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Introduction to Linear Equations
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Today, we're going to learn about linear equations in two variables. Can anyone remind me what's a linear equation in one variable?
Isnβt it something like x + 1 = 0?
Exactly! Now, when we move to two variables, it looks like this: ax + by + c = 0. Here, x and y can have multiple solutions. For example, in x + y = 10, if x is 4, what is y?
Oh, that would be y = 6!
Great! Remember, each combination of x and y that satisfies the equation is a part of its solution set.
Examples of Linear Equations
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Letβs review some examples. If I give you the linear equation 2x + 3y = 6, can someone rewrite it in the form ax + by + c = 0?
We can write it as 2x + 3y - 6 = 0!
Correct! Now, what are the values for a, b, and c?
a = 2, b = 3, and c = -6.
Very good! Let's try another: What about 4 = 5x - 3y?
We can rearrange it to 5x - 3y - 4 = 0.
Finding Solutions
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Now, if I have the equation x + 2y = 6, how can we find pairs of (x, y) that satisfy it?
We can choose values for x, like 0, and find y!
Exactly! Letβs start with x = 0. What do we get for y?
If x = 0, then 2y = 6, so y = 3. So one solution is (0,3).
Nicely done! Now, if we let y = 0 instead, what will we have?
Then x = 6, so another solution is (6, 0).
Perfect! Remember, there are infinitely many solutions.
Applying Linear Equations to Real-World Problems
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Letβs consider a real-world example: Two friends scored a total of 176 runs in a cricket match. How can we represent their scores as a linear equation?
We can call one score x and the other y, right? So x + y = 176!
Yes! Thatβs a linear equation in two variables. How can we visualize this?
Every point on a graph that satisfies the equation represents a possible score combination!
Exactly! Understanding these equations helps us see the relationship between variables in practical terms.
Introduction & Overview
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Quick Overview
Standard
In this section, we explore linear equations in two variables, discussing their general form (ax + by + c = 0), solution sets, and examples. The section highlights the characteristics of such equations and their representation on the Cartesian plane.
Detailed
Linear Equations in Two Variables
This section delves into the concept of linear equations in two variables. A linear equation can be generally expressed in the form ax + by + c = 0, where a and b are not both zero. It emphasizes the relationship between the variables and introduces the idea of solutions, which are ordered pairs (x, y). Unlike linear equations in one variable, linear equations in two variables can generate infinitely many solutions, as demonstrated through various examples. Additionally, the section provides exercises aimed at transforming equations into the standard form and encourages problem-solving as a method to understand the relationships between variables in real-life contexts.
Example 2 : Write each of the following as an equation in two variables:
(i) \( x = 7 \)
(ii) \( y = 3 \)
(iii) \( 5x = 15 \)
(iv) \( 4y = 20 \)
Solution :
(i) \( x = 7 \) can be written as \( 1 \cdot x + 0 \cdot y = 7 \), \( 0 \cdot x + 1 \cdot y + 7 = 0 \).
(ii) \( y = 3 \) can be written as \( 0 \cdot x + 1 \cdot y - 3 = 0 \).
(iii) \( 5x = 15 \) can be written as \( 5 \cdot x - 15 = 0 \).
(iv) \( 4y = 20 \) can be written as \( 0 \cdot x + 4 \cdot y - 20 = 0 \).
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Understanding Linear Equations
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Let us first recall what you have studied so far. Consider the following equation:
2x + 5 = 0
Its solution, i.e., the root of the equation, is 5/2. This can be represented on the number line as shown below:
While solving an equation, you must always keep the following points in mind:
- The solution of a linear equation is not affected when:
(i) the same number is added to (or subtracted from) both the sides of the equation.
(ii) you multiply or divide both the sides of the equation by the same non-zero number.
Detailed Explanation
In this chunk, we begin with the definition of linear equations. A linear equation is an equation of degree one, which means the highest power of the variable is one. The example provided, 2x + 5 = 0, showcases how to find solutions to such equations. The solution, in this case, is 5/2, which means if you substitute x with 5/2, the equation will hold true. Moreover, the two properties mentioned are crucial for manipulating equations without changing their solutions: adding or subtracting the same value from both sides maintains equality, as does multiplying or dividing both sides by a non-zero number.
Examples & Analogies
Imagine a balance scale in a grocery store. If you add or remove the same weight from both sides of the scale (letβs say, adding 2 kg of apples to both sides), the scale remains balanced. Similarly, in a linear equation, if you perform equivalent operations on both sides, you maintain the equality β just like the balanced scale.
Forming Linear Equations in Two Variables
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Let us now consider the following situation:
In a One-day International Cricket match between India and Sri Lanka played in Nagpur, two Indian batsmen together scored 176 runs. Express this information in the form of an equation.
Here, you can see that the score of neither of them is known, i.e., there are two unknown quantities. Let us use x and y to denote them. So, the number of runs scored by one of the batsmen is x, and the number of runs scored by the other is y. We know that
x + y =176, which is the required equation.
Detailed Explanation
This chunk illustrates how to create a linear equation with two variables based on a real-life situation. In the example, two cricket players' combined scores provide a scenario where each player's score is unknown. By letting x represent one player's score and y represent the other, we can form the equation x + y = 176. This demonstrates the essence of linear equations in two variables: they represent relationships where two unknown quantities can be solved together.
Examples & Analogies
Think of this like a mystery where you have two friends who both brought sandwiches to a picnic, but you only know that together, they brought 12 sandwiches. If you label the number of sandwiches one friend brought as 'x' and the other as 'y', then you can express this as x + y = 12. This equation helps you figure out how many sandwiches each friend could have brought as long as their total equals 12.
