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2 - Linear Equations in One Variable

Interactive Audio Lesson

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Introduction to Linear Equations

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Teacher
Teacher

Today, we will delve into linear equations in one variable. Can anyone tell me what they think a linear equation is?

Student 1
Student 1

Is it an equation that has a straight line when graphed?

Teacher
Teacher

Exactly! Linear equations are represented graphically as straight lines. They consist of variables raised to the first power. For example, `2x + 5 = 10`.

Student 2
Student 2

What does it mean for an equation to have one variable?

Teacher
Teacher

Great question! An equation with one variable means that it only contains one type of variable, like `x` or `y`. In our example `2x + 5 = 10`, `x` is our single variable.

Teacher
Teacher

To simplify, remember the acronym 'LIN' - **L**inear, **I**nvolves one variable, and **N**o exponents greater than one. Everyone got that?

Student 3
Student 3

Got it! LIN helps me remember.

Teacher
Teacher

Following our introduction, let’s see a simple problem. If `x + 3 = 7`, how do we find `x`?

Student 4
Student 4

We subtract 3 from both sides!

Teacher
Teacher

Correct! That gives us `x = 4`. Well done!

Solving Equations with Variables on Both Sides

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Teacher
Teacher

Now, let's tackle equations like `2x - 3 = x + 2`, which have variables on both sides. Who can suggest how we might approach this?

Student 2
Student 2

We could try moving the variables to one side.

Teacher
Teacher

Exactly! We subtract `x` from both sides. Let’s do it step by step: we get `2x - x - 3 = 2`.

Student 1
Student 1

So that simplifies to `x - 3 = 2`.

Teacher
Teacher

Correct! Now how do we solve for `x`?

Student 3
Student 3

Add 3 to both sides to find `x = 5`.

Teacher
Teacher

Fantastic! Remember the phrase 'SIMPLIFY AND SOLVE' to help you remember our approach!

Student 4
Student 4

I find that helpful!

Teacher
Teacher

Great! Let’s do another similar example. Try `5x + 7 = 3x - 5`.

Simplifying Equations

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Teacher
Teacher

Next, we will discuss simplifying equations before solving them. Can anyone give me an example of what that might look like?

Student 1
Student 1

What about when we have fractions? They can make it tough.

Teacher
Teacher

Absolutely! For example, consider the equation `6(x + 1/2) = 3`.

Student 2
Student 2

We could multiply everything by 2 to eliminate the fraction!

Teacher
Teacher

Exactly! Multiplying through by 2 gives us `12(x + 1) = 6`, simplifying our work considerably.

Student 3
Student 3

Doesn't that mean balancing the equation too?

Teacher
Teacher

Correct! Whatever we do to one side, we must do to the other to maintain the equality. Remember to always 'BALANCE' your operations.

Student 4
Student 4

Sounds good! I'll keep that in mind when I'm solving.

Introduction & Overview

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

Quick Overview

This section introduces linear equations in one variable, explaining their characteristics and methods for solving them.

Standard

In this section, students revisit algebraic expressions and equations, focusing specifically on linear equations in one variable. The section outlines how to identify linear expressions, demonstrates solving equations involving variables on both sides, and emphasizes techniques for simplifying equations to find solutions.

Detailed

Detailed Summary

In this section, we explore linear equations in one variable, which are characterized by their linear nature, meaning the variable involved has an exponent of one. We differentiate between algebraic expressions (like 5x and 2x - 3) and equations (where an equality sign = is present).

Key Aspects Covered:
- Definitions: Understanding what constitutes an algebraic equation and the significance of the left-hand side (LHS) and right-hand side (RHS) in an equation.
- Solving Techniques: Methods to solve equations with variables on both sides by manipulating expressions while maintaining equality. For instance, in the equation 2x - 3 = x + 2, students learn to isolate the variable through transposition and addition or subtraction of terms.
- Examples: Practical examples illustrate the solving process in detail, demonstrating essential algebraic manipulations.
- Equations with More Complexity: We also discuss simplifying more complicated equations, such as those containing fractions or requiring the use of the least common multiple (LCM) for solving denominators.

