Two Variables, One Line (Class 9): Notes और MCQ Practice

आसान भाषा में

जब किसी equation में दो variables (जैसे x और y) हों और उनकी degree 1 हो, तो उसे Linear Equation in Two Variables कहते हैं। इसका graph हमेशा एक सीधी रेखा (straight line) बनाता है। इस रेखा पर आने वाला हर point (ordered pair) इस equation का solution होता है, इसलिए एक equation के infinitely many solutions होते हैं।

Important Points & Formulas

• Standard Form: ax + by + c = 0 (जहाँ a, b, c real numbers हैं और a तथा b दोनों 0 नहीं हो सकते)।
• Origin से Pass होना: यदि c = 0 हो (ax + by = 0), तो line हमेशा origin (0, 0) से होकर गुज़रती है।
• Slope Formula: Slope (m) = Rise ÷ Run = (y₂ − y₁) ÷ (x₂ − x₁)
• Slope-Intercept Form: y = mx + d
• ax + by + c = 0 को बदलने पर: m = −a ÷ b और d = −c ÷ b (b ≠ 0)
• m = slope (steepness), d = y-intercept
• Nature of Slope:
• Positive (m > 0): Line बाएँ से दाएँ ऊपर (upward) जाती है।
• Negative (m < 0): Line बाएँ से दाएँ नीचे (downward) जाती है।
• Zero (m = 0): Horizontal line, x-axis के parallel (जैसे y = 5)।
• Undefined: Vertical line, y-axis के parallel (जैसे x = 2, क्योंकि denominator 0 होता है)।
• Pair of Equations: a₁x + b₁y + c₁ = 0 और a₂x + b₂y + c₂ = 0

Memory Tricks

1. Rise over Run: 'Rise' मतलब ऊपर उठना (y-axis) और 'Run' मतलब ज़मीन पर दौड़ना (x-axis)। इसलिए Slope = y का अंतर ÷ x का अंतर।
2. Zero Constant, Origin Instant: अगर equation में constant term c = 0 है, तो आँख बंद करके समझें कि line Origin (0, 0) से ही जाएगी!

Common Board Mistakes

1. Ordered Pair का क्रम बदलना: (x, y) में पहली value हमेशा x की और दूसरी y की होती है। (4, 3) में x = 4 और y = 3 रखना है, उल्टा नहीं।
2. Vertical Line के Slope में भ्रम: x = 2 जैसी vertical line का slope 0 नहीं, बल्कि Undefined होता है (क्योंकि division by 0 संभव नहीं है)। Slope 0 केवल horizontal line (y = 5) का होता है।

Two Variables, One Line - Class 9 Study Notes

1. Chapter in Short

This chapter introduces linear equations in two variables. In everyday life, many situations involve two related unknown quantities—such as buying two different kinds of fruit or buying adult and child tickets.

Here is what this chapter covers:

• Standard Form: Representing situations using the general form ax + by + c = 0.
• Solutions: Finding ordered pairs (x, y) that make an equation true. A single linear equation in two variables has infinitely many solutions.
• Graph of an Equation: Every linear equation in two variables forms a straight line when plotted on a graph. Every point on the line is a solution, and every solution is a point on the line.
• Slope (Gradient): Understanding how steep a line is using the ratio of vertical change (Rise) to horizontal change (Run).
• Slope-Intercept Form: Writing equations as y = mx + d, where m is the slope and d is the y-intercept.
• Pair of Linear Equations: Setting up two simultaneous equations to solve real-world problems.

2. Key Concepts Explained Simply

A. What is a Linear Equation in Two Variables?

When you buy items of two types, their total cost depends on two separate quantities.

• Everyday Example: Suppose mangoes cost ₹60 per kg and bananas cost ₹50 per kg. If you spend ₹280 in total, let x be the kilograms of mangoes and y be the kilograms of bananas.
• Cost of mangoes = 60x
• Cost of bananas = 50y
• Equation: 60x + 50y = 280

Because both variables x and y have an exponent (power) of 1, this is called a linear equation in two variables.

B. The Standard Form

Any linear equation in two variables can be arranged into the form:

ax + by + c = 0

• a is the coefficient of x.
• b is the coefficient of y.
• c is the constant term.
• a and b cannot both be zero.
• An equation in one variable, like 4x + 7 = 0, is also a special case of two variables: 4x + 0y + 7 = 0.

C. Solutions and Ordered Pairs

A solution is a pair of numbers (one for x and one for y) that makes the equation true. We write this as an ordered pair (x, y). The order matters: the first number is always x, and the second is always y.

• For any value you choose for x, you can solve for a corresponding value of y.
• Therefore, a linear equation in two variables has infinitely many solutions.

D. Geometric Meaning (Drawing the Line)

When you plot all the solutions (x, y) on a coordinate plane, they join together to form a single straight line.

• Easy way to draw a line: Find where it crosses the axes:
• Put x = 0 to find the point on the y-axis.
• Put y = 0 to find the point on the x-axis.
• Passing through Origin: If c = 0 in ax + by + c = 0, the equation becomes ax + by = 0. Substituting (0, 0) satisfies the equation, so any line with c = 0 always passes through the origin.

E. Understanding Slope (Steepness)

Think of riding a bicycle up a hill. The steeper the hill, the more effort it takes. In mathematics, steepness is called slope or gradient.

• Rise: The vertical change between two points (y2 − y1). It is positive if moving up, negative if moving down.
• Run: The horizontal change between two points (x2 − x1). It is positive if moving to the right, negative if moving to the left.
• Slope Formula: Slope = Rise ÷ Run = (y2 − y1) ÷ (x2 − x1).
• The slope is constant anywhere along a straight line.

