Question 1

Question 2(i)Express the following linear equations in the form ax + by + c = 0 and indicate the values of a, b and c in each case :2x + 3y = 9.35‾9.3\overline{5}9.35

Accepted Answer

Given,⇒ 2x + 3y = 9.35‾9.3\overline{5}9.35On re-arranging the equation, we have :⇒ 2x + 3y - 9.35‾9.3\overline{5}9.35 = 0⇒ (2)x + (3)y + (-9.35‾9.3\overline{5}9.35) = 0Comparing above equation with ax + by + c = 0, we get :a = 2, b = 3 and c = -9.35‾9.3\overline{5}9.35Hence, linear equation : 2x + 3y - 9.35‾9.3\overline{5}9.35 = 0, a = 2, b = 3 and c = -9.35‾9.3\overline{5}9.35.

Linear Equations in Two Variables - Interactive Study Notes | Bright Tutorials

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BRIGHT TUTORIALS

BRIGHT TUTORIALSCBSE Class IX | Academic Year 2026-2027
    
    9403781999Excellence in Education
  
  Mathematics | Linear Equations in Two VariablesWeb Content &bull; Interactive Notes
  
    Linear Equations in Two Variables &mdash; Interactive Study Guide
    
      Infinite Solutions
      A linear equation in two variables has infinitely many solutions. Each solution is a point on the line. The line extends forever in both directions.
      Key Concept: One equation, two unknowns = infinite solutions. You need TWO equations to get a unique solution (Class X topic).

Graphing Made Easy
      
        Rewrite as y = mx + c.
        Make a table: choose 3 values of x, calculate y for each.
        Plot the 3 points. If they&rsquo;re collinear, draw the line.

Special Lines
      y = 3: Horizontal line, 3 units above x-axis.
      x = &minus;2: Vertical line, 2 units left of y-axis.
      y = x: Line at 45&deg; through origin.

Quick Self-Check
      
        Is (2, 3) a solution of x + y = 5? (Yes: 2 + 3 = 5)
        Give 3 solutions of y = 2x. ((0,0), (1,2), (2,4))
        What is the equation of the x-axis? (y = 0)

Bright Tutorials | Hariom Nagar, Nashik Road | 9403781999 | brighttutorials.in

Question 2

Question 2(ii)Express the following linear equations in the form ax + by + c = 0 and indicate the values of a, b and c in each case :x - y5\dfrac{y}{5}5y​ - 10 = 0

Accepted Answer

Given,⇒ x - y5\dfrac{y}{5}5y​ - 10 = 0⇒ (1)x + (−15)y\Big(-\dfrac{1}{5}\Big)y(−51​)y + (-10) = 0Comparing above equation with ax + by + c = 0, we get :a = 1, b = -15\dfrac{1}{5}51​ and c = -10Hence, linear equation : x - y5\dfrac{y}{5}5y​ - 10 = 0, a = 1, b = −15-\dfrac{1}{5}−51​ and c = -10.

Linear Equations in Two Variables - Interactive Study Notes | Bright Tutorials

.web-topic { background: var(--bt-bg-light); border-radius: 10px; padding: 20px; margin: 20px 0; border: 1px solid var(--bt-border-light); }
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    .key-point { background: #eaf2f8; padding: 10px 15px; border-left: 4px solid var(--bt-primary); margin: 10px 0; border-radius: 4px; }
    .quick-quiz { background: #fff8e1; padding: 15px; border-radius: 8px; border: 2px solid #f39c12; margin: 15px 0; }
    .quick-quiz h4 { color: #e67e22; margin-top: 0; }

BRIGHT TUTORIALS

BRIGHT TUTORIALSCBSE Class IX | Academic Year 2026-2027
    
    9403781999Excellence in Education
  
  Mathematics | Linear Equations in Two VariablesWeb Content &bull; Interactive Notes
  
    Linear Equations in Two Variables &mdash; Interactive Study Guide
    
      Infinite Solutions
      A linear equation in two variables has infinitely many solutions. Each solution is a point on the line. The line extends forever in both directions.
      Key Concept: One equation, two unknowns = infinite solutions. You need TWO equations to get a unique solution (Class X topic).

