91Ó°ÊÓ

When working with algebra, the skill of algebraic manipulation is crucial. It's the process of reworking equations or expressions into different forms, usually to make them easier to understand or solve. In the context of solving linear equations, manipulation involves various operations such as adding, subtracting, multiplying, or dividing both sides of the equation by the same number.

Let's take our original equation, 4x - 5y = -2. To begin, one might simplify or rearrange terms to group like variables. Here, we want to solve for y, so we'll focus on isolating the y-terms. This often involves moving all terms that contain y to one side and everything else to the other side. In our case, we add 5y to both sides to get rid of the y from the right side. Now, we have 5y on the left side and 4x + 2 on the right. This is a neat example of algebraic manipulation which sets the stage for the step that follows – isolating the variable.

To find the value of a single variable, one must isolate it on one side of the equation. This means we perform operations to leave the variable by itself. In our exercise, once we've algebraically manipulated the equation to 5y = 4x + 2, we proceed to isolate y.

This process often involves division or multiplication, depending on the initial condition of the variable. Since our variable y is multiplied by 5, we'll divide the entire equation by 5. That gives us y = (4/5)x + (2/5). Isolating the variable is an essential step in solving equations as it gives us a clear view of the variable's value in terms of the other quantities in the equation. Remember, whatever operation you perform on one side of the equation must be performed on the other to maintain balance.

The process of equation rewriting is often used in algebra to change the form of an equation to make it easier to solve or to highlight a particular relationship. It is a form of algebraic manipulation that specifically targets the equation's format. Rewriting an equation can include expanding, factoring, combining like terms, or, as seen in our exercise, manipulating it to solve for a variable.

Once we have the equation in the form 5y = 4x + 2 from the algebraic manipulation step, we rewrite it to its simplest form for the variable y: y = (4/5)x + (2/5). This rewritten equation, in terms of x, is the solution to the original problem. It clearly shows the relationship between y and x, and is helpful for understanding how changes in one variable affect the other. Effective equation rewriting allows us to cleanly convert a complex relationship into an easily understandable one.