4.3.2 - Finding Multiple Solutions
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Nature of Solutions in Linear Equations
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Today, we are learning about the nature of solutions in linear equations, especially in two variables. How many solutions do you think a linear equation can have?
I think it should have one solution, like in one-variable equations.
But what if it has two variables? Maybe it's different.
Good observation! A linear equation in two variables can have infinitely many solutions. Let’s explore why. For example, the equation 2x + 3y = 12 has multiple pairs of (x, y) values that work.
So, we can find many pairs that satisfy the equation?
Exactly! Picking any value for x allows us to solve for y, leading to various solutions. Remember, if you think of x as your input, y can be derived from it. This is the essence of functions where one variable depends on another.
What if I pick x = 4? How do I find y then?
If x = 4, you substitute it back into the equation 2(4) + 3y = 12, which simplifies to 3y = 4, giving you y = 4/3. Therefore, (4, 4/3) is another solution!
In summary, linear equations in two variables create a scenario where substituting one variable determines the other, thus leading to infinite solutions.
Finding Multiple Solutions
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Now, let's practice finding solutions together! Let's explore the equation x + 2y = 6. Can anyone start by choosing a value for x?
I’ll choose x = 0.
Great choice! Substituting x = 0 gives us the equation 0 + 2y = 6. What is y?
That means 2y = 6, so y = 3!
Correct, so (0, 3) is one solution. Can someone try a different value for x?
How about x = 2?
Excellent! Substituting x = 2 gives us 2 + 2y = 6. What’s y now?
I get 2y = 4, so y = 2. So another solution is (2, 2)!
Awesome! Keep observing; you can substitute anything for x to find corresponding y values. This allows us to create as many pairs as we like!
Verifying Solutions
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Now, let’s discuss how we can verify if a pair of values is indeed a solution to an equation. If I give you (3, 0), how would you verify it for the equation x + 2y = 6?
We would just substitute x and y into the equation.
Exactly! Let’s do that together. How about you, Student_1?
So, 3 + 2(0) = 3, which does not equal 6.
Correct! Hence, (3, 0) is not a solution. Let’s try (2, 2) next. Would that work?
If I substitute, I get 2 + 2(2) = 2 + 4 = 6, so it does work!
Well done! Verifying solutions ensures we only consider valid pairs that satisfy the original equation. To summarize, always substitute back to confirm a candidate solution.
Introduction & Overview
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Quick Overview
Standard
The section elaborates on the concept that while a linear equation in one variable has a unique solution, a linear equation in two variables can yield infinitely many solutions. It explores methods to find various solutions by substituting values and verifies the solutions through calculations.
Detailed
Finding Multiple Solutions to Linear Equations in Two Variables
In this section, we delve into the nature of linear equations in two variables, determining how many solutions they possess. Unlike linear equations in one variable which offer a unique solution, linear equations involving two variables present the possibility of infinite solutions due to the presence of two variables, x and y. For instance, considering the equation 2x + 3y = 12, we see that specific pairs of (x, y) values, such as (3, 2), (0, 4), and (6, 0), satisfy this equation. Furthermore, one can generate more solutions by choosing specific values for one variable and subsequently solving for the other. For example, assigning x different values leads to corresponding y values through substitutions. Hence, the section concludes that a linear equation in two variables does not just have limited solutions but an infinite range based on varying values for either variable. This foundational understanding marks the different handling of solving linear equations compared to their one-variable counterparts.
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Understanding Solutions in Two Variables
Chapter 1 of 3
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Chapter Content
You have seen that every linear equation in one variable has a unique solution. What can you say about the solution of a linear equation involving two variables? As there are two variables in the equation, a solution means a pair of values, one for x and one for y which satisfy the given equation. Let us consider the equation 2x + 3y = 12. Here, x = 3 and y = 2 is a solution because when you substitute x = 3 and y = 2 in the equation above, you find that 2x + 3y = (2 × 3) + (3 × 2) = 12. This solution is written as an ordered pair (3, 2), first writing the value for x and then the value for y. Similarly, (0, 4) is also a solution for the equation above...
Detailed Explanation
In this chunk, we learn about solutions in linear equations with two variables. A linear equation can have multiple solutions rather than just one, as seen in equations involving only one variable. In our specific example, the equation 2x + 3y = 12 can be satisfied by different pairs of (x, y). Here, for x = 3, y must be 2 to satisfy the equation, while for x = 0, y must be 4. Written as ordered pairs, these solutions appear as (3, 2) and (0, 4). The term 'ordered pair' signifies the sequence they must be considered in: the first number refers to x, and the second refers to y.