Identifying Linear Equations Form
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So, any equation which can be put in the form ax + by + c = 0, where a, b and c are real numbers, and a and b are not both zero, is called a linear equation in two variables. This means that you can think of many such equations. Some examples of linear equations in two variables are:
- 2s + 3t = 5
- p + 4q = 7
- Οu + 5v = 9
- 3 = 2x β 7y.
Detailed Explanation
In this chunk, the standard form of a linear equation in two variables is introduced: ax + by + c = 0. Here, 'a', 'b', and 'c' can be any real numbers, but it is necessary that 'a' and 'b' are not both zero; otherwise, the equation would not be linear. This point emphasizes the versatility and applicability of linear equations, including various forms, highlighted through example equations that fit this standard form.
Examples & Analogies
Consider a school where students can choose between two activities: sports and arts. If we define 'x' as the number of students in sports and 'y' as those in arts, we can express different scenarios as equations. For example, if we know that 60 students choose sports and arts combined, we can write an equation like x + y = 60. Each combination of students in sports and arts gives us a valid linear equation, allowing us to analyze the preferences of students.
Rearranging Equations to Standard Form
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Note that you can put these equations in the form ax + by + c = 0, where you rearrange. For example:
- 2x + 3y = 4 can be written as 2x + 3y - 4 = 0.
- x β 4 = 3y can be written as x - 3y - 4 = 0.
Detailed Explanation
Routinely, equations need to be rearranged into a standard format to easily identify and solve for variables. This chunk provides instructions on transforming equations by moving all terms to one side. When we write equations in the form ax + by + c = 0, it helps us see their structure better and analyze them. This transformation is crucial for visualization techniques such as graphing.
Examples & Analogies
Imagine decluttering a room. You want all your toys (like x) and books (like y) on one side, and once you clear the space, you can see how many of each you have. Similarly, by rearranging the parts of an equation into a standard form, we clean up the mathematical space to make solving easier, just like organizing a messy room.
Examples of Rearranging Equations
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Example 1: Write each of the following equations in the form ax + by + c = 0 and indicate the values of a, b and c in each case:
(i) 2x + 3y = 4.37
(ii) x β 4 = 3y
(iii) 4 = 5x β 3y
(iv) 2x = y.
Detailed Explanation
This example gives students a practical task of rewriting various equations in standard form and identifying the coefficients a, b, and c. Each transformation requires careful handling of the equation to ensure that equality is maintained. For instance, taking 2x + 3y = 4.37, you subtract 4.37 from both sides, resulting in the standard form, clearly illustrating the components of the equation and their corresponding coefficients.
Examples & Analogies
Think of converting a recipe to suit your kitchen. If the recipe calls for three teaspoons of sugar and you know that it measures out in cups instead, you have to convert those measurements while ensuring the recipe remains unchanged. Rearranging equations is akin to adapting those measurements while keeping the outcome (or equality) consistent.
Linear Equations from Simple Statements
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Equations of the type ax + b = 0 are also examples of linear equations in two variables because they can be expressed as ax + 0y + b = 0. For example, 4 β 3x = 0 can be written as -3x + 0y + 4 = 0.
Detailed Explanation
This section reinforces that even simple equations can fit into the form of linear equations in two variables. The transformation allows students to consider equations like ax + b = 0 as linear equations by introducing an additional variable. This broadens the understanding of what qualifies as a linear equation, emphasizing the flexibility of the format.
Examples & Analogies
Imagine if you were solving for your monthly allowance based on chores done. If your allowance is ever so slightly different, such as 'no chores means no allowance,' you could write it down as 0 chores help you save, allowing for flexibility in inputs. This example illustrates how adding a variable may change the context but keeps the linear relationship intact.
Writing Equations from Known Values
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Example 2: Write each of the following as an equation in two variables:
(i) x = β5
(ii) y = 2
(iii) 2x = 3
(iv) 5y = 2.
Detailed Explanation
This chunk asks students to write given equations in the format of linear equations with two variables. It's crucial understanding since it helps establish that any linear relationship can be framed in the two-variable context, which accommodates various scenarios. The tasks require students to think critically about the relationships between variables even when they start in very simplified forms.
Examples & Analogies
Consider a store where you know how many shirts (x) are available and how many each cost (y). If you have 'x' shirts and the cost is 'y', writing this relationship might look simple, but expressing it in terms with two variables allows you to explore different pricing strategies together.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Linear Equation: A fundamental mathematical expression involving variables that can be represented graphically as a straight line.
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Infinitely Many Solutions: Each linear equation in two variables has an infinite set of solutions, depicted as points on a line in a Cartesian plane.
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Standard Form: The standard form of a linear equation, ax + by + c = 0, helps identify the coefficients and constant clearly.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example 1: To convert 2x + 3y = 4.37 into standard form, write it as 2x + 3y - 4.37 = 0.
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Example 2: If given x = 5 as a linear equation, it can be expressed in standard form as x + 0. y - 5 = 0.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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When x and y are combined, a line you will surely find.
π Fascinating Stories
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Imagine a river, x and y are its banks. They flow together forming a linear path, with points that connect them.
π§ Other Memory Gems
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A: Always, B: Bring, C: Cookies - A = coefficients of x, B = coefficients of y, C = constant term.
π― Super Acronyms
LINEAR
- L: = Line
- I: = In
- N: = Normal
- E: = Equation
- A: = All
- R: = Relations.
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Linear Equation
Definition:
An equation of the form ax + by + c = 0, where a and b are real numbers, and not both a and b are zero.
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Term: Variable
Definition:
A symbol (often x or y) representing a number in equations.
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Term: Solution Set
Definition:
The collection of all ordered pairs (x, y) that satisfy a given linear equation.
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Term: Ordered Pair
Definition:
A pair of numbers (x, y) representing the solution to a linear equation.