91Ó°ÊÓ

When we talk about algebraic manipulation, we're discussing the various ways we can transform an algebraic expression or equation to simplify it or to solve it for a particular variable. This process involves applying a series of arithmetic operations and properties of equality—such as adding, subtracting, multiplying, or dividing both sides of the equation by the same value to maintain equality.

For instance, consider the given equation from our exercise, \(5x + 9y = 18\). Here, we used algebraic manipulation by subtracting \(5x\) from both sides. This procedure didn't alter the equation's balance; instead, it rearranged the components, setting the stage to isolate the variable \(y\). As you practice algebra, you'll begin to recognize patterns and understand how to manipulate equations efficiently to reach a solution.

Isolating variables is a fundamental technique in solving algebraic equations. The primary goal is to get the variable of interest, in this case \(y\), by itself on one side of the equation—resulting in something that looks like \(y =\) (some expression), which tells us what \(y\) equals in terms of the other variables or numbers.

To achieve this isolation, we use the principles of algebraic manipulation, like when we subtracted \(5x\) from both sides of our given equation to eventually express \(y\) as a function of \(x\). The key is to perform actions that simplify the equation without changing its solutions, an aspect that requires not just mathematical operations, but a good dose of logical reasoning and problem-solving skills as well.

Solving an equation systematically consists of a sequence of steps that guide you from a complex problem to a simpler one until you find the value of the unknown variable.

The initial step often requires you to carry out algebraic manipulation—like rearranging terms to group like variables or constants together. Next, you might need to perform operations that help in isolating the variable you're solving for. In our example, after subtracting \(5x\) from both sides of \(5x + 9y = 18\), we find that \(9y = 18 - 5x\).

Following this, we divide everything by 9, which is the coefficient of \(y\), simplifying the equation to find that \(y = \frac{{18 - 5x}}{9}\). Notice how each step moves toward making the equation simpler until the variable \(y\) is by itself. It's important to do each step correctly and check your work to ensure accuracy.