Examples & Analogies
Imagine scoring in basketball, where x represents points scored by one player and y by another. The total points scored need to equal 12. If player A scores 3 points, then player B must score 2 points to reach the total of 12 (as in our example). However, if player A scores 0 points, player B would need to score all 12 points! Just like this basketball game, many combinations of scores (or solutions) can reach the total goal.
Finding More Solutions
Chapter 2 of 3
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Chapter Content
You have seen at least two solutions for 2x + 3y = 12, i.e., (3, 2) and (0, 4). Can you find any other solution? Do you agree that (6, 0) is another solution? Verify the same. In fact, we can get many solutions in the following way. Pick a value of your choice for x (say x = 2) in 2x + 3y = 12. Then the equation reduces to 4 + 3y = 12, which gives you y = 8/3, providing another solution (2, 8/3) for 2x + 3y = 12. Similarly, choosing x = -5 leads us to y = 22/3, yielding another solution (-5, 22/3). So there is no end to different solutions of a linear equation in two variables. That is, a linear equation in two variables has infinitely many solutions.
Detailed Explanation
In this chunk, we continue exploring how to find different solutions for the linear equation 2x + 3y = 12. Starting with already known solutions, we can substitute different x values to yield corresponding y values. Choosing x = 2 gives us y = 8/3, creating a new ordered pair (2, 8/3). Similarly, if we select x = -5, we can find another y value of 22/3, leading us to the ordered pair (-5, 22/3). This illustrates that we can generate an infinite number of pairs (solutions) just by varying x.
Examples & Analogies
Consider a garden where you can plant various types of flowers in certain pairs (like x flowers of type A and y flowers of type B) while aiming for a specific total number of flowers. If you plant 2 of type A, you can calculate exactly how many of type B you still need for the total. However, you could also decide to plant fewer of type A and adjust the number of type B accordingly. As a result, many combinations will still achieve your target total, much like finding countless solutions in our equation.
Examples of Finding Multiple Solutions
Chapter 3 of 3
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Chapter Content
Example 3: Find four different solutions of the equation x + 2y = 6. By inspection, x = 2, y = 2 is a solution because for x = 2, y = 2, x + 2y = 2 + 4 = 6. Now, let us choose x = 0. With this value of x, the given equation reduces to 2y = 6 which has the unique solution y = 3. So x = 0, y = 3 is also a solution of x + 2y = 6. Similarly, taking y = 0, the given equation reduces to x = 6. So, x = 6, y = 0 is a solution of x + 2y = 6 as well. Finally, let us take y = 1. The given equation now reduces to x + 2 = 6, whose solution is given by x = 4. Therefore, (4, 1) is also a solution of the given equation. So four of the infinitely many solutions of the given equation are: (2, 2), (0, 3), (6, 0) and (4, 1).
Detailed Explanation
In this final chunk, we review how we can derive multiple solutions for the equation x + 2y = 6. Starting by plugging in x = 2, we quickly determine corresponding values of y, achieving a solution of (2, 2). If we instead set x to 0, we find y must equal 3, forming another solution (0, 3). Each of these steps reveals a new pair until we have reached four distinct solutions. This method illustrates a pattern and shows the systematic approach to finding multiple solutions.
Examples & Analogies
Think of organizing a meeting where there are a total of six participants represented by x (individuals) and pairs of tasks represented by y (each requiring two people). Depending on how you assign individuals to tasks, you can achieve a viable combination of participants to fulfill each role, resulting in multiple configurations. Just like our equation, each arrangement leads to a unique outcome however we describe it!
Key Concepts
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Linear Equations Have Infinite Solutions: A linear equation in two variables can have infinitely many (x, y) pairs that satisfy it.
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Finding Solutions: By assigning a value to one variable, we can easily find the corresponding value of the other variable.
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Verifying Solutions: Any proposed solution can be verified by substituting the values back into the original equation.
Examples & Applications
For the equation 3x + 2y = 12, we find solutions such as (0, 6), (4, 0), and (2, 3).
In the equation x + y = 5, choosing x = 1 results in y = 4, giving the solution (1, 4).
Memory Aids
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Rhymes
When linear's the game, infinite pairs you’ll gain!
Stories
Imagine a road stretching forever, where every point is a treasure, just like solutions on a line of a linear equation.
Memory Tools
First pick x, then solve y, a linear equation’s by your side!
Acronyms
S.O.L. - Substitute, Obtain, and List solutions!
Flash Cards
Glossary
- Linear Equation
An equation of the form ax + by + c = 0 where a, b, and c are real numbers, and not both a and b are zero.
- Solution
A pair of values (x, y) that satisfy the equation.
- Ordered Pair
A pair of numbers used to represent a point in a two-dimensional coordinate system.
- Infinitely Many Solutions
A characteristic of a linear equation in two variables that allows for an endless number of valid (x, y) pairs.
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