Graphing Made Easy
      
        Rewrite as y = mx + c.
        Make a table: choose 3 values of x, calculate y for each.
        Plot the 3 points. If they&rsquo;re collinear, draw the line.

Special Lines
      y = 3: Horizontal line, 3 units above x-axis.
      x = &minus;2: Vertical line, 2 units left of y-axis.
      y = x: Line at 45&deg; through origin.

Quick Self-Check
      
        Is (2, 3) a solution of x + y = 5? (Yes: 2 + 3 = 5)
        Give 3 solutions of y = 2x. ((0,0), (1,2), (2,4))
        What is the equation of the x-axis? (y = 0)

Bright Tutorials | Hariom Nagar, Nashik Road | 9403781999 | brighttutorials.in

Question 3

Question 2(v)Express the following linear equations in the form ax + by + c = 0 and indicate the values of a, b and c in each case :2x = -5y

Accepted Answer

Given,⇒ 2x = -5yOn re-arranging the equation, we have :⇒ 2x + 5y = 0⇒ (2)x + (5)y + (0) = 0Comparing above equation with ax + by + c = 0, we get :a = 2, b = 5 and c = 0Hence, linear equation : 2x + 5y + 0 = 0, a = 2, b = 5 and c = 0.

Linear Equations in Two Variables - Interactive Study Notes | Bright Tutorials

.web-topic { background: var(--bt-bg-light); border-radius: 10px; padding: 20px; margin: 20px 0; border: 1px solid var(--bt-border-light); }
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    .key-point { background: #eaf2f8; padding: 10px 15px; border-left: 4px solid var(--bt-primary); margin: 10px 0; border-radius: 4px; }
    .quick-quiz { background: #fff8e1; padding: 15px; border-radius: 8px; border: 2px solid #f39c12; margin: 15px 0; }
    .quick-quiz h4 { color: #e67e22; margin-top: 0; }

BRIGHT TUTORIALS

BRIGHT TUTORIALSCBSE Class IX | Academic Year 2026-2027
    
    9403781999Excellence in Education
  
  Mathematics | Linear Equations in Two VariablesWeb Content &bull; Interactive Notes
  
    Linear Equations in Two Variables &mdash; Interactive Study Guide
    
      Infinite Solutions
      A linear equation in two variables has infinitely many solutions. Each solution is a point on the line. The line extends forever in both directions.
      Key Concept: One equation, two unknowns = infinite solutions. You need TWO equations to get a unique solution (Class X topic).

Graphing Made Easy
      
        Rewrite as y = mx + c.
        Make a table: choose 3 values of x, calculate y for each.
        Plot the 3 points. If they&rsquo;re collinear, draw the line.

Special Lines
      y = 3: Horizontal line, 3 units above x-axis.
      x = &minus;2: Vertical line, 2 units left of y-axis.
      y = x: Line at 45&deg; through origin.

Quick Self-Check
      
        Is (2, 3) a solution of x + y = 5? (Yes: 2 + 3 = 5)
        Give 3 solutions of y = 2x. ((0,0), (1,2), (2,4))
        What is the equation of the x-axis? (y = 0)

Bright Tutorials | Hariom Nagar, Nashik Road | 9403781999 | brighttutorials.in

Question 4

Question 2(vi)Express the following linear equations in the form ax + by + c = 0 and indicate the values of a, b and c in each case :3x + 2 = 0

Accepted Answer

Given,.⇒ 3x + 2 = 0On re-arranging the equation, we have :⇒ 3x + 2 = 0⇒ (3)x + (0)y + 2 = 0Comparing above equation with ax + by + c = 0, we get :a = 3, b = 0 and c = 2Hence, linear equation : 3x + 0.y + 2 = 0, a = 3, b = 0 and c = 2.

