Curriculum
Course: Mathematics Grade 9
Login

Curriculum

Mathematics Grade 9

Mathematics Grade 9

0/12

Worksheets Chapter Wise

0/0

0/0
Text lesson

Chapter 4: Linear Equations in Two Variables

 

<!– All styles are 100% inline. No class=, no , no , no flex/grid/position –>

Class 9 Mathematics · Chapter 4

Linear Equations in Two Variables

Understand the form ax + by + c = 0, find infinitely many solutions, and graph straight lines on the Cartesian plane with full worked examples, practice sets, and exam tips.

📘 Grade 9   📐 Algebra · Coordinate Geometry

“The principal use of the Analytic Art is to bring Mathematical Problems to Equations and to exhibit those Equations in the most simple terms that can be.” — Edmund Halley

Contents at a Glance

4.1 Introduction
4.2 What Is a Linear Equation?
4.3 Solutions & the Graph
4.4 Practice & Summary

📖  

4.1 Introduction

In earlier classes you worked with linear equations in one variable — equations like x + 1 = 0 or 2y + 3 = 0. Every such equation has exactly one solution, which can be plotted as a single point on the number line.

Now we step up to equations that involve two unknown quantities. This happens naturally in real life: if two cricket players together score 176 runs but we do not know either score individually, we cannot write a one-variable equation. We need two variables — say x and y — giving us the equation x + y = 176.

In this chapter we ask three key questions: Does such an equation have a solution? Is that solution unique? What does the complete set of solutions look like on the Cartesian plane?

📐  

4.2 What Is a Linear Equation in Two Variables?

📌 Definition

An equation that can be written in the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not both zero, is called a linear equation in two variables.

The two variables are usually named x and y, though any letters work. Here are examples and how they fit the standard form:

Original Equation Standard Form (ax + by + c = 0) a b c
2x + 3y = 4.37 2x + 3y − 4.37 = 0 2 3 −4.37
x − 4 = √3 y x − √3 y − 4 = 0 1 −√3 −4
4 = 5x − 3y 5x − 3y − 4 = 0 5 −3 −4
2x = y 2x − y + 0 = 0 2 −1 0

💡 Key Insight: Single-variable equations are a special case

An equation like 3x + 2 = 0 is actually a linear equation in two variables in disguise: it can be written as 3x + 0·y + 2 = 0. Here b = 0, but a ≠ 0, so the definition still holds.

🔍  

4.3 Solutions of a Linear Equation

A solution of a linear equation in two variables is an ordered pair (x, y) that makes the equation true when substituted. For example, for the equation 2x + 3y = 12:

✅ (3, 2) — IS a solution

2(3) + 3(2) = 6 + 6 = 12 ✓

✅ (0, 4) — IS a solution

2(0) + 3(4) = 0 + 12 = 12 ✓

❌ (1, 4) — NOT a solution

2(1) + 3(4) = 2 + 12 = 14 ≠ 12 ✗

📜 Theorem: Infinitely Many Solutions

A linear equation in two variables always has infinitely many solutions.

Proof idea: For any real value you choose for x, say x = k, the equation reduces to a linear equation in one variable (in y), which always has exactly one solution. Since x can take infinitely many real values, there are infinitely many ordered pair solutions. ∎

Practical method to find solutions:

Step Action Example (2x + 3y = 12)
1 Set x = 0, solve for y 3y = 12 → y = 4 → (0, 4)
2 Set y = 0, solve for x 2x = 12 → x = 6 → (6, 0)
3 Choose any other x, solve for y x = 3 → 6 + 3y = 12 → y = 2 → (3, 2)

📊  

Graph of a Linear Equation — It’s Always a Straight Line

Every solution (x, y) is a point on the Cartesian plane. When you plot all solutions of a linear equation, they form a straight line. The graph below represents 2x + 3y = 12 using a table-cell coordinate grid:

5
4
3
2
1
0
-1

               
  ●            
    ●          
        ●      
          ●    
            ●  
              ●

−1 0 1 2 3 4 5 6

● Purple dots = verified solutions of 2x + 3y = 12 → (0,4), (3,2), (6,0)   ● Yellow dots = additional points on the same line

📌 Graphical Theorem (Two-Way Correspondence)

Every point on the graph of a linear equation is a solution of that equation, AND every solution corresponds to a point on the graph. The graph is always a straight line extending infinitely in both directions.

⚠️  

Special Cases: Lines Parallel to Axes

When one coefficient is zero, the graph is a horizontal or vertical line:

📏 y = k (Horizontal Line)

Equation: 0·x + 1·y − k = 0

Example: y = 3

y is always 3; x can be anything.