By mastering these techniques, students will be equipped to tackle various mathematical problems involving linear equations effectively.

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Audio Book

Dive deep into the subject with an immersive audiobook experience.

Introduction to Linear Equations

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In the earlier classes, you have come across several algebraic expressions and equations. Some examples of expressions we have so far worked with are: 5x, 2x – 3, 3x + y, 2xy + 5, xyz + x + y + z, x2 + 1, y + y2.

Some examples of equations are: 5x = 25, 2x – 3 = 9, 6z + 10 = −2. You would remember that equations use the equality (=) sign; it is missing in expressions. Of these given expressions, many have more than one variable. For example, 2xy + 5 has two variables. We, however, restrict to expressions with only one variable when we form equations. Moreover, the expressions we use to form equations are linear. This means that the highest power of the variable appearing in the expression is 1.

Detailed Explanation

This section introduces the concept of linear equations, which are equations that represent a straight line when graphed on a coordinate system. It distinguishes between algebraic expressions (like 5x or 2x - 3) and equations (such as 5x = 25) because equations have an equality sign. Linear expressions are specifically those where the highest power of the variable (e.g., x or y) is 1.

Examples & Analogies

Think of linear equations like budget planning. Just as you cannot spend more than you have (you must keep the sides balanced), in linear equations, the left side must equal the right side, representing a balanced financial situation. If you earn x dollars and spend it on y items, the equation representing this situation would be linear because the highest power of your earnings (x) and items (y) is one.

Linear vs. Non-linear Expressions

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These are linear expressions: 2x, 2x + 1, 3y – 7, 12 – 5z, (x – 4) + 10.

These are not linear expressions: x2 + 1, y + y2, 1 + z + z2 + z3 (since highest power of variable > 1). Here we will deal with equations with linear expressions in one variable only. Such equations are known as linear equations in one variable.

Detailed Explanation

This section clarifies the difference between linear expressions and non-linear expressions. It gives examples of each, highlighting that linear expressions can only have variables to the first power (for example, 2x), while non-linear expressions include variables raised to higher powers (such as x²). The focus will remain on working with linear equations that involve only one variable.

Examples & Analogies

Imagine planting seeds: if you have 5 seeds (2x can represent this) to plant, that’s straightforward. But if you have 5 seed packets each containing different quantities (like x² where x can be 1, 2, or 3) that complicates the garden plan. Linear equations help keep our gardening straightforward by using simple counts.

Understanding Solutions of Equations

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(a) An algebraic equation is an equality involving variables. It has an equality sign. The expression on the left of the equality sign is the Left Hand Side (LHS). The expression 2x – 3 = 7 on the right of the equality sign is the Right Hand Side (RHS). For x = 5, LHS = 2 × 5 – 3 = 7 = RHS. On the other hand, x = 10 is not a solution of the equation.

Detailed Explanation

Here, we learn how to identify the 'left-hand side' (LHS) and 'right-hand side' (RHS) of an equation. Solving an equation involves finding values for the variable that make both sides equal, termed the solutions. For example, if we substitute x with 5 in the equation 2x - 3 = 7, we find that both sides equal 7, meaning x = 5 is a solution. However, if we try x = 10, we get different values for LHS and RHS, so it's not a solution.

Examples & Analogies

Think of an equation as a seesaw – both sides must balance for it to work. If you put 5 kg on one side of the seesaw (LHS) and 7 kg on the other (RHS), it won't balance. If you figure out that 5 kg of items on one side matches with the 7 kg somehow, that’s your solution. Finding the right balance is key!

Solving Equations with Variables on Both Sides

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An equation is the equality of the values of two expressions. In the equation 2x – 3 = x + 2, the two expressions are 2x – 3 and x + 2. We now discuss how to solve such equations which have expressions with the variable on both sides.

Detailed Explanation

This part talks about solving equations where variables appear on both sides. In the example 2x - 3 = x + 2, we learn to treat both sides equally and perform operations such as adding or subtracting the same number or variable term to find a solution.