Linear Equations in Two Variables - Interactive Study Notes | Bright Tutorials

.web-topic { background: var(--bt-bg-light); border-radius: 10px; padding: 20px; margin: 20px 0; border: 1px solid var(--bt-border-light); }
    .web-topic h3 { color: var(--bt-primary); margin-top: 0; }
    .key-point { background: #eaf2f8; padding: 10px 15px; border-left: 4px solid var(--bt-primary); margin: 10px 0; border-radius: 4px; }
    .quick-quiz { background: #fff8e1; padding: 15px; border-radius: 8px; border: 2px solid #f39c12; margin: 15px 0; }
    .quick-quiz h4 { color: #e67e22; margin-top: 0; }

BRIGHT TUTORIALS

BRIGHT TUTORIALSCBSE Class IX | Academic Year 2026-2027
    
    9403781999Excellence in Education
  
  Mathematics | Linear Equations in Two VariablesWeb Content &bull; Interactive Notes
  
    Linear Equations in Two Variables &mdash; Interactive Study Guide
    
      Infinite Solutions
      A linear equation in two variables has infinitely many solutions. Each solution is a point on the line. The line extends forever in both directions.
      Key Concept: One equation, two unknowns = infinite solutions. You need TWO equations to get a unique solution (Class X topic).

Graphing Made Easy
      
        Rewrite as y = mx + c.
        Make a table: choose 3 values of x, calculate y for each.
        Plot the 3 points. If they&rsquo;re collinear, draw the line.

Special Lines
      y = 3: Horizontal line, 3 units above x-axis.
      x = &minus;2: Vertical line, 2 units left of y-axis.
      y = x: Line at 45&deg; through origin.

Quick Self-Check
      
        Is (2, 3) a solution of x + y = 5? (Yes: 2 + 3 = 5)
        Give 3 solutions of y = 2x. ((0,0), (1,2), (2,4))
        What is the equation of the x-axis? (y = 0)

Bright Tutorials | Hariom Nagar, Nashik Road | 9403781999 | brighttutorials.in

Question 5

Question 2(vii)Express the following linear equations in the form ax + by + c = 0 and indicate the values of a, b and c in each case :y - 2 = 0

Accepted Answer

Given,⇒ y - 2 = 0⇒ (0)x + (1)y + (-2) = 0Comparing above equation with ax + by + c = 0, we get :a = 0, b = 1 and c = -2Hence, linear equation : 0.x + 1.y - 2 = 0, a = 0, b = 1 and c = -2.

Linear Equations in Two Variables - Interactive Study Notes | Bright Tutorials

.web-topic { background: var(--bt-bg-light); border-radius: 10px; padding: 20px; margin: 20px 0; border: 1px solid var(--bt-border-light); }
    .web-topic h3 { color: var(--bt-primary); margin-top: 0; }
    .key-point { background: #eaf2f8; padding: 10px 15px; border-left: 4px solid var(--bt-primary); margin: 10px 0; border-radius: 4px; }
    .quick-quiz { background: #fff8e1; padding: 15px; border-radius: 8px; border: 2px solid #f39c12; margin: 15px 0; }
    .quick-quiz h4 { color: #e67e22; margin-top: 0; }

BRIGHT TUTORIALS

BRIGHT TUTORIALSCBSE Class IX | Academic Year 2026-2027
    
    9403781999Excellence in Education
  
  Mathematics | Linear Equations in Two VariablesWeb Content &bull; Interactive Notes
  
    Linear Equations in Two Variables &mdash; Interactive Study Guide
    
      Infinite Solutions
      A linear equation in two variables has infinitely many solutions. Each solution is a point on the line. The line extends forever in both directions.
      Key Concept: One equation, two unknowns = infinite solutions. You need TWO equations to get a unique solution (Class X topic).

Graphing Made Easy
      
        Rewrite as y = mx + c.
        Make a table: choose 3 values of x, calculate y for each.
        Plot the 3 points. If they&rsquo;re collinear, draw the line.

Special Lines
      y = 3: Horizontal line, 3 units above x-axis.
      x = &minus;2: Vertical line, 2 units left of y-axis.
      y = x: Line at 45&deg; through origin.

Quick Self-Check
      
        Is (2, 3) a solution of x + y = 5? (Yes: 2 + 3 = 5)
        Give 3 solutions of y = 2x. ((0,0), (1,2), (2,4))
        What is the equation of the x-axis? (y = 0)

Bright Tutorials | Hariom Nagar, Nashik Road | 9403781999 | brighttutorials.in

Question 6

Question 2(viii)Express the following linear equations in the form ax + by + c = 0 and indicate the values of a, b and c in each case :5 = 2x

Accepted Answer

Given,⇒ 5 = 2xOn re-arranging the equation, we have :⇒ -2x + 5 = 0⇒ (-2)x + (0)y + (5) = 0Comparing above equation with ax + by + c = 0, we get :a = -2, b = 0 and c = 5Hence, linear equation : -2x + 0.y + 5 = 0, a = -2, b = 0 and c = 5.