Solutions: (0, 3), (1, 3), (−5, 3), (100, 3)…

3   — — —  
2          
1          
0          

📏 x = k (Vertical Line)

Equation: 1·x + 0·y − k = 0

Example: x = 2

x is always 2; y can be anything.

Solutions: (2, 0), (2, 1), (2, −3), (2, 7)…

3     |    
2     |    
1     |    
0     |    
  0 1 2 3 4

✏️  

Worked Examples (10 Fully Solved)

EXAMPLE 1   Convert to standard form: 5x − 7 = 3y

Solution: Move all terms to the left side:
5x − 3y − 7 = 0
So a = 5, b = −3, c = −7.

EXAMPLE 2   Write y = 7 as a two-variable equation

Solution: Introduce x with coefficient 0:
0·x + 1·y − 7 = 0, i.e., 0x + y − 7 = 0
Here a = 0, b = 1, c = −7. (Note: a and b are NOT both zero, so it qualifies.)

EXAMPLE 3   Find four solutions of x + 2y = 6

Solution:
x = 0 → 2y = 6 → y = 3 → (0, 3)
y = 0 → x = 6 → (6, 0)
x = 2 → 2 + 2y = 6 → 2y = 4 → y = 2 → (2, 2)
y = 1 → x + 2 = 6 → x = 4 → (4, 1)

EXAMPLE 4   Verify whether (2, 1) satisfies 2x + 3y = 7

Solution: Substitute x = 2, y = 1:
LHS = 2(2) + 3(1) = 4 + 3 = 7 = RHS ✓
Therefore (2, 1) IS a solution.

EXAMPLE 5   Find two solutions of 2x + 5y = 0

Solution:
x = 0 → 5y = 0 → y = 0 → (0, 0)
x = 5 → 10 + 5y = 0 → 5y = −10 → y = −2 → (5, −2)
Note: (0,0) is the only solution when both x = 0 and y = 0 are tried together.

EXAMPLE 6   Two solutions for 3y + 4 = 0

Solution: Rewrite: 0·x + 3y + 4 = 0 → y = −4/3 for every x.
Since x is free: (0, −4/3) and (1, −4/3) are two solutions.
This graphs as a horizontal line at y = −4/3.

EXAMPLE 7   Real-life equation: Tickets cost ₹5 (children) and ₹10 (adults); total = ₹80. Write a linear equation.

Solution: Let x = number of children’s tickets, y = number of adult tickets.
5x + 10y = 80, or in standard form: 5x + 10y − 80 = 0 (a=5, b=10, c=−80).
One solution: x = 0 → y = 8, meaning 8 adult tickets only.

EXAMPLE 8   Find k if (1, −1) is a solution of kx − 2y = 6

Solution: Substitute x = 1, y = −1:
k(1) − 2(−1) = 6
k + 2 = 6
k = 4

EXAMPLE 9   Find the y-intercept and x-intercept of 4x + 3y = 12

Solution:
y-intercept (set x = 0): 3y = 12 → y = 4 → point (0, 4)
x-intercept (set y = 0): 4x = 12 → x = 3 → point (3, 0)
These two points are enough to draw the entire line.

EXAMPLE 10   A number is 3 more than twice another. Express as a linear equation in two variables and find two solutions.

Solution: Let the numbers be x and y. Then y = 2x + 3, or in standard form: 2x − y + 3 = 0.
x = 0 → y = 3 → (0, 3)
x = 1 → y = 5 → (1, 5)

📝  

Practice Set A — Standard Form (Basic)

Write each equation in the form ax + by + c = 0 and state a, b, c.

# Question Answer (click to reveal)
1 3x + 4y = 7
Show Answer

3x + 4y − 7 = 0 | a=3, b=4, c=−7
2 x = 5y + 2
Show Answer

x − 5y − 2 = 0 | a=1, b=−5, c=−2
3 7 = 3x − 2y
Show Answer

3x − 2y − 7 = 0 | a=3, b=−2, c=−7
4 y = 4
Show Answer

0·x + 1·y − 4 = 0 | a=0, b=1, c=−4
5 πx + √2 y = 0
Show Answer

πx + √2 y + 0 = 0 | a=π, b=√2, c=0

📝  

Practice Set B — Finding Solutions

Find two solutions for each equation using the intercept method.

