2.3. Reducing Equations to Simpler Form

Example 16: Solve +1=6x+1 x−3 3 6 Why 6? Because it is the smallest multiple (or LCM) of the given denominators.

In this example, we are given an equation with fractions. The equation says that some expression related to 'x' equals another. The fractions have denominators of 3 and 6. To solve the equation more easily, we can find a number called the Least Common Multiple (LCM) for the denominators. Here, the LCM of 3 and 6 is 6. This step will help eliminate the fractions when we multiply through by this number.

Imagine you are trying to find a common time to meet with friends who have different schedules. You need to find a time that works for everyone. Similarly, finding the LCM is like finding that common time, so we can work with the equation more easily.

Solution: Multiplying both sides of the equation by 6, we get 6(6x+1) = 6(x−3).

To eliminate the fractions, we multiplied every term in the equation by 6. This operation keeps the equation balanced because whatever we do to one side, we must do to the other. After this multiplication, the equation will be easier to work with since there won't be any fractions left. The left side becomes 6 times '6x + 1' while the right side becomes 6 times '(x - 3)'.

Think of a balance scale. If you have weights on both sides and you want to add more weight to one side, you must add the same amount to the other side to keep it balanced. In the same way, we multiply both sides by the same number (6 here) to keep the equations balanced.

or 2 (6x + 1) + 6 = x - 3 or 12x + 2 + 6 =x - 3 (opening the brackets).

After multiplying through, we opened the parentheses and simplified the equation further. Opening the brackets means distributing the multiplied factor (in this case, 6) to each term inside the parenthesis. When we did this, we collected like terms to get a simpler representation of our equation.

Imagine unwrapping a present that is enclosed in several boxes. Opening the outer boxes allows you to see the smaller boxes inside – it makes the whole thing simpler to understand. Similarly, opening the brackets helps us simplify the equation and see the terms clearly.

or 12x + 8 = x - 3 or 12x - x + 8 = -3.

After combining like terms on both sides of the equation, we brought all 'x' terms to one side and constant terms to the other side. This process involves arithmetic where we add or subtract terms that are similar. This gives us a clearer view of our equation and moves us closer to finding the solution for 'x'.

Imagine sorting your toys into two piles: one for cars and another for action figures. Combining like terms is like gathering all similar items into one place, making it easier to count or organize them just as we do with 'x' terms and constants in an equation.

or 11x = -3 - 8 or 11x = -11 or x = -1 (required solution).

Finally, we isolate 'x' by performing operations that allow us to get 'x' by itself on one side. This includes dividing both sides by the coefficient of 'x'. Here, we divided by 11 to find that 'x' equals -1. This is the solution to our equation.

Think of trying to find out how many cookies someone has when you already know the total number of cookies in a jar. To find that specific count, you separate the unknown amount from the total. This process of isolating is similar to how we isolate 'x' to find its value.

Check: LHS = +1= +1 = + = = 3 3 3 3 3 3 LHS = RHS. (as required).

After finding the solution, it's essential to verify if it is correct by substituting 'x' back into the original equation. We do this by replacing 'x' with -1 and checking if both sides of the equation are equal. If they are, we confirm that our solution is valid.

Just like checking your work on a math test, verifying your solution ensures that the answer is correct. It’s like a final proof that keeps you confident about your findings.

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