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Learn it your way
A short class with Sarah and her students. The other ways are under More.
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Today, we will start with the concept of reducing equations to simpler forms. Can anyone tell me why it’s important to simplify equations?
It makes them easier to solve!
Exactly! Simplifying reduces complexity. For instance, consider the equation 6x + 1 = x - 3. Wouldn't it be easier if there were no fractions?
Yes! How do we remove them?
Great question! One method is to find the least common multiple, or LCM, of the denominators. In our case, we would multiply both sides by 6.
Now let's look at how we can apply the LCM. For 6x + 1/3 = x - 3/6, the LCM is 6. So, what happens when we multiply through by 6?
We would do 6(6x + 1/3) = 6(x - 3/6)?
Correct! And what do we get on the left?
It becomes 12x + 2!
Exactly! Now it’s simplified. Now, we can solve for x easily.
After eliminating fractions, what’s the next step?
We should combine like terms!
Correct! For example, let's simplify the equation 12x + 8 = x - 3. What do we get when we move x terms to one side?
11x + 8 = -3!
Exactly! Further simplifications lead us to our solution.
Let's solve an example together. Reduce the equation 5x - 2(2x - 7) = 2(3x - 1). What’s our first step?
First, we open the brackets!
Right! And we’ll also combine like terms. What do we end up with?
We should end with a linear equation!
Exactly! This practice is vital in solving real-world problems.
To wrap up, can someone list the main steps we took today?
Find the LCM and multiply, then combine like terms.
Check our solutions, and make sure LHS equals RHS.
Perfect! Remember, simplifying helps us grasp harder problems too.
To find the LCM, don't delay, multiply the smallest, that’s the way!
Once in math land, LCM was a helpful knight, clearing fractions left and right!
Use ‘Reduce, Multiply, Combine’ to help you remember the steps!
UCMC: Use, Combine, Multiply, Check – for solving equations!
Term
What does LCM stand for?
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Term
What is a linear equation?
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An equation composed of variables and constants, linked by an equality sign.
An equation in which the highest power of the variable is one.
The smallest multiple that is evenly divisible by each of the denominators.
The process of simplifying expressions by adding or subtracting terms with the same variable raised to the same power.
This section explains how to reduce equations, particularly those involving fractions and variables, to simpler forms for easier solution finding.
In this section, we will learn about techniques for simplifying equations, especially those that contain fractions. Strategies like finding the least common multiple (LCM) and rearranging terms will be discussed. Several examples illustrate how to approach these equations systematically.
This section focuses on methods to simplify and solve equations, particularly those containing fractions and multiple terms. Simplifying equations can often involve opening brackets, rearranging terms, and finding a common denominator. The examples provided guide students through these processes, showing how to multiply both sides of an equation by the least common multiple (LCM) of denominators to eliminate fractions, and how to effectively combine like terms for clarity. Through systematic steps, students learn to transform complicated expressions into a linear form that is easier to handle.
Solution: Multiply both sides of the equation by 6,
or
or
or
or
(required solution)
Solution: Multiply both sides of the equation by 12,
or
or
or
(required solution)
LHS =
RHS =
Thus, LHS = RHS = (as required)
Example : Solve
Solution:
Let’s open the brackets.
The equation is
Check:
Therefore, required solution is: