📐 Linear Equations in Two Variables - Class 9

Complete notes with graphical methods, examples, and formula sheet

1. Introduction to Linear Equations in Two Variables

Linear equations are algebraic expressions that represent straight lines when plotted on a graph. They are fundamental in mathematics and have numerous real-world applications in science, economics, and engineering.

📖 What is a Linear Equation in Two Variables?

A linear equation in two variables is 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. The variables are x and y.

Examples: 2x + 3y = 6, x - y = 5, 3x + 4y - 12 = 0

1.1 Why Two Variables?

In Class 8, we learned about linear equations in one variable (like 2x + 5 = 0).
Now we're extending to two variables (x and y), which gives us more possibilities.
Two variables allow us to represent relationships between two quantities.
They help us model real-world situations more accurately.
The graph of a linear equation in two variables is always a straight line.

2. Understanding Linear Equations

2.1 Standard Form

Standard form: ax + by + c = 0
a is the coefficient of x
b is the coefficient of y
c is the constant term
At least one of a or b must be non-zero (otherwise it's not an equation in two variables).

📝 Example: Identifying Linear Equations

Q: Which of the following are linear equations in two variables?

a) 2x + 3y = 5

b) x² + y = 4

c) 3x - 2y + 7 = 0

d) xy + 5 = 0

e) y = 2x - 3

Solution:

a) ✓ Linear (can be written as 2x + 3y - 5 = 0)

b) ✗ Not linear (x² means power is 2, not 1)

c) ✓ Linear (already in standard form)

d) ✗ Not linear (xy means product of variables)

e) ✓ Linear (can be written as 2x - y - 3 = 0)

2.2 Other Forms of Linear Equations

General form: ax + by = c (like 2x + 3y = 6)
Slope-intercept form: y = mx + c (where m is slope, c is y-intercept)
Point-slope form: y - y₁ = m(x - x₁)
Intercept form: x/a + y/b = 1
All these forms can be converted to each other.

⚠️ Important: Linear vs Non-Linear

Linear equations have:

• Variables with power 1 only (x, y, not x², y², √x, etc.)

• No products of variables (no xy, x²y, etc.)

• Straight line graphs

Non-linear equations: Have higher powers, products, or other operations

3. Solutions of Linear Equations

A solution of a linear equation in two variables is a pair of values (x, y) that satisfies the equation.

📖 What is a Solution?

A solution is an ordered pair (x, y) that makes the equation true when substituted.

For example, if (2, 3) is a solution of 2x + y = 7, then:

2(2) + 3 = 4 + 3 = 7 ✓ (True, so it's a solution)

3.1 Finding Solutions

Choose any value for one variable (say x).
Substitute it into the equation.
Solve for the other variable (y).
The pair (x, y) is a solution.
A linear equation in two variables has infinitely many solutions.

📝 Example: Finding Solutions

Q: Find three solutions of 2x + y = 6

Solution:

Method: Choose values for x, then find corresponding y values

When x = 0:

2(0) + y = 6

y = 6

Solution: (0, 6)

When x = 1:

2(1) + y = 6

2 + y = 6

y = 4

Solution: (1, 4)

When x = 3:

2(3) + y = 6

6 + y = 6

y = 0

Solution: (3, 0)

Three solutions: (0, 6), (1, 4), (3, 0)

3.2 Infinite Solutions

A linear equation in two variables has infinitely many solutions.
For every value of x, there's a corresponding value of y.
All these solutions lie on a straight line when graphed.
We can find as many solutions as we want by choosing different x values.

🔑 Key Points About Solutions

Every point on the line is a solution
Every solution is a point on the line
Solutions are written as ordered pairs (x, y)
Order matters: (2, 3) ≠ (3, 2)
We can verify solutions by substitution

4. Graphical Representation

The graph of a linear equation in two variables is always a straight line. Every point on the line represents a solution of the equation.

4.1 Steps to Draw a Graph

Step 1: Find at least two solutions of the equation (three is better for accuracy).
Step 2: Draw the Cartesian plane with X and Y axes.
Step 3: Plot the points corresponding to the solutions.
Step 4: Join the points with a straight line using a ruler.
Step 5: Extend the line on both sides with arrows.
Step 6: Label the line with the equation.

📝 Example: Drawing Graph

Q: Draw the graph of x + y = 5

Solution:

Step 1: Find three solutions

When x = 0: y = 5 → Point (0, 5)

When x = 2: y = 3 → Point (2, 3)

When x = 5: y = 0 → Point (5, 0)

Step 2: Make a table

x | 0 | 2 | 5

y | 5 | 3 | 0

Step 3: Plot points (0,5), (2,3), (5,0) on graph

Step 4: Join them with a straight line

Result: A straight line passing through these points represents x + y = 5

4.2 Important Observations

The graph is always a straight line (hence called "linear").
Two points are enough to draw a line, but three points ensure accuracy.
If three points don't lie on a straight line, check your calculations!
The line extends infinitely in both directions.
Every point on the line satisfies the equation.
Points not on the line do NOT satisfy the equation.

⚠️ Tips for Graphing

• Always choose simple values for x (like 0, 1, 2, -1, etc.) to make calculations easier.

• Use graph paper for accuracy.

• Choose an appropriate scale based on the values.

• Label the axes and mark the scale clearly.

• Use a ruler to draw the line - it must be straight!

5. Equations of Lines Parallel to Axes

Some special linear equations represent lines parallel to the coordinate axes.

5.1 Line Parallel to X-axis

Equation form: y = k (where k is a constant)
This means y-coordinate is constant for all points.
The line is horizontal (parallel to X-axis).
Example: y = 3 (all points have y = 3, like (0,3), (1,3), (5,3), etc.)
The X-axis itself is the line y = 0.
Distance from X-axis = |k|

5.2 Line Parallel to Y-axis

Equation form: x = k (where k is a constant)
This means x-coordinate is constant for all points.
The line is vertical (parallel to Y-axis).
Example: x = 2 (all points have x = 2, like (2,0), (2,3), (2,-1), etc.)
The Y-axis itself is the line x = 0.
Distance from Y-axis = |k|

Feature	Parallel to X-axis	Parallel to Y-axis
Equation	y = k	x = k
Direction	Horizontal	Vertical
Variable Constant	y is constant	x is constant
Example	y = 5	x = 3
Points	(..., 5), any x with y=5	(3, ...), any y with x=3

📝 Example: Lines Parallel to Axes

Q1: Draw the graph of y = 4

Solution:

y = 4 means y-coordinate is always 4

Points: (-2, 4), (0, 4), (1, 4), (3, 4)

This is a horizontal line parallel to X-axis, 4 units above it.

Q2: Draw the graph of x = -2

Solution:

x = -2 means x-coordinate is always -2

Points: (-2, 0), (-2, 1), (-2, -3), (-2, 5)

This is a vertical line parallel to Y-axis, 2 units to the left of it.

5.3 The Coordinate Axes

X-axis equation: y = 0 (horizontal line through origin)
Y-axis equation: x = 0 (vertical line through origin)
These are special cases of lines parallel to axes.
They pass through the origin (0, 0).

6. Finding Equation from Graph

If a graph is given, we can find its equation by identifying two or more points on the line.

6.1 Method

Step 1: Identify at least two points on the line from the graph.
Step 2: If the line is parallel to an axis, the equation is immediate (x = k or y = k).
Step 3: Otherwise, substitute the points into the general form ax + by + c = 0.
Step 4: Solve the resulting equations to find a, b, and c.
Step 5: Write the final equation.

📝 Example: Finding Equation from Points

Q: Find the equation of a line passing through (1, 2) and (3, 4)

Solution:

Using two-point form, we can see both points satisfy certain patterns.

Let's check if they satisfy y = mx + c

From (1, 2): 2 = m(1) + c → 2 = m + c ... (i)

From (3, 4): 4 = m(3) + c → 4 = 3m + c ... (ii)

Subtracting (i) from (ii):

4 - 2 = 3m + c - (m + c)

2 = 2m

m = 1

Substituting m = 1 in (i):

2 = 1 + c

c = 1

Equation: y = x + 1 or x - y + 1 = 0

7. Real-Life Applications

7.1 Where Are Linear Equations Used?

Economics: Cost-revenue analysis, profit calculations, demand-supply curves.
Physics: Motion equations (distance = speed × time), conversion formulas.
Business: Calculating total cost, pricing strategies, break-even analysis.
Chemistry: Concentration calculations, reaction rates.
Daily Life: Taxi fares, mobile bills, shopping discounts.
Engineering: Structural designs, electrical circuits.

📝 Example: Real-Life Application

Problem: A taxi charges ₹20 as fixed charge and ₹10 per km. Write an equation for total fare.

Solution:

Let x = distance in km

Let y = total fare in ₹

Total fare = Fixed charge + Variable charge

y = 20 + 10x

Equation: y = 10x + 20 or 10x - y + 20 = 0

Using the equation:

For 5 km: y = 10(5) + 20 = 50 + 20 = ₹70

For 10 km: y = 10(10) + 20 = 100 + 20 = ₹120

7.2 More Application Problems

Age problems: If father's age is twice son's age.
Money problems: Total value of notes of different denominations.
Speed-time-distance: Train problems, meeting point problems.
Work problems: Combined work, efficiency calculations.
Mixture problems: Mixing liquids of different concentrations.

8. Problem-Solving Strategies

8.1 Steps to Solve Word Problems

Step 1: Read the problem carefully and identify what is given and what is to be found.
Step 2: Assign variables to unknown quantities (usually x and y).
Step 3: Translate the word problem into a mathematical equation.
Step 4: Find solutions or draw the graph as required.
Step 5: Verify your answer and check if it makes sense in the context.
Step 6: Write the final answer with proper units.

📝 Example: Complete Problem

Problem: The sum of two numbers is 15. Express this as a linear equation and find three solutions.

Solution:

Let the two numbers be x and y

Given: Sum = 15

Equation: x + y = 15

Finding three solutions:

When x = 0: y = 15 → Solution (0, 15)

When x = 5: y = 10 → Solution (5, 10)

When x = 10: y = 5 → Solution (10, 5)

Three solutions: (0, 15), (5, 10), (10, 5)

Interpretation:

(0, 15) means first number is 0, second is 15

(5, 10) means first number is 5, second is 10

(10, 5) means first number is 10, second is 5

📚 Quick Formula Sheet - Linear Equations in Two Variables

Standard Form

ax + by + c = 0

Where a, b, c are real numbers

a and b cannot both be zero

Graph is always a straight line

Other Forms

General: ax + by = c

Slope-intercept: y = mx + c

Point-slope: y-y₁ = m(x-x₁)

Intercept: x/a + y/b = 1

All forms are equivalent

Finding Solutions

1. Choose value for x

2. Substitute in equation

3. Solve for y

4. Write as ordered pair (x, y)

Infinite solutions exist!

Graphing Steps

1. Find 3 solutions (points)

2. Draw coordinate axes

3. Plot the points

4. Join with straight line

Use ruler for accuracy

Lines Parallel to Axes

Parallel to X-axis:

y = k (horizontal)

Parallel to Y-axis:

x = k (vertical)

k is constant distance

The Axes Equations

X-axis: y = 0

Y-axis: x = 0

Origin: (0, 0)

Special lines through origin

Verification

To check if (a, b) is solution:

Substitute x=a and y=b

If equation true → Solution ✓

If equation false → Not solution ✗

Key Properties

• Infinitely many solutions

• Graph is straight line

• 2 points determine a line

• All points on line = solutions

Remember: Linear = Straight!

💡 Study Tips for Linear Equations

• Always verify your solutions by substituting back into the equation.

• When graphing, choose simple values (0, 1, 2, -1) for easier calculations.

• Three points are better than two for accuracy when drawing graphs.

• Use graph paper for neat and accurate graphs.

• Remember: The graph must be a STRAIGHT line - use a ruler!

• For word problems, carefully identify what each variable represents.

• Check if your answer makes sense in the context of the problem.

• Practice different forms of equations - be comfortable converting between them.

🔑 Common Mistakes to Avoid

Don't confuse (x, y) with (y, x) - order matters!
Don't assume only positive solutions - negative values are valid too.
Don't forget to extend the line with arrows on both sides.
Don't use only two points without verification - use three for safety.
Remember: y = 5 is horizontal, x = 5 is vertical (not the opposite!).
Don't forget units when solving real-life problems.
Check that your line is straight - if points don't align, recheck calculations.

🎯 Quick Practice Questions

1. Find 3 solutions: 3x + 2y = 12

2. Graph the equation: 2x - y = 4

3. What is the equation of a line parallel to X-axis at distance 7 units above it?

4. Check if (2, 5) is a solution of x + 2y = 12

5. A notebook costs ₹x and a pen costs ₹y. If total cost of 3 notebooks and 2 pens is ₹50, write the equation.

Answers:

1. (0,6), (2,3), (4,0) [or any valid solutions]

2. Plot points like (0,-4), (2,0), (3,2) and join

3. y = 7

4. 2 + 2(5) = 2 + 10 = 12 ✓ Yes, it's a solution!

5. 3x + 2y = 50