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Interactive Audio Lesson
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Understanding Solutions
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Today we're going to discuss linear equations in two variables. Can anyone tell me why we consider pairs of numbers for solutions?
Because there are two variables, we need two values that satisfy the equation?
Exactly! For example, in the equation 2x + 3y = 12, if I say x = 3, what could y be?
If x is 3, then y would be 2, right? Because 2(3) + 3(2) equals 12.
Spot on! So, (3, 2) is one solution. But there are infinitely many solutions as we can choose different values for x. What happens if x = 0?
Then y becomes 4!
Great! So (0, 4) is another solution. This shows that a linear equation in two variables has multiple pairs that satisfy its conditions.
Remember, each solution can be visualized as a point in the Cartesian plane.
Finding More Solutions
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Let's continue with our previous example. If I select x=1 for the equation 2x + 3y = 12, what do we get for y?
So we substitute and solve it: 2(1) + 3y = 12, which gives us y = rac{10}{3}.
Right! So (1, rac{10}{3}) is also a solution. Notice how we continuously generate new pairs? That's the beauty of linear equations.
Can we find solutions by throwing in negative values for x?
Absolutely! If x = -1, whatβs y?
We substitute: 2(-1) + 3y = 12 so y = rac{14}{3}.
Perfect! Each of these solutions leads us to different points on the graph, illustrating the concept of infinite solutions visually.
Application of Solutions in Equations
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Imagine you have a scenario where you need to budget money between two funds. If $x is the amount in Fund A and $y is the amount in Fund B, and they need to equal a total of $500, how would we express that?
We could say x + y = 500, giving us another equation!
Exactly! Just like our previous equations, this also has infinite solutions. If x is $200, whatβs y?
Then y would be $300, right?
Correct! Can anyone tell me why this helps in planning?
Since we have multiple solutions, we can allocate funds flexibly based on our priorities!
Exactly! Each point represents a possible budgeting strategy, allowing us to visualize options effectively.
Checking Validity of Solutions
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Let's see if (2, 1) is a solution for 2x + 3y = 12. Who can check it?
Substituting x and y gives us 2(2) + 3(1) = 4 + 3 = 7, which is not equal to 12.
Great effort! So, (2, 1) does not satisfy the equation. How about (3, 2)?
Substituting shows we get 2(3) + 3(2) = 6 + 6 = 12! So, it works!
Correct! A solid way to validate solutions is by substitution. Always be sure to check the answers you've derived.
Remember, with infinite solutions, checking is essential to ensure you have the right pairs.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In this section, we carefully analyze the solutions of linear equations involving two variables, highlighting that each equation can yield infinitely many pairs fitting the equation. We explore various methods of finding solutions through examples and exercises, reinforcing the concept that each solution corresponds to a point on the Cartesian plane.
Detailed
Solution of a Linear Equation
This section discusses the solutions to linear equations in two variables. Unlike linear equations in one variable that possess a unique solution, a linear equation in two variables provides us not just with one solution, but rather infinitely many pairs of values satisfying the equation form.
For example, the equation
$$2x + 3y = 12$$ is examined, demonstrating how different pairs of $(x, y)$ sets, such as (3, 2) and (0, 4), fulfill the equation's requirements. Students learn to verify candidates' solutions through substitution and are encouraged to explore choosing values for one variable to find corresponding values for the other variable.
The section includes practical examples to derive multiple solutions from a single linear equation, explaining methods like setting one variable to zero to find intercepts on the Cartesian plane. Through engaging exercises and examples, learners understand that each valid solution corresponds to a point on the linear equation's graph, revealing the infinite nature of solutions in this context.
Example : Find four different solutions of the equation \(x + 3y = 12\).
Solution: By inspection, \(x = 6, y = 2\) is a solution because for \(x = 6, y = 2\)
\[ 6 + 3 \times 2 = 12 \]
Now, let us choose \(x = 0\). With this value of \(x\), the equation reduces to \(3y = 12\) which has the unique solution \(y = 4\) so, \(x = 0, y = 4\) is also a solution of the given equation.
Similarly, taking \(y = 0\), the given equation reduces to \(x = 12\). So, \(x = 12, y = 0\) is a solution of \(x + 3y = 12\). Finally, let us take \(y = 1\). The given equation now reduces to \(x + 3 = 12\), whose solution is \(x = 9\). Therefore, \((9, 1), (6, 2), (0, 4), (12, 0)\) is also a solution of the given equation. So four of the infinitely many solutions of the given equation are:
\[(6, 2), (0, 4), (12, 0), (9, 1)\]
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Understanding Solutions for Linear Equations
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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.
