Professional Courses
Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.
Categories
Interactive Games
Fun, engaging games to boost memory, math fluency, typing speed, and English skillsβperfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
Enroll to start learning
Youβve 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 Variables on Both Sides
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Today, we're going to learn about solving equations with variables on both sides. Can anyone give an example of such an equation?
How about `2x - 3 = x + 2`?
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?
We can subtract `x` from both sides?
Good thinking! That's a great first step. When we subtract `x`, what do we get on each side?
We get `2x - x - 3 = 2`.
Right! Now, can someone simplify this further?
That gives us `x - 3 = 2`.
Excellent! Now, whatβs the next step?
Add 3 to both sides, so we get `x = 5`.
Perfect! Remember, we always need to do the same operation on both sides to keep the equation balanced.
Multiplying to Simplify
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Letβs look at a more complex example: `5x + 7/2 = x - 14`. Whatβs our first step?
We could multiply by 2 to eliminate the fraction!
Exactly! Multiplying both sides by 2 gives us a simpler equation. What do we get?
That becomes `10x + 7 = 2x - 28`.
Great! Now, how do we isolate `x` from here?
We can subtract `2x` from both sides.
Correct! And what does that lead us to?
It simplifies to `8x + 7 = -28`.
Now, whatβs our next move?
We subtract `7` from both sides to get `8x = -35`.
Fantastic! Finally, how do we find `x`?
We divide by `8`, giving us `x = -35/8`.
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
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
Solving Equations with the Variable on Both Sides
An equation represents the equality of the values of two expressions, such as in the example 2x - 3 = 7. Here, the left-hand side (LHS) and the right-hand side (RHS) can also include variables, as demonstrated in 2x - 3 = x + 2.
To solve these equations, we need to isolate the variable. The section introduces how to manipulate both sides of the equation. For example, in the equation 2x - 3 = x + 2, steps such as subtracting x from both sides help us to eventually isolate x as shown in the solution where it simplifies to x = 5.
Additional examples, such as 5x + 7/2 = x - 14, illustrate multiplying each side of the equation to simplify the expressions further. The section emphasizes understanding the operations performed on both sides to maintain equality and reach the solution effectively.
Example 2: Solve \[ 3x - 4 = 2x + 5 \]
Solution: We have
\[ 3x - 4 = 2x + 5 \]
or
\[ 3x - 2x = 5 + 4 \]
or
\[ x = 9 \]
(solution)
Youtube Videos
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Understanding Equations with Variables on Both Sides
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
An equation is the equality of the values of two expressions. In the equation 2x β 3 = 7, the two expressions are 2x β 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 β 3 = x + 2 has expressions with a variable on both sides; the expression on the LHS is (2x β 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
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Example 1: Solve 2x β 3 = x + 2. Solution: We have 2x = x + 2 + 3 or 2x = x + 5 or 2x β 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
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Example 2: Solve 5x + = x β14. Solution: Multiply both sides of the equation by 2. We get (2Γ(5x + 7/2)) = (2Γ(x β 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.
π΅ Rhymes Time
-
To keep the balance, let's not lose, do the same to both, that's the right move!
π 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.
π§ Other Memory Gems
-
Remember: I.S.O.L.A.T.E - Isolate, Subtract, Or, Leave Alone The Equation!
π― Super Acronyms
MATH - Move All Terms Homogeneously.
Flash Cards
Review key concepts with flashcards.
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.