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 skills—perfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
Enroll to start learning
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 Substitution Method
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Today we'll dive into the substitution method for solving simultaneous equations. Does anyone know what simultaneous equations are?
Are they equations that have two variables?
Exactly! They are systems with two or more equations. The substitution method is a technique where we solve one equation for a variable and then substitute that into another equation. Can anyone share why we might use this method?
Maybe when it's easier to isolate one variable?
Great observation! Let's remember this: when one variable is easily isolated, we solve for that first. This way, our calculations become simpler.
Steps of the Substitution Method
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Let's break down the steps. First, we rearrange one equation. Can someone explain how we would actually do that?
We isolate the variable on one side of the equation?
Correct! Then we substitute this into the other equation. So now, what do we do next?
We solve for the remaining variable!
Exactly! And once we have that value, what’s our final step?
We use it to find the other variable!
Fantastic! Each step flows into the next. Remember – rearranging, substituting, solving, and then back substituting.
Example Problem Using Substitution
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Now let’s solve an example together. We have y = 2x + 1 and 3x + y = 13. What’s our first step?
Substitute y into the second equation?
Correct! Let’s do that. What does it look like when we substitute?
3x + (2x + 1) = 13, right?
Spot on! Now simplify it for me.
5x + 1 = 13, so then 5x = 12 and x = 2.4.
Perfect! Now how do we find y?
We plug x back into y = 2(2.4) + 1, and that gives us y = 5.8.
Exactly! Our solution is (2.4, 5.8). Well done, everyone!
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
The substitution method allows Algebra students to solve simultaneous equations by expressing one variable in terms of another and substituting this expression into a second equation. This section explores step-by-step instructions, provides examples, and outlines when this method is most effective.
Detailed
Substitution Method
The substitution method is a powerful technique used to solve simultaneous equations, specifically when one equation can be easily rearranged to express one variable in terms of the other. This method is particularly useful when one of the variables is isolated, making it straightforward to substitute into the other equation.
Key Steps in the Substitution Method
- Rearranging the First Equation: Begin by rearranging one of the equations to solve for one variable.
- Substituting into the Second Equation: Substitute the rearranged expression into the second equation, which results in a single-variable equation.
- Solving for the Variable: Solve the resulting equation for the variable.
- Finding the Other Variable: Use the value obtained to find the other variable by substituting back into the rearranged equation.
Example of the Substitution Method
Given two equations: y = 2x + 1 and 3x + y = 13, we can substitute the first equation into the second:
- Substitute: 3x + (2x + 1) = 13
- Combine and simplify: 5x + 1 = 13 => 5x = 12 => x = 2.4
- Lastly, find y: y = 2(2.4) + 1 = 5.8. So, the solution set is (x, y) = (2.4, 5.8).
This method is essential not only for algebraic solutions but also serves as a foundation for understanding more complex algebraic concepts such as functions and systems of linear equations.
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Overview of the Substitution Method
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
The Substitution Method is used when one equation is easily rearranged to express one variable in terms of the other.
Detailed Explanation
The Substitution Method is a strategy for solving simultaneous equations. It works best when one equation can be manipulated to isolate one of the variables. For example, if we have two equations, we can choose one equation and rearrange it to make one variable (like y) a function of the other variable (like x). Once we have this relationship, we can substitute that expression into the second equation, allowing us to solve for one variable directly.
Examples & Analogies
Imagine you have a recipe that calls for a certain ratio of ingredients. If you know the amount of one ingredient you have, you can determine how much of the other ingredient you need by substituting that known amount into the recipe's equation.
Steps of the Substitution Method
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Steps:
1. Rearrange one equation to express one variable in terms of the other.
2. Substitute into the second equation.
3. Solve the resulting single-variable equation.
4. Substitute back to find the second variable.
Detailed Explanation
The steps of the Substitution Method involve four main actions: First, you rearrange one of the original equations to isolate one variable. Next, you substitute this expression into the other equation to form a single-variable equation. After solving this equation for the isolated variable, you then substitute back to find the other variable, completing the solution process.
Examples & Analogies
Consider a situation where you're trying to balance the ingredients in a smoothie recipe. If you know how much fruit (x) you have and you want to find out how much liquid (y) you need, you can rearrange the recipe to express the liquid in terms of fruit. Once you know that, you can figure out how much liquid to add by simply substituting your known amount of fruit into the equation.
Example of the Substitution Method
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Example: Solve:
𝑦 = 2𝑥 + 1
3𝑥 + 𝑦 = 13
Solution: Substitute 𝑦 = 2𝑥 + 1 into the second equation:
3𝑥 + (2𝑥 + 1) = 13
5𝑥 + 1 = 13 ⇒ 𝑥 = 2.4
Then, 𝑦 = 2(2.4) + 1 = 5.8
Solution: 𝑥 = 2.4, 𝑦 = 5.8
Detailed Explanation
Let's work through this example step by step. We start with the equations: y = 2x + 1, which is already isolated for y, and our second equation: 3x + y = 13. We substitute the expression for y from the first equation into the second equation. This gives us a single-variable equation: 3x + (2x + 1) = 13. Combining like terms leads to 5x + 1 = 13. We then solve for x by isolating it: 5x = 12 ⇒ x = 2.4. Finally, we substitute back to find y by putting x = 2.4 into the first equation: y = 2(2.4) + 1, which gives y = 5.8. Thus, we find the solution as x = 2.4 and y = 5.8.
Examples & Analogies
Think of it like solving a mystery where you know one part of a story but need to find the missing part. If you have a clue that says 'the total is 13' and another clue that says 'I’m two times something plus one', you use the second clue to fill in the gaps of the first clue, leading you to uncover both parts of the story!
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Substitution Method: A technique to solve simultaneous equations by isolating a variable and substituting it into another equation.
-
Rearranging: The process of manipulating an equation to isolate a variable.
-
Solution Set: The values for which all equations are simultaneously true.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
Solve y = 2x + 1 and 3x + y = 13, yielding x = 2.4, y = 5.8.
-
Convert the equations 2y - x = 4 and y = 3x + 5 into a single-variable equation using substitution.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
To solve through substitutions, we rearrange, find a new equation; replace and get, results that will amaze!
📖 Fascinating Stories
-
Once there were two friends, X and Y, who wanted to find their paths together. X shared his secret formula, Y plugged it in, and together they found their joint journey at (2.4, 5.8).
🧠 Other Memory Gems
-
R-S-S-B: Rearrange, Substitute, Solve, Back substitute.
🎯 Super Acronyms
SUSS
- Substitution
- Use
- Simplify
- Solve.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Simultaneous Equations
Definition:
Equations that share variables and are solved together.
-
Term: Substitution Method
Definition:
A method of solving simultaneous equations by rearranging one equation and substituting into another.
-
Term: Isolate
Definition:
To rearrange an equation to have one variable alone on one side.