# Equation Two Solutions
1 x + y = 10
Show Answer

(0, 10) and (10, 0)
2 3x − 2y = 6
Show Answer

(0, −3) and (2, 0)
3 5x + y = 15
Show Answer

(0, 15) and (3, 0)
4 x = 3y
Show Answer

(0, 0) and (3, 1)
5 2x + 3y = 0
Show Answer

(0, 0) and (3, −2)

📝  

Practice Set C — Verify Solutions & Find k
# Question Answer
1 Is (3, 1) a solution of 2x + 5y = 11?
Show Answer

2(3)+5(1)=6+5=11=RHS ✓ Yes, it is.
2 Is (4, 0) a solution of x − 2y = 4?
Show Answer

4−0=4=RHS ✓ Yes, it is.
3 Find k if (2, 1) satisfies 2x + 3y = k
Show Answer

2(2)+3(1)=4+3=7 → k=7
4 Find k if (k, 3) satisfies 5x − 2y = 4
Show Answer

5k−6=4 → 5k=10 → k=2
5 Is (0, 2) a solution of x − 2y = 4?
Show Answer

0−4=−4 ≠ 4 ✗ Not a solution.

📝  

Practice Set D — Higher Order (Challenge)
# Challenge Question Answer
1 The perimeter of a rectangle is 48 cm. Express length (l) and breadth (b) as a linear equation.
Show Answer

2l + 2b = 48 → l + b = 24 → l + b − 24 = 0
2 For y = 3x + 5, does the equation have a unique, two, or infinitely many solutions? Why?
Show Answer

Infinitely many — for every real value of x, there is exactly one corresponding y, giving uncountably many ordered pairs.
3 Write 4 solutions for 3x + 2y = 18 and plot the x-intercept and y-intercept.
Show Answer

(0, 9), (6, 0), (2, 6), (4, 3). x-intercept=(6,0), y-intercept=(0,9).
4 If a and b are both zero in ax + by + c = 0 and c ≠ 0, what happens?
Show Answer

The equation becomes 0 + 0 + c = 0, i.e., c = 0, a contradiction. So no valid equation exists — hence the rule that a and b cannot both be zero.
5 A line passes through (0, 5) and (2, 1). Write its equation in standard form.
Show Answer

Slope = (1−5)/(2−0) = −2. Using y = mx + c: y = −2x + 5. Standard form: 2x + y − 5 = 0.

⚡  

8-Point Exam Quick-Check

Review these 8 critical points before your exam:

1  

Standard form is ax + by + c = 0 where a and b are NOT both zero.
2  

A solution is an ordered pair (x, y) that satisfies the equation.
3  

Every linear equation in two variables has infinitely many solutions.
4  

The graph is always a straight line on the Cartesian plane.
5  

x = k is a vertical line; y = k is a horizontal line.
6  

Setting x = 0 gives the y-intercept; y = 0 gives the x-intercept.
7  

Every point on the line is a solution; every solution is a point on the line (two-way correspondence).
8  

To find k: substitute the known point into the equation and solve — do not guess!

Chapter Summary

📌 Definition

ax + by + c = 0 with a, b not both zero. Involves two unknown variables, infinitely representable on the Cartesian plane.

♾️ Solutions

Infinitely many ordered pairs (x, y) satisfy any linear equation in two variables. Found by substituting any value for one variable and solving for the other.

📈 Graph

Always a straight line. Two points determine it (use the intercepts). x = k is vertical; y = k is horizontal.

🔑 Key Technique

To find intercepts: set x = 0 for y-intercept, set y = 0 for x-intercept. These two points are sufficient to draw the full line.

⚠️ Special Forms

Equations like 3x + 2 = 0 (single variable) are special cases of two-variable equations with b = 0. They graph as vertical lines.

🌍 Real Life

Runs scored by two batsmen, cost of items, perimeters of shapes — all modelled naturally by linear equations in two variables.

This comprehensive study guide on Linear Equations in Two Variables — Chapter 4 of Class 9 Mathematics — covers everything students need for CBSE board exams and competitive entrance tests. Topics include the standard form ax + by + c = 0, identifying coefficients a, b, c, finding ordered pair solutions, plotting graphs on the Cartesian plane, identifying x-intercepts and y-intercepts, understanding that every linear equation has infinitely many solutions, and tackling word problems that translate real-world situations into linear equations. Practice Sets A through D with worked answers make this the ultimate revision resource for Grade 9 maths students in India, following the NCERT Class 9 Mathematics syllabus (Reprint 2025–26).

Layer 1
Login Categories
Mathematics Science
Privacy Policy Refund Policy Disclaimer Terms & Conditions Contact Us