Learn
Games

4.3.3 - Examples of Solutions

You've not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take mock test.

Interactive Audio Lesson

Listen to a student-teacher conversation explaining the topic in a relatable way.

Introduction to Solutions of Linear Equations

Unlock Audio Lesson

Signup and Enroll to the course for listening the Audio Lesson

0:00
Teacher
Teacher

Today, we're going to explore the solutions of linear equations in two variables. Can anyone tell me what a solution actually means?

Student 1
Student 1

I think it's the values that make the equation true?

Teacher
Teacher

Exactly! When we have an equation like \(2x + 3y = 12\), a solution is any pair of values for \(x\) and \(y\) that satisfies this equation.

Student 2
Student 2

So, can we find multiple pairs that work?

Teacher
Teacher

Yes! In fact, for linear equations in two variables, there are infinitely many solutions.

Finding Solutions

Unlock Audio Lesson

Signup and Enroll to the course for listening the Audio Lesson

0:00
Teacher
Teacher

Let's take the equation \(2x + 3y = 12\). Can anyone suggest a value for \(x\) that we might use?

Student 3
Student 3

What if we try \(x = 0\)?

Teacher
Teacher

Great choice! Plugging \(x = 0\) in, we get \(3y = 12\), which gives us \(y = 4\). So one solution is \((0, 4)\).

Student 4
Student 4

Can we find another pair using a different value?

Teacher
Teacher

Absolutely! If we choose \(x = 3\), then \(2(3) + 3y = 12\) leads us to \(y = 2\). Another solution is \((3, 2)\).

Generalizing Solutions

Unlock Audio Lesson

Signup and Enroll to the course for listening the Audio Lesson

0:00
Teacher
Teacher

Now, seeing how we can substitute different values generates new solutions. Who would like to suggest another value?

Student 1
Student 1

How about we set \(x = 6\)?

Teacher
Teacher

Excellent! Plugging \(x = 6\) into the equation gives us \(2(6) + 3y = 12\), leading to \(y = 0\). Thus, we have another solution: \((6, 0)\).

Student 2
Student 2

So we can get all sorts of points just by changing \(x\)!

Teacher
Teacher

Exactly! And this is just one way to find solutions. You can start with either variable!

Identifying Valid Solutions

Unlock Audio Lesson

Signup and Enroll to the course for listening the Audio Lesson

0:00
Teacher
Teacher

Now, let’s verify if certain pairs are indeed solutions for the equation \(x - 2y = 4\). Who can help me check \((2, 0)\)?

Student 3
Student 3

I think we need to plug in \(x = 2\) and \(y = 0\). So, \(2 - 0 = 4\) is false.

Teacher
Teacher

Correct! That pair does not work. What about \((4, 0)\)?

Student 4
Student 4

For \(4 - 2(0) = 4\), that works!

Teacher
Teacher

Great job! Validating solutions helps us identify which pairs are feasible.

Summary of Solutions

Unlock Audio Lesson

Signup and Enroll to the course for listening the Audio Lesson

0:00
Teacher
Teacher

Let's summarize! What have we learned about solutions for linear equations in two variables?

Student 1
Student 1

They can have many solutions, often represented as pairs!

Student 2
Student 2

You can find solutions by choosing values for either \(x\) or \(y\)!

Teacher
Teacher

Exactly, and validating each pair ensures accuracy. Well done everyone!

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

This section explores the solutions to linear equations in two variables, emphasizing the idea that each equation typically has infinitely many solutions.

Standard

In this section, students learn that linear equations in two variables often have multiple solutions, represented as ordered pairs. They can substitute values for one variable to find corresponding values for another, enabling the identification of various solutions.

Detailed

Detailed Summary

In this section, we focus on the concept of solutions for linear equations in two variables. Unlike linear equations in one variable, which have a unique solution, linear equations in two variables can yield infinitely many solutions. For example, the equation \(2x + 3y = 12\) has multiple solutions like \((3, 2)\) and \((0, 4)\). To find more solutions, students can assign a specific value to \(x\) or \(y\) and solve for the other variable, illustrating the rich set of points that satisfy the given equation.

Through examples like \(x + 2y = 6\), students are guided to find several valid ordered pairs, emphasizing that one can generate an endless list of solutions. The importance of ordered pairs, the role of substitution, and understanding the graphical representation of these solutions on the Cartesian plane are vital concepts introduced in this section.