Linear Equations in Two Variables - Interactive Study Notes | Bright Tutorials

.web-topic { background: var(--bt-bg-light); border-radius: 10px; padding: 20px; margin: 20px 0; border: 1px solid var(--bt-border-light); }
    .web-topic h3 { color: var(--bt-primary); margin-top: 0; }
    .key-point { background: #eaf2f8; padding: 10px 15px; border-left: 4px solid var(--bt-primary); margin: 10px 0; border-radius: 4px; }
    .quick-quiz { background: #fff8e1; padding: 15px; border-radius: 8px; border: 2px solid #f39c12; margin: 15px 0; }
    .quick-quiz h4 { color: #e67e22; margin-top: 0; }

BRIGHT TUTORIALS

BRIGHT TUTORIALSCBSE Class IX | Academic Year 2026-2027
    
    9403781999Excellence in Education
  
  Mathematics | Linear Equations in Two VariablesWeb Content &bull; Interactive Notes
  
    Linear Equations in Two Variables &mdash; Interactive Study Guide
    
      Infinite Solutions
      A linear equation in two variables has infinitely many solutions. Each solution is a point on the line. The line extends forever in both directions.
      Key Concept: One equation, two unknowns = infinite solutions. You need TWO equations to get a unique solution (Class X topic).

Graphing Made Easy
      
        Rewrite as y = mx + c.
        Make a table: choose 3 values of x, calculate y for each.
        Plot the 3 points. If they&rsquo;re collinear, draw the line.

Special Lines
      y = 3: Horizontal line, 3 units above x-axis.
      x = &minus;2: Vertical line, 2 units left of y-axis.
      y = x: Line at 45&deg; through origin.

Quick Self-Check
      
        Is (2, 3) a solution of x + y = 5? (Yes: 2 + 3 = 5)
        Give 3 solutions of y = 2x. ((0,0), (1,2), (2,4))
        What is the equation of the x-axis? (y = 0)

Bright Tutorials | Hariom Nagar, Nashik Road | 9403781999 | brighttutorials.in

Question 7

Question 1Which one of the following options is true, and why?y = 3x + 5 has(i) a unique solution(ii) only two solutions(iii) infinitely many solutions

Accepted Answer

Given, linear equation y = 3x + 5We know that,y = 3x + 5 is a linear equation in two variables in the form of ax + by + c = 0Substituting x = 0, in y = 3x + 5, we get :⇒ y = 3(0) + 5 = 0 + 5 = 5.∴ (0, 5) is one solution.Substituting x = 1, in y = 3x + 5, we get :⇒ y = 3(1) + 5 = 3 + 5 = 8.∴ (1, 8) is another solution.Substituting x = 2, in y = 3x + 5, we get :⇒ y = 3(2) + 5 = 6 + 5 = 11.∴ (2, 11) is another solution.Clearly, for different values of x, we get different values of y.Thus, y = 3x + 5 has infinitely many solutions.Hence, Option (iii) is the correct answer.

Linear Equations in Two Variables - Interactive Study Notes | Bright Tutorials

.web-topic { background: var(--bt-bg-light); border-radius: 10px; padding: 20px; margin: 20px 0; border: 1px solid var(--bt-border-light); }
    .web-topic h3 { color: var(--bt-primary); margin-top: 0; }
    .key-point { background: #eaf2f8; padding: 10px 15px; border-left: 4px solid var(--bt-primary); margin: 10px 0; border-radius: 4px; }
    .quick-quiz { background: #fff8e1; padding: 15px; border-radius: 8px; border: 2px solid #f39c12; margin: 15px 0; }
    .quick-quiz h4 { color: #e67e22; margin-top: 0; }

BRIGHT TUTORIALS

BRIGHT TUTORIALSCBSE Class IX | Academic Year 2026-2027
    
    9403781999Excellence in Education
  
  Mathematics | Linear Equations in Two VariablesWeb Content &bull; Interactive Notes
  
    Linear Equations in Two Variables &mdash; Interactive Study Guide
    
      Infinite Solutions
      A linear equation in two variables has infinitely many solutions. Each solution is a point on the line. The line extends forever in both directions.
      Key Concept: One equation, two unknowns = infinite solutions. You need TWO equations to get a unique solution (Class X topic).

