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What is the Pair of Straight Lines?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
A pair of straight lines is simply two straight lines represented by a single equation. Instead of having two separate equations like y = 2x + 1 and y = -x + 3, we combine them into one equation that describes both lines together.
Simple Example
Quick Example
Imagine you have two roads in your city crossing each other. Each road can be described by an equation. When we talk about the 'pair of straight lines' for these roads, we are looking at one combined equation that represents both roads at the same time, showing how they exist together.
Worked Example
Step-by-Step
Let's find the combined equation of two lines: (x - y + 1) = 0 and (2x + y - 3) = 0.
Step 1: Understand that the combined equation is formed by multiplying the individual equations together.
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Step 2: Set up the multiplication: (x - y + 1)(2x + y - 3) = 0.
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Step 3: Multiply the terms systematically. First, multiply 'x' by each term in the second bracket: x(2x) + x(y) + x(-3) = 2x^2 + xy - 3x.
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Step 4: Next, multiply '-y' by each term in the second bracket: -y(2x) -y(y) -y(-3) = -2xy - y^2 + 3y.
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Step 5: Finally, multiply '1' by each term in the second bracket: 1(2x) + 1(y) + 1(-3) = 2x + y - 3.
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Step 6: Combine all the terms obtained in Steps 3, 4, and 5: (2x^2 + xy - 3x) + (-2xy - y^2 + 3y) + (2x + y - 3) = 0.
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Step 7: Group like terms and simplify: 2x^2 + (xy - 2xy) - y^2 + (-3x + 2x) + (3y + y) - 3 = 0.
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Step 8: The simplified combined equation is: 2x^2 - xy - y^2 - x + 4y - 3 = 0.
Answer: The combined equation for the pair of straight lines (x - y + 1) = 0 and (2x + y - 3) = 0 is 2x^2 - xy - y^2 - x + 4y - 3 = 0.
Why It Matters
Understanding pairs of straight lines is crucial for designing structures in engineering, like bridges or buildings, where beams cross. In computer graphics and AI, it helps define boundaries and paths for objects. Even in medical imaging, doctors use these concepts to analyze shapes and patterns in scans, helping them diagnose problems accurately.
Common Mistakes
MISTAKE: Forgetting to multiply ALL terms when combining two linear equations. | CORRECTION: Use the distributive property carefully, multiplying each term from the first equation by every term in the second equation.
MISTAKE: Incorrectly combining like terms, especially with positive and negative signs. | CORRECTION: Always pay close attention to the signs (+/-) when adding or subtracting coefficients of similar terms (like 'xy' and '-2xy').
MISTAKE: Confusing the equation of a pair of straight lines with the equation of a single straight line. | CORRECTION: A single straight line equation is usually linear (like ax + by + c = 0), while a pair of straight lines will have a quadratic form (like Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0).
Practice Questions
Try It Yourself
QUESTION: Find the combined equation of the lines x = 0 and y = 0. | ANSWER: xy = 0
QUESTION: What is the combined equation for the lines (x - 2) = 0 and (y + 3) = 0? | ANSWER: xy + 3x - 2y - 6 = 0
QUESTION: Find the equation representing the pair of lines passing through the origin, one with slope 1 and the other with slope -1. (Hint: A line through origin with slope 'm' is y = mx) | ANSWER: x^2 - y^2 = 0
MCQ
Quick Quiz
Which of the following equations represents a pair of straight lines?
x + y = 5
x^2 + y^2 = 25
x^2 - 4y^2 = 0
y = x^2 + 1
The Correct Answer Is:
C
Option C, x^2 - 4y^2 = 0, can be factored as (x - 2y)(x + 2y) = 0, which gives two linear equations: x - 2y = 0 and x + 2y = 0. The other options represent a single line, a circle, and a parabola respectively.
Real World Connection
In the Real World
Think about the framework of a modern building or the criss-crossing beams in a flyover bridge being built in your city. Engineers use the concept of pairs of straight lines to model and analyze how these structural elements intersect and support each other, ensuring the design is strong and safe. It's like planning the layout of roads on a map or even designing circuit boards for your phone!
Key Vocabulary
Key Terms
LINEAR EQUATION: An equation whose graph is a straight line, like ax + by + c = 0. | QUADRATIC EQUATION: An equation where the highest power of the variable is 2. | DISTRIBUTIVE PROPERTY: The rule that a(b + c) = ab + ac, used for multiplying expressions. | ORIGIN: The point (0,0) where the x-axis and y-axis intersect on a graph. | COEFFICIENT: A number multiplied by a variable in an algebraic term.
What's Next
What to Learn Next
Great job learning about pairs of straight lines! Next, you can explore 'Homogeneous Equation of a Pair of Straight Lines'. This will help you understand special cases where both lines pass through the origin, which is a common scenario in many advanced math and science problems.