Youtube Videos

Linear Equations In Two Variables | All Examples | Chapter 4 | SEED 2024-2025
Linear Equations In Two Variables | All Examples | Chapter 4 | SEED 2024-2025
Linear Equations In 2 Variables Class 9 in One Shot 🔥 | Class 9 Maths Chapter 4 Complete Lecture
Linear Equations In 2 Variables Class 9 in One Shot 🔥 | Class 9 Maths Chapter 4 Complete Lecture
Example (3) Chapter:4 Linear Equations in Two Variables | Ncert Maths Class 9
Example (3) Chapter:4 Linear Equations in Two Variables | Ncert Maths Class 9
Example 4 Chapter 4 Linear Equations in Two Variables class 9, Math New NCERT
Example 4 Chapter 4 Linear Equations in Two Variables class 9, Math New NCERT
Linear Equations In Two Variables | Exercise 4.1 | Chapter 4 | SEED 2024-2025
Linear Equations In Two Variables | Exercise 4.1 | Chapter 4 | SEED 2024-2025
Linear Equations in Two Variables Class 9 in One Shot Revision in 15 Min | Class 9 Maths Chapter 4
Linear Equations in Two Variables Class 9 in One Shot Revision in 15 Min | Class 9 Maths Chapter 4
⭕ Pair of Linear Equations in Two Variables - Top 20 Most Important Questions | Class 10 Maths Ch 3🎯
⭕ Pair of Linear Equations in Two Variables - Top 20 Most Important Questions | Class 10 Maths Ch 3🎯
Example (2) Chapter:4 Linear Equations in Two Variables | Ncert Maths Class 9
Example (2) Chapter:4 Linear Equations in Two Variables | Ncert Maths Class 9
Linear Equation In Two Variables | One Shot | Class 9 Maths
Linear Equation In Two Variables | One Shot | Class 9 Maths
Linear Equations In Two Variables | Introduction | Chapter 4 | SEED 2024-2025
Linear Equations In Two Variables | Introduction | Chapter 4 | SEED 2024-2025

Audio Book

Dive deep into the subject with an immersive audiobook experience.

Understanding Solutions

Unlock Audio Book

Signup and Enroll to the course for listening the Audio Book

You have seen that every linear equation in one variable has a unique solution. What can you say about the solution of a linear equation involving two variables? As there are two variables in the equation, a solution means a pair of values, one for x and one for y which satisfy the given equation. Let us consider the equation 2 x + 3y = 12. Here, x = 3 and y = 2 is a solution because when you substitute x = 3 and y = 2 in the equation above, you find that 2x + 3y =(2 × 3) + (3 × 2) = 12. This solution is written as an ordered pair (3, 2), first writing the value for x and then the value for y.

Detailed Explanation

In this chunk, we learn that solutions to linear equations in two variables consist of ordered pairs. This means that for any valid solution, both 'x' and 'y' must satisfy the equation when substituted. For example, the equation 2x + 3y = 12 has a solution where x equals 3 and y equals 2. By substituting these values back into the equation, we verify the solution is correct because it balances the equation. The ordered pair format (3, 2) helps to denote this relationship clearly.

Examples & Analogies

Imagine you are sharing candies with a friend. If you have a total of 12 candies, and you decide to take 3 candies for yourself, then your friend would automatically get 9 candies (since 12 - 3 = 9). In this case, you and your friend's candy counts can be thought of as solutions to the equation: your candies as 'x' and your friend's candies as 'y'. Just like in the equation, you can find many combinations of candies (solutions) such that the total is always 12.

Multiple Solutions

Unlock Audio Book

Signup and Enroll to the course for listening the Audio Book

Similarly, (0, 4) is also a solution for the equation above. On the other hand, (1, 4) is not a solution of 2 x + 3y = 12, because on putting x = 1 and y = 4 we get 2 x + 3y = 14, which is not 12. Note that (0, 4) is a solution but not (4, 0). You have seen at least two solutions for 2 x + 3y = 12, i.e., (3, 2) and (0, 4). Can you find any other solution? Do you agree that (6, 0) is another solution? Verify the same.

Detailed Explanation

In this chunk, we explore the nature of solutions for linear equations involving two variables. We see that while (3, 2) and (0, 4) are solutions to the equation, other pairs like (1, 4) do not satisfy the equation. This means that not all combinations of x and y will work – a solution must fulfill the equation. Continuing the process, we find that more combinations like (6, 0) also work, demonstrating that there are indeed multiple solutions to a linear equation in two variables.

Examples & Analogies

Think of the solutions to our candy-sharing scenario again. If you decide to take 0 candies, your friend gets all 12. If you take 6, your friend has 6 left. But if you take 1 candy while claiming 4 for your friend, that doesn't add up to 12, showing that only specific amounts of candies for both of you make sense to stay true to the total. There are many ways to distribute these candies while keeping the total fixed, analogous to finding different pairs (solutions).

Finding More Solutions

Unlock Audio Book

Signup and Enroll to the course for listening the Audio Book

In fact, we can get many many solutions in the following way. Pick a value of your choice for x (say x = 2) in 2 x + 3y = 12. Then the equation reduces to 4 + 3 y = 12, which is a linear equation in one variable. On solving this, you get y = 8/3. So (2, 8/3) is another solution of 2 x + 3y = 12. Similarly, choosing x = –5, you find that the equation becomes –10 + 3 y = 12. This gives y = 22/3. So, (–5, 22/3) is another solution of 2x + 3y = 12.

Detailed Explanation

Here, we learn a method for finding other solutions to a linear equation in two variables. By selecting any value for x, one can find the corresponding value of y by rearranging the equation. For instance, if we choose x = 2, we reformulate our equation to solve for y, resulting in another ordered pair solution of (2, 8/3). Similarly, by choosing x as a negative number, we can find other pairs that also satisfy the equation. This method illustrates the infinitely many solutions that these equations can have.

Examples & Analogies

Let's apply this to a purchase scenario. If you have a budget fixed at $12 for snacks, and you decide to spend $4 on chips (i.e., choosing x = 4), then you have $8 left to spend (which can be thought of as the y value). If instead, you decided to buy something worth $2, you can recalculate the leftover budget. Each choice creates a new combination of purchases (x, y) while always totaling to that fixed budget. This mirrors how we can vary x and get several corresponding y values.

Finding Specific Solutions

Unlock Audio Book

Signup and Enroll to the course for listening the Audio Book

Example 3 : Find four different solutions of the equation x + 2y = 6. Solution : By inspection, x = 2, y = 2 is a solution because for x = 2, y = 2 x + 2y =2 + 4 = 6. Now, let us choose x = 0. With this value of x, the given equation reduces to 2 y = 6 which has the unique solution y = 3. So x = 0, y = 3 is also a solution of x + 2y = 6. Similarly, taking y = 0, the given equation reduces to x = 6. So, x = 6, y = 0 is a solution of x + 2y = 6 as well. Finally, let us take y = 1. The given equation now reduces to x + 2 = 6, whose solution is given by x = 4. Therefore, (4, 1) is also a solution of the given equation.

Detailed Explanation

In this example, we actively find multiple solutions to the linear equation x + 2y = 6. The chunk illustrates step-by-step how to choose different values for x and calculate the corresponding y values. For instance, by inspecting the equation, we find that (2, 2) works and later verify that (0, 3), (6, 0), and (4, 1) are also valid solutions. This method highlights how selecting values effectively allows us to explore the endless possibilities of solutions for the equation.

Examples & Analogies

Imagine trying to distribute a total of 6 jewels among you and a friend where you take x jewels and your friend takes twice that amount. If you keep trying different values for how many jewels you want (x), you can discover exactly how many jewels your friend ends up with (y). Each combination you come up with denotes a possible distribution scenario (solution), showing there are many ways to reach that total of 6 jewels.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Ordered pairs represent solutions to linear equations in two variables.

  • A linear equation can have infinitely many solutions.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • {'example': 'Find four solutions for the equation \(x + 2y = 6\).', 'solution': 'The solutions can be derived as follows:\n1. Let \(x = 0\), then \(2y = 6 \Rightarrow y = 3\), giving us \((0, 3)\).\n2. Let \(y = 0\), then \(x = 6\), yielding \((6, 0)\).\n3. Let \(x = 2\), then \(2y = 4 \Rightarrow y = 2\), thus \((2, 2)\).\n4. Let \(y = 1\), then \(x + 2 = 6 \Rightarrow x = 4\), giving \((4, 1)\).'}

  • {'example': 'Verify if the pairs \((0, 2)\) and \((4, 0)\) are solutions for the equation \(x - 2y = 4\).', 'solution': 'For \((0, 2)\): \(0 - 2(2) = -4 \neq 4\) (not a solution).\nFor \((4, 0)\): \(4 - 2(0) = 4 \) (is a solution).'}

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • Solutions are pairs, like x and y, plug in the numbers, watch them fly!

📖 Fascinating Stories

  • Imagine x is a path, y is a tree. Together they create a solution, just wait and see!

🧠 Other Memory Gems

  • X+Y=Pairs for a linear equation's cares!

🎯 Super Acronyms

SOLUTION

  • Set values
  • Learn if they work
  • Obtain
  • True or false
  • Identify Oh! Note!

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Linear Equation

    Definition:

    An equation of the form \(ax + by + c = 0\), where \(a\), \(b\), and \(c\) are real numbers, and \(a\) and \(b\) are not both zero.

  • Term: Ordered Pair

    Definition:

    A pair of numbers \((x, y)\) that represents a point on the Cartesian plane.

  • Term: Infinitely Many Solutions

    Definition:

    Refers to the behavior of linear equations in two variables, where numerous pairs \((x, y)\) can satisfy the equation.