Detailed Explanation
In mathematics, when we talk about a linear equation in one variable, like x + 2 = 5, there is only one value for x that makes the equation true. Similarly, a linear equation in two variables (like 2x + 3y = 12) requires a pair (x, y) that works in the equation. This means that if you select a value for x, there will be a corresponding value for y that satisfies the equation, resulting in infinite pairs of solutions, as you can choose many values for x, each leading to a valid y.
Examples & Analogies
Think of it like making a smoothie. If you decide to use 2 bananas, the number of strawberries you can add will depend on how many total fruits you want. If you want a total of 6 fruits, you can have 2 bananas and 4 strawberries (2, 4), or maybe 0 bananas and 6 strawberries (0, 6). Each combination forms a solution to your fruit smoothie recipe, just as pairs (x, y) form the solutions to linear equations.
Finding Solutions for a Given Equation
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Let us consider the equation 2 x + 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 Γ 3) + (3 Γ 2) = 12.
Detailed Explanation
To determine if a pair (x, y) is a solution for the equation 2x + 3y = 12, we substitute the values into the equation. For instance, plugging x = 3 and y = 2 gives us 2(3) + 3(2) = 6 + 6 = 12, which is correct. This confirms that (3, 2) is indeed a solution. This process can similarly be repeated for any other value pair to check its validity as a solution.
Examples & Analogies
Imagine you are buying apples and oranges. The equation could represent the total cost of those fruits. If apples cost $2 each and oranges cost $3 each, and you spend $12, the (number of apples, number of oranges) pairs that make this true (like (3, 2) for 3 apples and 2 oranges) are your solutions! By checking different combinations, you'll find all the ways to spend $12 on these fruits.
Infinitely Many Solutions
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In fact, we can get many many solutions in the following way. Pick a value of your choice for x (say x = 2) in 2 x + 3y = 12. Then the equation reduces to 4 + 3y = 12, which is a linear equation in one variable.
Detailed Explanation
This chunk highlights that you can generate solutions to a linear equation in two variables by choosing a value for one of the variables, then solving for the other. For example, if we choose x = 2, we adjust the equation to find y. This method reveals that not just two, but countless combinations suit the equation as long as they meet the criteria set by it, emphasizing the nature of linear equations.
Examples & Analogies
Consider designing a garden. If you decide how much space to allocate to flowers (x), you can determine how many shrubs (y) to plant based on your total garden area limit. As you try different plots of land for flowers (maybe x = 5 square feet), you recalculate how many shrubs you can plant within your total dedicated area, trying on the go to see how to maximize your garden.
Finding Specific Solutions
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Example 3: Find four different solutions of the equation x + 2y = 6. Solution: By inspection, x = 2, y = 2 is a solution because for x = 2, y = 2 x + 2y = 2 + 4 = 6.
Detailed Explanation
In this example, we are tasked with finding multiple solutions for the equation x + 2y = 6. We start with one solution (2, 2). Then, by substituting different values for x (like 0 or 6), we can determine corresponding values for y, thus finding additional solutions such as (0, 3) and (6, 0). This shows how different combinations yield various answers, reinforcing that there are infinitely many solutions.
Examples & Analogies
Think about making a budget for your monthly expenses. If you know you have $6 to spend, you can decide whether to spend $2 on food and $2 on entertainmentβthatβs one solution (2 food, 2 entertainment). You might also choose to spend all on food (6 food, 0 entertainment) or divide it in another way. Each budgeting choice represents solving the equation for different (x, y) pairs.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Linear Equation: An equation of the first degree involving two variables.
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Ordered Pair: A solution represented as (x, y) satisfying the equation.
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Infinite Solutions: The nature of linear equations in two variables leading to endless solutions.
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Substitution Method: A technique used to determine solutions by replacing a variable.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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{'example': 'Find different solutions for the equation 2x + 3y = 12.', 'solution': 'Solutions include (3, 2), (0, 4), (6, 0), (2, 4/3), etc.'}
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{'example': 'Verify if (2, 1) is a solution for 2x + 3y = 12.', 'solution': 'Substituting gives 2(2) + 3(1) = 4 + 3 = 7, not a solution.'}
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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For every linear line you see, pairs of values must come free.
π Fascinating Stories
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Alice and Bob share a total of 100 apples. For every apple Alice gets, Bob gets two. They can trade any way they like, but the total remains constant!
π§ Other Memory Gems
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SPLIT: Substitute, Pair, Linear, Infinite, Test if valid.
π― Super Acronyms
SPL
- Solutions
- Pairs
- Lines.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Linear Equation
Definition:
An equation involving two variables that can be written in the form ax + by + c = 0.
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Term: Ordered Pair
Definition:
A pair of numbers (x, y) that represents a solution to a linear equation.
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Term: Infinite Solutions
Definition:
A scenario where a linear equation in two variables can yield endless valid pairs (x, y).
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Term: Substitution
Definition:
A method of replacing a variable with a number or another expression in order to solve an equation.