91Ó°ÊÓ

Solving equations is an essential skill in mathematics. It involves finding the value of a variable that makes an equation true. Imagine equations as a balance with two sides. Our goal is to make sure both sides weigh the same, or are equal, by manipulating the elements.
Here’s a quick way to think about it:

• Our task is to perform operations that simplify the equation without altering its equality, like adding, subtracting, multiplying, or dividing both sides by the same number.
• Begin by analyzing the equation to see what needs to be done to isolate the variable.

In the provided example, the equation is:
\(-0.4y - 12 = -20\). We first focus on simplifying it by getting the term with the variable on its own side. Understanding these mechanics will make solving equations feel like second nature.

Isolating a variable means to get the variable by itself on one side of the equation. This is crucial for finding its exact value.
You can think of it as peeling away layers to reveal the core (in this case, the variable we’re solving for). The two most common operations for isolating a variable are addition/subtraction and multiplication/division. Consider our example equation:
First, we handle any constant terms around the variable by adding 12 to both sides:
\[-0.4y + 12 - 12 = -20 + 12\]which simplifies to:
\[-0.4y = -8\].
Now, to isolate \(y\), divide both sides by \(-0.4\):
\[y = \frac{-8}{-0.4} = 20\].This approach allows the equation to stay balanced and provides the value of \(y\) that satisfies the equation.

Just like double-checking your work, equation verification ensures that the solution found is indeed correct. This involves substituting the solution back into the original equation. If both sides of the equation are equal after the substitution, your solution is verified.
In our scenario with \(y = 20\), we substitute it back:

• Original equation: \(-0.4y - 12 = -20\).
• Substitute \(y = 20\): \(-0.4 \times 20 - 12\).

Completing the calculation:

• \(-8 - 12 = -20\).

The two sides balance perfectly, confirming that indeed 20 is the correct solution. This step is very important in problem-solving, as it boosts confidence in your solution accuracy.