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A short class with Sarah and her students. The other ways are under More.
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Welcome class! Today we'll be discussing linear equations. Can anyone tell me what you think a linear equation is?
I think it's about relationships in math. Like how things change together?
Yeah! Like how cost goes up with more items.
Great observations! A linear equation indeed describes relationships that have a constant rate of change. For example, if you buy more copies of a book, the total price increases linearly!
So it forms a straight line on a graph?
Exactly! When plotted, these equations form straight lines on a coordinate plane. Remember, linear means 'line' in Latin.
What do the 'a', 'b', and 'c' mean in the equations?
Great question! In the form ax + b = 0, 'a' is the coefficient representing the slope, and 'b' is the constant. In the two-variable equation ax + by = c, 'a', 'b', and 'c' are real numbers. Do you want to know more about how these parts work together?
Yes, please!
Let’s look into some examples! Who can give me the general form of a linear equation in one variable?
Is it like ax + b = 0?
That's right! And what about in two variables?
Um, ax + by = c?
Perfect! Let’s compute a basic example. If we have 3x - 7 = 11, what would be our first step?
Add 7 to both sides!
Correct! This gives us 3x = 18, what do we do next?
Divide by 3 to get x = 6?
Exactly! You’re all doing great. Remember to substitute back to check your solution.
Now that we understand linear equations, let’s talk about where we see them in real life. Can anyone think of an example?
What about budgeting?
Excellent! If a phone plan costs $10 plus an extra $0.50 for every minute, how can we write this as a linear equation?
C = 0.5m + 10, right?
Correct, C is the total cost and m is the minutes used. Linear equations can model many situations, like motion and profit calculations.
How do you interpret those equations?
By analyzing what the slope and intercept represent in each context! And remember, the slope indicates how steep that relationship is.
Line upon line, straight and true, ax and by will guide you through.
Once upon a time, a taxi driver charged a base fee and a per-mile rate. His fares kept track on a graph, forming a straight line through all the rides.
For the slope and intercept, remember 'S' for slope and 'I' for intercept when using the equation y = mx + b.
To recall ax + by = c, think 'A,B,C' for 'All Before Change'.
Term
What is a linear equation?
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Term
What does 'a' represent in ax + b = 0?
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Term
What is the y-intercept in y = mx + b?
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An algebraic equation where each term is either a constant or the product of a constant and a single variable.
The rate of change of a line represented in a linear equation, usually denoted as 'm'.
The point where a line crosses the y-axis, represented as 'c' in slope-intercept form.
Linear equations represent relationships with constant rates of change, forming straight lines when graphed.
A linear equation is defined as an equation where each term is either a constant or involves a single variable, resulting in a straight line on a graph. They can be represented in one or two variables, allowing for various methods of solving and interpreting real-world problems.
In mathematics, a linear equation is an algebraic formulation that outlines a relationship where the power of the variables involved is one. It can be expressed in the general form of a single variable equation as ax + b = 0, and for two variables as ax + by = c, where a, b, and c are real numbers. When plotted on a coordinate plane, these equations yield straight lines. Understanding linear equations is crucial in algebra as they represent countless real-world scenarios, from budgeting to scientific calculations. They serve as a foundation for more advanced concepts in algebra and mathematics, enabling students to develop problem-solving skills and appreciation for mathematical relationships.
Example
The equation 2x + y = 4 is a linear equation in two variables.
To find solutions, you can choose a value for one variable and solve for the other. For instance:
If x = 1, then 2(1) + y = 4, which means 2 + y = 4, so y = 2. A solution is (1, 2).
If x = 0, then 2(0) + y = 4, which means 0 + y = 4, so y = 4. A solution is (0, 4).Plotting these points and connecting them with a straight line gives you the graph of the equation.