F. Slope-Intercept Form: y = mx + d

When an equation is written in the form y = mx + d:

• m is the slope of the line.
• d is the y-intercept (the value of y where the line crosses the y-axis at (0, d)).
• Types of Slope:
• Positive slope (m > 0): The line goes upward from left to right (as x increases, y increases).
• Negative slope (m < 0): The line goes downward from left to right (as x increases, y decreases).
• Zero slope (m = 0): Horizontal line parallel to the x-axis (e.g., y = 5).
• Undefined slope: Vertical line parallel to the y-axis (e.g., x = 2, because Run = 0 and division by zero is undefined).

G. Pair of Linear Equations in Two Variables

Sometimes a situation requires finding two unknowns that satisfy two conditions at the same time (e.g., finding the cost of an adult ticket and a child ticket). This leads to a pair of equations:

a1 x + b1 y + c1 = 0 and a2 x + b2 y + c2 = 0.

A solution to this pair is an ordered pair (x, y) that satisfies both equations simultaneously.

3. Important Definitions, Formulas, and Facts

• Standard Form: ax + by + c = 0, where a, b, c are real numbers, and a and b are not both 0.
• Slope (m):
• Slope = Rise ÷ Run
• Slope = (y2 − y1) ÷ (x2 − x1)
• Slope-Intercept Form: y = mx + d
• Converting ax + by + c = 0 to Slope-Intercept Form (when b ≠ 0):
• y = −(a ÷ b)x − (c ÷ b)
• Slope m = −a ÷ b
• Intercept d = −c ÷ b
• Axis Equations:
• Equation of the x-axis: y = 0
• Equation of the y-axis: x = 0
• Horizontal Line: y = k (slope m = 0, parallel to x-axis)
• Vertical Line: x = k (slope is undefined, parallel to y-axis)
• Origin Condition: Any line of the form ax + by = 0 (where c = 0) passes through (0, 0).
• Equivalence: Multiplying the equation ax + by = c by a non-zero constant k gives kax + kby = kc, which has the exact same set of solutions.

4. Solved Examples

Example 1: Writing in Standard Form

Question: Express the equation x ÷ 2 − 4 = (3 ÷ 2)y in the standard form ax + by + c = 0, and identify the values of a, b, and c.

Solution:

1. Move all terms to the left-hand side:

(1 ÷ 2)x − (3 ÷ 2)y − 4 = 0

1. Comparing with ax + by + c = 0:

• a = 1 ÷ 2
• b = −3 ÷ 2
• c = −4

(Note: You can also multiply the entire equation by 2 to get integer coefficients: x − 3y − 8 = 0, where a = 1, b = −3, c = −8).

Example 2: Finding Unknown Constants

Question: In the equations 2mx + 3y = 7 and 4x + ny = −10, if the ordered pair (2, −1) is a solution to both equations, find the values of m and n.

Solution:

Since (2, −1) is a solution, substitute x = 2 and y = −1 into each equation.

1. For the first equation:

2m(2) + 3(−1) = 7

4m − 3 = 7

4m = 7 + 3

4m = 10

m = 10 ÷ 4 = 2.5 (or 5 ÷ 2)

1. For the second equation:

4(2) + n(−1) = −10

8 − n = −10

−n = −10 − 8

−n = −18

n = 18

Answer: m = 2.5 (or 5 ÷ 2) and n = 18.

Example 3: Calculating the Slope Between Two Points

Question: Find the slope of the line passing through the points A(−4, 5) and B(−1, 2).

Solution:

Let (x1, y1) = (−4, 5) and (x2, y2) = (−1, 2).

1. Rise = y2 − y1 = 2 − 5 = −3
2. Run = x2 − x1 = −1 − (−4) = −1 + 4 = 3
3. Slope = Rise ÷ Run = −3 ÷ 3 = −1

Answer: The slope of the line is −1 (a negative slope, meaning the line slopes downwards from left to right).

Example 4: Real-Life Slope Application

Question: An accessibility ramp for wheelchairs is designed with a slope requirement of 1 cm vertical rise for every 12 cm horizontal length. If the total height of the staircase is 18 cm, what must be the horizontal length of the ramp?

Solution:

1. Formula: Slope = Vertical Rise ÷ Horizontal Length
2. Given: Slope = 1 ÷ 12, Vertical Rise = 18 cm
3. Set up the equation:

1 ÷ 12 = 18 ÷ Horizontal Length

1. Solve for Horizontal Length:

Horizontal Length = 18 × 12 = 216 cm

Answer: The horizontal length of the ramp should be 216 cm (or 2.16 metres).

5. Common Mistakes and Quick Revision Points

• Watch the Signs When Moving Terms: When rewriting an equation like 5x = 3y into ax + by + c = 0, remember that moving 3y across the equals sign changes its sign: 5x − 3y + 0 = 0 (so b = −3, not +3).
• Don't Swap Coordinates: An ordered pair is always (x, y). If given (4, 3), substitute x = 4 and y = 3, never the other way around.
• Number of Solutions: A single linear equation in two variables does not have just one solution—it has infinitely many solutions.
• Zero Slope vs. Undefined Slope:
• A horizontal line (y = k) has a slope of 0 (Rise is 0).
• A vertical line (x = k) has an undefined slope (Run is 0, and division by zero is not allowed).
• Checking if a Point Lies on a Line: A point (u, v) lies on the line if and only if putting x = u and y = v makes LHS equal to RHS.