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2.2 - Solving Equations having the Variable on both Sides

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Introduction to Variables on Both Sides

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Teacher
Teacher

Today, we're going to learn about solving equations with variables on both sides. Can anyone give an example of such an equation?

Student 1
Student 1

How about `2x - 3 = x + 2`?

Teacher
Teacher

Exactly! In this case, both sides contain the variable `x`. The goal is to isolate the variable. Can anyone tell me how we might begin solving this?

Student 2
Student 2

We can subtract `x` from both sides?

Teacher
Teacher

Good thinking! That's a great first step. When we subtract `x`, what do we get on each side?

Student 3
Student 3

We get `2x - x - 3 = 2`.

Teacher
Teacher

Right! Now, can someone simplify this further?

Student 4
Student 4

That gives us `x - 3 = 2`.

Teacher
Teacher

Excellent! Now, what’s the next step?

Student 1
Student 1

Add 3 to both sides, so we get `x = 5`.

Teacher
Teacher

Perfect! Remember, we always need to do the same operation on both sides to keep the equation balanced.

Multiplying to Simplify

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Teacher
Teacher

Let’s look at a more complex example: `5x + 7/2 = x - 14`. What’s our first step?

Student 2
Student 2

We could multiply by 2 to eliminate the fraction!

Teacher
Teacher

Exactly! Multiplying both sides by 2 gives us a simpler equation. What do we get?

Student 3
Student 3

That becomes `10x + 7 = 2x - 28`.

Teacher
Teacher

Great! Now, how do we isolate `x` from here?

Student 4
Student 4

We can subtract `2x` from both sides.

Teacher
Teacher

Correct! And what does that lead us to?

Student 1
Student 1

It simplifies to `8x + 7 = -28`.

Teacher
Teacher

Now, what’s our next move?

Student 2
Student 2

We subtract `7` from both sides to get `8x = -35`.

Teacher
Teacher

Fantastic! Finally, how do we find `x`?

Student 3
Student 3

We divide by `8`, giving us `x = -35/8`.

Teacher
Teacher

Absolutely right! This method of manipulating both sides is key in solving these equations.

Introduction & Overview

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

Quick Overview

This section focuses on solving equations with variables on both sides, detailing methods and providing examples.

Standard

In this section, we learn how to solve equations where variables are present on both sides. We explore various methods to manipulate these equations, illustrated by examples that show the step-by-step process to isolate the variable and obtain solutions.

Detailed

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Audio Book

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Understanding 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 = 7, the two expressions are 2x – 3 and 7. In most examples that we have come across so far, the RHS is just a number. But this need not always be so; both sides could have expressions with variables. For example, the equation 2x – 3 = x + 2 has expressions with a variable on both sides; the expression on the LHS is (2x – 3) and the expression on the RHS is (x + 2).

Detailed Explanation

In mathematics, an equation states that two expressions are equal to each other. For instance, in the equation given (2x - 3 = x + 2), we have a variable (x) on both sides of the equation. This means that we need to find a value for x that makes both the left-hand side (LHS) and right-hand side (RHS) equal. Unlike simpler equations where one side might just be a number, these types of equations require us to manipulate the expressions to isolate the variable.

Examples & Analogies

Imagine you have two jars of candies. The first jar has a variable number of candies minus a few, represented by the expression 2x - 3, while the second jar has a different expression representing a variable number of candies plus a couple (x + 2). To find out how many candies are in each jar, you must figure out the value of x that makes both jars have the same amount.

Example 1: 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) or x = 5 (solution). Here we subtracted from both sides of the equation, not a number (constant), but a term involving the variable.

Detailed Explanation

To solve the equation 2x - 3 = x + 2, we want to get all terms involving x on one side and constants on the other. Starting with the equation, we move -3 over to the right side, which gives us 2x = x + 5. To isolate x, we subtract x from both sides, giving us x = 5. This means that if we replace x with 5 in the original equation, both sides will equal each other, confirming it's the correct solution.

Examples & Analogies

Think of this as an evenly balanced scale. You start off with some weights on one side (2x - 3) and want to balance them with weights on the other side (x + 2). By doing proper adjustments (subtracting), you eventually find the exact weight (value of x) that balances the scale.

Example 2: Another Equation

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Example 2: Solve 5x + = x −14. Solution: Multiply both sides of the equation by 2. We get (2×(5x + 7/2)) = (2×(x − 14)). After simplification, we reach 10x + 7 = 3x - 28.

Detailed Explanation

The equation 5x + 7 = x - 14 includes fractions. To eliminate the fraction, we can multiply the entire equation by 2 (the denominator). This gives us integers throughout, making it easier to solve. After multiplying and simplifying, we isolate x by moving 3x to the left side by subtracting it, yielding 7x + 7 = -28. Finally, we solve for x to get x = -5.

Examples & Analogies

Imagine a baking recipe where you need to double all ingredients. The term with x in your recipe represents the number of cups of flour. By multiplying the entire recipe by 2 (just like we multiplied the equation), you ensure that everything stays proportional. By the end, you find exactly how many cups you need, represented by the solution for x.

Definitions & Key Concepts

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

Key Concepts

  • Isolating Variables: The process of getting the variable on one side of the equation.

  • Maintaining Equality: Understanding that operations must be balanced on both sides.

  • Step-by-Step Solutions: Breaking down each operation to find the solution.

Examples & Real-Life Applications

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

Examples

  • Example 1: Solve 2x - 3 = x + 2, leading to x = 5.

  • Example 2: Solve 5x + 7/2 = x - 14, leading to x = -35/8.

Memory Aids

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

🎵 Rhymes Time

  • To keep the balance, let's not lose, do the same to both, that's the right move!

📖 Fascinating Stories

  • Once there was a magician named Variable who loved to play games. He would always keep EQUALITY balanced by doing the same trick on both sides of his magic equation.

🧠 Other Memory Gems

  • Remember: I.S.O.L.A.T.E - Isolate, Subtract, Or, Leave Alone The Equation!

🎯 Super Acronyms

MATH - Move All Terms Homogeneously.

Flash Cards

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Glossary of Terms

Review the Definitions for terms.

  • Term: Equation

    Definition:

    A statement that asserts the equality of two expressions, typically containing variables.

  • Term: Variable

    Definition:

    A symbol representing an unknown quantity in mathematics, often denoted by letters such as x, y, etc.