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4.2.1 - Examples of Linear Equations in Two Variables

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Understanding Linear Equations

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0:00
Teacher
Teacher

Today, we will learn about linear equations in two variables. Can anyone put what they remember about linear equations in general into words?

Student 1
Student 1

I remember that linear equations can be in the form of ax + b = 0!

Teacher
Teacher

Great start! In two variables, it's similar; we write them as ax + by + c = 0. Does anyone know what a, b, and c represent?

Student 2
Student 2

The letters a and b are coefficients of x and y, right? What about c?

Teacher
Teacher

Exactly! 'c' is the constant term. Remember: for an equation to be linear in two variables, both coefficients a and b cannot be zero simultaneously. A memory aid to keep this in mind is the acronym ABE: 'A' for 'a', 'B' for 'b', 'E' for 'Equation'.

Expressing Equations in Standard Form

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Teacher
Teacher

Now, let's practice converting equations to the standard form. If I give you the equation 2x + 3y = 9, how would we express it in the form ax + by + c = 0?

Student 3
Student 3

I think we'd subtract 9 from both sides, making it 2x + 3y - 9 = 0.

Teacher
Teacher

Perfect! Now, what are the values of a, b, and c in that case?

Student 4
Student 4

So a = 2, b = 3, and c = -9.

Teacher
Teacher

Exactly! Keep in mind that this form lets us easily identify our linear equation characteristics.

Solution Characteristics

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0:00
Teacher
Teacher

Can anyone tell me what we mean by the solution of a linear equation in two variables?

Student 1
Student 1

Are solutions pairs of x and y that satisfy the equation?

Teacher
Teacher

Yes, precisely! Each point (x, y) on the line of the equation is a solution. How many solutions do we think a linear equation in two variables possesses?

Student 2
Student 2

I think it has infinitely many solutions!

Teacher
Teacher

Correct! This means that as long as a point lies on the line, it satisfies the equation. Use the memory aid 'SOL' – 'Solutions On the Line' to remember this!

Finding Specific Solutions

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0:00
Teacher
Teacher

Let’s work on finding specific solutions. If we take the equation x + 2y = 6, what would be a solution if we set x = 0?

Student 3
Student 3

We get 0 + 2y = 6, so y = 3! That gives us the solution (0, 3).

Teacher
Teacher

Great! Now, if we set y = 0 instead, what do we find?

Student 4
Student 4

Then the equation becomes x = 6, meaning (6, 0) is another solution!

Teacher
Teacher

Exactly! This technique of setting one variable to zero helps us find solutions easily. Remember the tip: 'O' for 'Zeroes' when searching for quick solutions!

Introduction & Overview

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Quick Overview

This section introduces linear equations in two variables, highlighting their characteristics, forms, and the concept of solutions.

Standard

Linear equations in two variables are equations that can be expressed in the form ax + by + c = 0, where the solutions are pairs of values (x, y) that satisfy the equation. This section details how to convert equations into this standard form and explores the nature of solutions, emphasizing infinite solutions for such equations.

Detailed

Detailed Summary

In this section, we explored the definition and representation of linear equations in two variables. A linear equation can generally be expressed in the form:

$$ ax + by + c = 0 $$

where a, b, and c are real numbers, and both a and b cannot be zero simultaneously. Each equation corresponds to a geometric representation on a Cartesian plane, where every point (x, y) that lies on the line defined by the equation represents a solution.

The section also covered how to express different forms of equations in this standard linear form, illustrated by several examples. Additionally, it discussed the infinite number of solutions such equations possess, emphasizing that any point on the line is a solution, and methods to identify specific solutions through substitution were provided.

This exploration lays the foundation for understanding the graphical representation of linear equations and their real-world applications.

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Audio Book

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Introduction to Linear Equations in Two Variables

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In a One-day International Cricket match between India and Sri Lanka played in Nagpur, two Indian batsmen together scored 176 runs. Express this information in the form of an equation. Here, you can see that the score of neither of them is known, i.e., there are two unknown quantities. Let us use x and y to denote them. So, the number of runs scored by one of the batsmen is x, and the number of runs scored by the other is y. We know that
x + y = 176,
which is the required equation.

Detailed Explanation

In this example, we translate a real-world situation into a mathematical equation. Two players have scores that add up to a total (176 runs). We denote unknown scores with variables x and y. By defining the total equation, we articulate the relationship between both players' scores using an algebraic expression, leading to the equation x + y = 176.

Examples & Analogies

Think of it like sharing a pizza between two friends. If you know that together they ordered a pizza with a total of 8 slices, but don’t know how many slices each person took, you could let x represent the slices taken by one friend and y the slices taken by the other. Thus, you can express the idea as x + y = 8.

Form of a Linear Equation in Two Variables

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This is an example of a linear equation in two variables. It is customary to denote the variables in such equations by x and y, but other letters may also be used. Some examples of linear equations in two variables are: 1.2s + 3t = 5, p + 4q = 7, πu + 5v = 9 and 3 = 2x – 7y. Note that you can put these equations in the form 1.2s + 3t – 5 = 0, p + 4q – 7 = 0, πu + 5v – 9 = 0 and 2x – 7y – 3 = 0, respectively.

Detailed Explanation

A linear equation in two variables is an equation that can be expressed in the form ax + by + c = 0. Here, 'a', 'b', and 'c' are constants, and 'x' and 'y' are the variables. This format helps in identifying relationship patterns. For example, components like 1.2s + 3t = 5 can be rearranged into canonical form, which is useful for further analysis and graphing.

Examples & Analogies

Imagine you are budgeting your monthly purchases. Let x represent the money spent on food and y represent the money spent on entertainment. Your budget can be written as an equation like 50x + 30y = 1000. Rearranging it into the standard form helps you understand how much you can allocate towards each aspect.

Characteristics of Linear Equations

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So, any equation which can be put 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. This means that you can think of many many such equations.

Detailed Explanation

For an equation to be classified as a linear equation in two variables, it must meet specific criteria: it should be expressible in the standard form provided, and at least one of the coefficients a or b must be non-zero. This emphasizes that a linear equation sketches out straight-line relationships which can either intersect, be parallel, or meet at a common point in geometric representations.

Examples & Analogies

Consider designing a simple mobile plan where the costs depend on the minutes used (x) and the texts sent (y). If your plan states that each minute costs $2 and each text costs $1, your total expense can be expressed with an equation like 2x + 1y = total_cost, demonstrating how different usages impact overall spending.

Transforming Equations into Standard Form

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Example 1 : Write each of the following equations in the form ax + by + c = 0 and indicate the values of a, b and c in each case:
(i) 2x + 3y = 4.37 (ii) x – 4 = 3y (iii) 4 = 5 x – 3y (iv) 2 x = y

Detailed Explanation

In this example, the task is to rearrange several equations to match the standard form ax + by + c = 0 and identify coefficients a, b, and c. By manipulating the equations through addition, subtraction, or rearrangement, students learn how to express equations in a consistent format that facilitates solving and graphing.

Examples & Analogies

Think of it like reorganizing a messy closet. You can’t find anything until you sort it out. By putting everything in its specific place, or in this case, standard format, it becomes much easier to see what you have and make decisions on what to do next.

Identification of Linear Equations

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Equations of the type ax + b = 0 are also examples of linear equations in two variables because they can be expressed as
ax + 0. y + b = 0.

Detailed Explanation

Equations that can be reformulated into the standard form with one variable having a coefficient of zero are still considered linear equations in two variables. This highlights the flexibility of interpreting linear relationships, even when one variable is absent, as the equation still represents a straight line when graphed.

Examples & Analogies

Imagine a scenario where you’re tracking your savings. If you know your savings goal (like $100 by a certain date), you can express your savings progress as an equation: 200 - y = 0, where y represents how much you've saved. Even though this equation seems to deal with a single variable in appearance, it effectively describes a relationship involving a secondary component (time, in this case).

Definitions & Key Concepts

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

Key Concepts

  • Linear Equations: Expressed in the form ax + by + c = 0, emphasizing variable relationships.

  • Infinitely Many Solutions: Any point on the line of an equation represents a solution.

Examples & Real-Life Applications

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

Examples

  • {'example': 'Represent the equation 2x + 3y = 9 in standard form.', 'solution': '2x + 3y - 9 = 0 \quad (a = 2, b = 3, c = -9)'}

  • {'example': 'Find solutions for the equation x + 2y = 6 when x = 0 and y = 0.', 'solution': '(0, 3) and (6, 0)'}

Memory Aids

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

🎵 Rhymes Time

  • Linear lines, straight and fine, solutions infinite, in each design.

📖 Fascinating Stories

  • Imagine a treasure map divided by lines. Each dot on this line shows a location to find treasure – they're all solutions to the treasure equation!

🧠 Other Memory Gems

  • SOL - Solutions On the Line helps remember that every point on the line is a solution.

🎯 Super Acronyms

ABE - a for 'a', b for 'b', E for 'Equation' helps recall the coefficients of linear equations.

Flash Cards

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Glossary of Terms

Review the Definitions for terms.

  • Term: Linear Equation

    Definition:

    An equation that can be expressed in the form ax + by + c = 0, where a, b, c are real numbers and not both a, b are zero.

  • Term: Variables

    Definition:

    Symbols (like x and y) that represent unknown values in equations.

  • Term: Solution

    Definition:

    A pair of values (x, y) that satisfies the equation.

  • Term: Cartesian Plane

    Definition:

    A two-dimensional number line where solutions of equations are represented as points.