Graphing Made Easy
      
        Rewrite as y = mx + c.
        Make a table: choose 3 values of x, calculate y for each.
        Plot the 3 points. If they&rsquo;re collinear, draw the line.

Special Lines
      y = 3: Horizontal line, 3 units above x-axis.
      x = &minus;2: Vertical line, 2 units left of y-axis.
      y = x: Line at 45&deg; through origin.

Quick Self-Check
      
        Is (2, 3) a solution of x + y = 5? (Yes: 2 + 3 = 5)
        Give 3 solutions of y = 2x. ((0,0), (1,2), (2,4))
        What is the equation of the x-axis? (y = 0)

Bright Tutorials | Hariom Nagar, Nashik Road | 9403781999 | brighttutorials.in

Question 8

Question 2(iii)Write four solutions for the following equations:x = 4y

Accepted Answer

Given, equation :⇒ x = 4ySubstituting x = 0, in the given equation, we get :⇒ 0 = 4y⇒ y = 04\dfrac{0}{4}40​⇒ y = 0∴ Solution is (0, 0).Substituting x = 1, in the given equation, we get :⇒ 1 = 4y⇒ y = 14\dfrac{1}{4}41​.∴ Solution is (1,14)\Big(1, \dfrac{1}{4}\Big)(1,41​).Substituting x = 4, in the given equation, we get :⇒ 4 = 4y⇒ y = 44\dfrac{4}{4}44​⇒ y = 1∴ Solution is (4, 1).Substituting x = -4, in the given equation, we get :⇒ -4 = 4y⇒ y = −44\dfrac{-4}{4}4−4​⇒ y = -1∴ Solution is (-4, -1).Hence, solutions for equation x = 4y are (0, 0), (1,14)(1, \dfrac{1}{4}\Big)(1,41​), (4, 1) and (-4, -1).

Linear Equations in Two Variables - Interactive Study Notes | Bright Tutorials

.web-topic { background: var(--bt-bg-light); border-radius: 10px; padding: 20px; margin: 20px 0; border: 1px solid var(--bt-border-light); }
    .web-topic h3 { color: var(--bt-primary); margin-top: 0; }
    .key-point { background: #eaf2f8; padding: 10px 15px; border-left: 4px solid var(--bt-primary); margin: 10px 0; border-radius: 4px; }
    .quick-quiz { background: #fff8e1; padding: 15px; border-radius: 8px; border: 2px solid #f39c12; margin: 15px 0; }
    .quick-quiz h4 { color: #e67e22; margin-top: 0; }

BRIGHT TUTORIALS

BRIGHT TUTORIALSCBSE Class IX | Academic Year 2026-2027
    
    9403781999Excellence in Education
  
  Mathematics | Linear Equations in Two VariablesWeb Content &bull; Interactive Notes
  
    Linear Equations in Two Variables &mdash; Interactive Study Guide
    
      Infinite Solutions
      A linear equation in two variables has infinitely many solutions. Each solution is a point on the line. The line extends forever in both directions.
      Key Concept: One equation, two unknowns = infinite solutions. You need TWO equations to get a unique solution (Class X topic).

Graphing Made Easy
      
        Rewrite as y = mx + c.
        Make a table: choose 3 values of x, calculate y for each.
        Plot the 3 points. If they&rsquo;re collinear, draw the line.

Special Lines
      y = 3: Horizontal line, 3 units above x-axis.
      x = &minus;2: Vertical line, 2 units left of y-axis.
      y = x: Line at 45&deg; through origin.

