91Ó°ÊÓ

Algebraic equations are a fundamental component of mathematics that involve variables, numbers, and operations. They are statements of equality, indicating that the value on one side of the equals sign is the same as the value on the other side. In the case of the exercise we're looking at, the equation is 9x - 14 = -77.

The goal when dealing with algebraic equations is to find the value of unknown variables that make the equation true. Through a series of steps, we manipulate the equation to make the variable the subject of the equation, which is what we mean when we talk about 'solving' it. This process often involves adding, subtracting, multiplying, and dividing both sides of the equation by the same amounts, to maintain the balance and integrity of the original equation.

To 'isolate the variable' means to rearrange the equation so that the variable you're solving for is alone on one side of the equation. Think of the equation as a balance scale: whatever you do to one side, you must do to the other to keep it balanced.

Tip for Isolating the Variable

When you encounter terms added or subtracted from the variable, you can perform the opposite operation to both sides to remove them. In our sample exercise, the variable x is initially not isolated because it is part of the term 9x with 14 subtracted from it. By adding 14 to both sides (9x - 14 + 14 = -77 + 14), we remove the -14 and move closer to isolating x. Continuing this process eventually leaves x on one side of the equal sign, which signifies that the variable has been isolated.

Once we believe we have solved an equation, it's important to verify that our solution is accurate. This process is known as 'equation checking'. It requires substituting the solution back into the original equation to ensure that the left and right sides remain equal.

For our given exercise, after determining that x = -7 is the solution, we check it by replacing x with -7 in the original equation: 9(-7) - 14 = -77. Simplifying the left side gives us -77 = -77, confirming that our solution is correct. This verification step is crucial because it ensures that any mistakes made in the earlier steps are caught before they can lead to misunderstandings or incorrect conclusions in further calculations.