<!– 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
|
|
| 4.2 What Is a Linear Equation? |
|
| 4.3 Solutions & the Graph |
|
|
|
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:
| ● 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)…
|
|
|
📏 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!
|
|
|
|
📌 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).
