91Ó°ÊÓ

Understanding equation manipulation is a fundamental skill in algebra, particularly when solving for a specific variable. When you encounter an equation like 2x + 3y = 7, think of it as a scale in balance. To 'manipulate' is essentially to perform operations that will maintain the balance while transforming the equation into a more useful form. Always remember, whatever operation you perform on one side, you must also perform on the other to keep the equation balanced.

Starting with 2x + 3y = 7, you want to isolate y. The operation chosen here is to subtract 2x from both sides, resulting in 3y = 7 - 2x. Notice that this step moves all terms with x to one side and the terms with y to the other, inching closer to isolating y. This is a smooth transition to the next concept of the exercise, handling functions of x.

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs. When we talk about functions of x, we are referring to equations where x is the input and the output is the result of the equation’s operations applied to x.

In our example, after manipulating the equation to 3y = 7 - 2x, we want to express y as a function of x, which means we want to find a formula that shows y's dependency on x. The process of solving for y, as shown in the steps, helps us create this function. By dividing both sides of the equation by 3, we reformat it to y = (7 - 2x)/3, and we've successfully written y as a function of x, often notated as f(x).

The ability to isolate a variable within an equation is a critical algebraic skill, often required in various fields of mathematics and sciences. When we isolate a variable, we are rearranging the equation to get one variable alone on one side of the equality sign.

The technique involves using addition, subtraction, multiplication, division, and other algebraic operations to achieve this goal. Tailoring our approach to the problem at hand, in 3y = 7 - 2x, we need to isolate y. By dividing each term by the numerical coefficient of y (which is 3 in this case), we perform our final manipulation: (3y)/3 = (7 - 2x)/3. Simplifying this, we end with y = (7 - 2x)/3, thereby isolating y and allowing us to clearly see how changes in x will affect y.