Quick Self-Check
      
        Is (2, 3) a solution of x + y = 5? (Yes: 2 + 3 = 5)
        Give 3 solutions of y = 2x. ((0,0), (1,2), (2,4))
        What is the equation of the x-axis? (y = 0)

Bright Tutorials | Hariom Nagar, Nashik Road | 9403781999 | brighttutorials.in

Question 9

Question 3Check which of the following are solutions of the equation x – 2y = 4 and which are not:(i) (0, 2)(ii) (2, 0)(iii) (4, 0)(iv) (2,42)(\sqrt{2}, 4\sqrt{2})(2​,42​)(v) (1, 1)

Accepted Answer

(i) Substituting (0, 2), in L.H.S. of the equation x - 2y = 4, we get :⇒ x - 2y = 0 - 2(2) = 0 - 4 = -4.Since, L.H.S. ≠ R.H.S.Hence, (0, 2) is not a solution for equation x – 2y = 4.(ii) Substituting (2, 0), in L.H.S. of the equation x - 2y = 4, we get :⇒ x - 2y = 2 - 2(0) = 2 - 0 = 2Since, L.H.S. ≠ R.H.S.Hence, (2, 0) is not a solution for equation x – 2y = 4.(iii) Substituting (4, 0), in L.H.S. of the equation x - 2y = 4, we get :⇒ x - 2y = 4 - 2(0) = 4 - 0 = 4Since, L.H.S. = R.H.S.Hence, (4, 0) is a solution for equation x – 2y = 4.(iv) Substituting (2,42\sqrt{2}, 4\sqrt{2}2​,42​), in L.H.S. of the equation x - 2y = 4, we get :⇒ x - 2y = 2−2(42)=2−82=−72\sqrt{2} - 2(4\sqrt{2}) = \sqrt{2} - 8\sqrt{2} = -7\sqrt{2}2​−2(42​)=2​−82​=−72​.Since, L.H.S. ≠ R.H.S.Hence, (2,42)(\sqrt{2}, 4\sqrt{2})(2​,42​) is not a solution for equation x – 2y = 4.(v) Substituting, (1, 1), in L.H.S. of the equation x - 2y = 4, we get :⇒ x - 2y = 1 - 2(1) = 1 - 2 = -1Since, L.H.S. ≠ R.H.S.Hence, (1, 1) is not a solution for equation x – 2y = 4.

Linear Equations in Two Variables - Interactive Study Notes | Bright Tutorials

.web-topic { background: var(--bt-bg-light); border-radius: 10px; padding: 20px; margin: 20px 0; border: 1px solid var(--bt-border-light); }
    .web-topic h3 { color: var(--bt-primary); margin-top: 0; }
    .key-point { background: #eaf2f8; padding: 10px 15px; border-left: 4px solid var(--bt-primary); margin: 10px 0; border-radius: 4px; }
    .quick-quiz { background: #fff8e1; padding: 15px; border-radius: 8px; border: 2px solid #f39c12; margin: 15px 0; }
    .quick-quiz h4 { color: #e67e22; margin-top: 0; }

BRIGHT TUTORIALS

BRIGHT TUTORIALSCBSE Class IX | Academic Year 2026-2027
    
    9403781999Excellence in Education
  
  Mathematics | Linear Equations in Two VariablesWeb Content &bull; Interactive Notes
  
    Linear Equations in Two Variables &mdash; Interactive Study Guide
    
      Infinite Solutions
      A linear equation in two variables has infinitely many solutions. Each solution is a point on the line. The line extends forever in both directions.
      Key Concept: One equation, two unknowns = infinite solutions. You need TWO equations to get a unique solution (Class X topic).

Graphing Made Easy
      
        Rewrite as y = mx + c.
        Make a table: choose 3 values of x, calculate y for each.
        Plot the 3 points. If they&rsquo;re collinear, draw the line.

Special Lines
      y = 3: Horizontal line, 3 units above x-axis.
      x = &minus;2: Vertical line, 2 units left of y-axis.
      y = x: Line at 45&deg; through origin.

Quick Self-Check
      
        Is (2, 3) a solution of x + y = 5? (Yes: 2 + 3 = 5)
        Give 3 solutions of y = 2x. ((0,0), (1,2), (2,4))
        What is the equation of the x-axis? (y = 0)

Bright Tutorials | Hariom Nagar, Nashik Road | 9403781999 | brighttutorials.in

Linear Equations in Two Variables

Exercise 41

Exercise 42