91Ó°ÊÓ

An algebraic equation is a mathematical statement that shows the equality of two expressions, typically involving at least one variable. In our exercise, \( \frac{y}{4} + 7 = 0 \) is an example of a simple linear equation where the variable \( y \) is what we want to find. Solving these equations generally involves manipulating the equation to find the value of the variable that makes the equation true. To achieve this, various fundamental operations such as addition, subtraction, multiplication, and division are used in conjunction with properties of equality.

When tackling such problems, it's essential to perform the same operations on both sides of the equation to maintain balance—similar to a scale, where whatever you do on one side, you must do on the other to keep it level. Understanding this principle is crucial as it is at the core of solving algebraic equations effectively.

The primary goal in solving an algebraic equation is to isolate the variable—getting the variable by itself on one side of the equation. This process often involves reversing the operations acting on the variable. If the variable is being multiplied by a number, as in our example, where \( y \) is divided by 4, the reverse operation would be multiplication.

Let’s look closer at the given problem. We start by eliminating the additional number on the variable’s side, which is 7, through subtraction. Next, since \( y \) is divided by 4, we isolate it by doing the opposite operation—multiplying both sides by 4. Once \( y \) stands alone on one side of the equation, we have successfully isolated the variable, resulting in the simple statement \( y = -28 \) where \( y \) is clearly defined. Remembering to 'do unto one side as you do unto the other' while isolating the variable is a cornerstone concept that makes solving equations clearer and more systematic.

A crucial step after solving an equation is to verify the solution to ensure its accuracy; this is called equation verification. You want to check that the value you found for the variable indeed makes the original equation true. To do this, we substitute the solution back into the original equation and simplify.

In our example, substituting \( y = -28 \) into \( \frac{y}{4} + 7 \) and simplifying should result in the other side of the equation, which is 0. Performing the check, we see that \( \frac{-28}{4} + 7 \) indeed simplifies to 0, proving that our solution \( y = -28 \) satisfies the original equation. This confirmation is the final touch that ensures the solution is not only theoretically valid but also practically applicable, giving confidence in the accuracy of the work.