Examples & Analogies

Consider a balancing scale at a market. If the left side shows 2 apples minus 3 coins and the right shows one apple plus 2 coins, to know how many apples are there in total, you would shift items across the scale to keep it balanced, just like reorganizing the equation to isolate the variable.

Step-by-Step Example of Solving an Equation

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Example 1: Solve 2x – 3 = x + 2
Solution: We have 2x = x + 2 + 3 or 2x = x + 5 or 2x – x = 5 (subtracting x from both sides). Here we subtract x from both sides to get x = 5.

Detailed Explanation

This chunk walks through a practical example of solving an equation step-by-step. The solution shows how to manipulate the equation by subtracting x from both sides, simplifying it until we find that x equals 5. This method ensures we maintain the balance of the equation while isolating the variable.

Examples & Analogies

Think of it as organizing a box of toys. If you have 2 cars but need to know how many more you have than 1 stuffed animal you’re putting in another box, subtracting the animals (variables) from your toy collection lets you clearly see your total cars left.

Reducing Equations to Simpler Forms

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Example 16: Solve + 1 = 6x + 1
Solution: Multiplying both sides of the equation by 6. LHS becomes (2(6x + 1)) + 6 = x – 3.

Detailed Explanation

In this example, we learn how to simplify complex equations using multiplication. By ensuring both sides of the equation are treated with equal operations, we can reduce the equation to a simpler form for easier solving. This process may involve combining like terms and balancing both sides.

Examples & Analogies

Imagine having multiple packs of candies divided among friends. If you multiply the number of packs by how many candies are in each to see totals in hand, simplifying the equation works similarly. You'll know how many more packs are needed based on the balance of what remains.

What Have We Discussed?

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  1. An algebraic equation is an equality involving variables. 2. Linear equations in one variable have the highest power of a variable as 1. 3. Solutions of equations can be found by balancing both sides. 4. Sometimes equations need simplification before solving.

Detailed Explanation

The final chunk summarizes everything discussed in the section, reinforcing key points about linear equations, the nature of solutions, and importance of simplifying equations before solving them.

Examples & Analogies

Recap with a sports analogy: In a game, understanding the rules (equations), how to score points (solutions), and when to reset your strategies (simplifying) makes you a better player overall. Each part clearly fits together for you to win the game.

Definitions & Key Concepts

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

Key Concepts

  • Linear Equation: An equation where the highest power of the variable is one.

  • Variables: Symbols representing unknown values in equations.

  • LHS and RHS: The left and right sides of an equation that must be equal.

Examples & Real-Life Applications

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

Examples

  • To solve the equation 2x + 3 = 7, we subtract 3 from both sides, resulting in 2x = 4, and then divide by 2 to get x = 2.

  • In 3(x - 2) = 9, we first divide by 3 to get x - 2 = 3, and then add 2 to solve for x = 5.

Memory Aids

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

🎵 Rhymes Time

  • Linear equations might sound a chore, but with 'solve and balance', you'll never bore!

📖 Fascinating Stories

  • Once upon a time, in Equatia, there lived a wise old mathematician who said, 'To solve an equation, keep both sides equal - treat them as best friends.'

🧠 Other Memory Gems

  • Remember 'SOLVE': Separate, Operate, Leave variable alone, Verify solution, Eliminate mistakes.

🎯 Super Acronyms

Use 'L.E.S.S.' to remember - Linear equations, Equal sign, Simplifying expressions, Solve for variable.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Linear Equation

    Definition:

    An equation involving a variable raised to the first power.

  • Term: Variable

    Definition:

    A symbol used to represent an unknown value in an equation.

  • Term: Expression

    Definition:

    A combination of numbers, variables, and operations without an equality sign.

  • Term: LeftHand Side (LHS)

    Definition:

    The expression on the left side of the equality sign in an equation.

  • Term: RightHand Side (RHS)

    Definition:

    The expression on the right side of the equality sign in an equation.