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A short class with Sarah and her students. The other ways are under More.
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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?
Isn’t 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.
Let’s 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.
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! Let’s 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.
Let’s 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! That’s 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.
A fundamental mathematical expression involving variables that can be represented graphically as a straight line.
Each linear equation in two variables has an infinite set of solutions, depicted as points on a line in a Cartesian plane.
The standard form of a linear equation, ax + by + c = 0, helps identify the coefficients and constant clearly.
When x and y are combined, a line you will surely find.
Imagine a river, x and y are its banks. They flow together forming a linear path, with points that connect them.
A: Always, B: Bring, C: Cookies - A = coefficients of x, B = coefficients of y, C = constant term.
LINEAR: L = Line, I = In, N = Normal, E = Equation, A = All, R = Relations.
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What is a linear equation?
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What does it mean if an equation has infinitely many solutions?
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What are ordered pairs?
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An equation of the form ax + by + c = 0, where a and b are real numbers, and not both a and b are zero.
A symbol (often x or y) representing a number in equations.
The collection of all ordered pairs (x, y) that satisfy a given linear equation.
A pair of numbers (x, y) representing the solution to a linear equation.
This section introduces linear equations in two variables, emphasizing their structure, solutions, and various forms.
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.
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)
(ii)
(iii)
(iv)
Solution : (i) can be written as , .
(ii) can be written as .
(iii) can be written as .
(iv) can be written as .