Var
Infinite
Graph
Line
a,b
Not Both 0
Parallel x
y=k
Questions and Answers from Chapter
Short Questions (1 Mark)
Q1. Is 2x + 3y = 4.37 a linear equation in two variables?
Answer: Yes.
Q2. Is x – 4 = \sqrt{3} y a linear equation?
Answer: Yes.
Q3. Is 4 = 5x – 3y a linear equation?
Answer: Yes.
Q4. Is 2x = y a linear equation?
Answer: Yes.
Q5. For 2x + 3y = 4.37, a=?
Answer: 2.
Q6. For x – 4 = \sqrt{3} y, b=?
Answer: -\sqrt{3}.
Q7. For 4 = 5x – 3y, c=?
Answer: -4.
Q8. For 2x = y, a=?
Answer: 2.
Q9. Is x^2 + x = 2 linear in two?
Answer: No.
Q10. Is 3x + 2 = 1 linear in two?
Answer: No.
Q11. Is y - \sqrt{2} = 0 linear in two?
Answer: No.
Q12. Is x + y = 5 linear in two?
Answer: Yes.
Q13. One solution of x + y = 5?
Answer: (0,5).
Q14. Graph of linear is?
Answer: Line.
Q15. Equation parallel to x-axis?
Answer: y=k.
Q16. Equation parallel to y-axis?
Answer: x=k.
Q17. For \pi x + y = 9, one solution?
Answer: (0,9).
Q18. For x = 4y, one solution?
Answer: (0,0).
Q19. Number of solutions?
Answer: Infinite.
Q20. Is a=b=0 allowed?
Answer: No.
Medium Questions (3 Marks)
Q1. Write four solutions for x + y = 5.
Answer: (0,5),(1,4),(2,3),(3,2).
Q2. Write four solutions for \pi x + y = 9.
Answer: (0,9),(1,9-\pi),(2,9-2\pi),(3,9-3\pi).
Q3. Write four solutions for x = 4y.
Answer: (0,0),(4,1),(8,2),(12,3).
Q4. Find two solutions for x + 2y = 0.
Answer: (0,0),(2,-1).
Q5. Find two solutions for y – 2x = – 2.
Answer: (0,-2),(1,0).
Q6. Find two solutions for 4x + 3y = 12.
Answer: (0,4),(3,0).
Q7. Find two solutions for y = 3x + 5.
Answer: (0,5),(1,8).
Q8. Find two solutions for 2x + 5y = 0.
Answer: (0,0),(5,-2).
Q9. Find two solutions for 3y + 4 = 0.
Answer: y=-4/3, any x.
Q10. Find two solutions for x – y = 4.
Answer: (4,0),(5,1).
Q11. Draw graph of x + y = 7.
Answer: Points (0,7),(7,0), line.
Q12. Draw graph of x – y = 2.
Answer: (2,0),(0,-2), line.
Q13. Draw graph of 2x + 3y = 6.
Answer: (0,2),(3,0), line.
Q14. Draw graph of 3x + 2y = 14.
Answer: (0,7),( 14/3,0 ), line.
Q15. Draw graph of 2y = 4x – 6.
Answer: (0,-3),(3,3), line.
Q16. Draw graph of y = x.
Answer: (0,0),(1,1), line.
Q17. Draw graph of y = x + 3.
Answer: (0,3),(1,4), line.
Q18. Draw graph of x = -2.
Answer: Vertical line at x=-2.
Q19. Draw graph of y = 4.
Answer: Horizontal line at y=4.
Q20. Find if (2,0) solution of x – 2y = 4.
Answer: No.
Long Questions (6 Marks)
Q1. Write standard form and a,b,c for 2x + 3y = 4.37.
Answer: 2x + 3y - 4.37 = 0, a=2, b=3, c=-4.37.
Q2. Write standard form and a,b,c for x – 4 = \sqrt{3} y.
Answer: x - \sqrt{3} y - 4 = 0, a=1, b=-\sqrt{3}, c=-4.
Q3. Write standard form and a,b,c for 4 = 5x – 3y.
Answer: 5x - 3y - 4 = 0, a=5, b=-3, c=-4.
Q4. Write standard form and a,b,c for 2x = y.
Answer: 2x - y = 0, a=2, b=-1, c=0.
Q5. Find four solutions for x + y = 5.
Answer: Choose x=0, y=5; x=1, y=4; x=2, y=3; x=3, y=2.
Q6. Find four solutions for \pi x + y = 9.
Answer: x=0, y=9; x=1, y=9-\pi; x=2, y=9-2\pi; x=3, y=9-3\pi.
Q7. Find four solutions for x = 4y.
Answer: y=0, x=0; y=1, x=4; y=2, x=8; y=3, x=12.
Q8. Find two solutions for x + 2y = 0.
Answer: (0,0), (2,-1). Verify by substitution.
Q9. Find two solutions for y – 2x = – 2.
Answer: (0,-2), (1,0). Verify.
Q10. Find two solutions for 4x + 3y = 12.
Answer: (0,4), (3,0). Verify.
Q11. Find two solutions for y = 3x + 5.
Answer: (0,5), (-1,2). Verify.
Q12. Find two solutions for 2x + 5y = 0.
Answer: (0,0), (5,-2). Verify.
Q13. Find two solutions for 3y + 4 = 0.
Answer: y=-4/3, any x like (0,-4/3), (1,-4/3).
Q14. Find two solutions for x – y = 4.
Answer: (4,0), (5,1). Verify.
Q15. Draw graph of x + y = 7 and find solutions.
Answer: Points (0,7),(7,0), line. Any on line solution.
Q16. Draw graph of x – y = 2 and find solutions.
Answer: (2,0),(0,-2), line.
Q17. Draw graph of 2x + 3y = 6 and find solutions.
Answer: (0,2),(3,0), line.
Q18. Draw graph of 3x + 2y = 14 and find solutions.
Answer: (0,7),( 14/3,0 ), line.
Q19. Draw graph of 2y = 4x – 6 and find solutions.
Answer: (0,-3),(3,3), line.
Q20. Draw graph of y = x and find solutions.
Answer: (0,0),(1,1), line through origin.
Interactive Knowledge Quiz
Test your understanding of Linear Equations in Two Variables
Quick Revision Notes
Form
- ax + by + c = 0
Solutions
- Infinite
- On line
Graph
- Straight line
Exam Strategy Tips
- Rewrite standard
- Find solutions
- Plot graphs
- Verify points
- Special lines
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