Answer:
make use of the properties of equality of of inverse functions and operations
Step-by-step explanation:
In algebra, each of the operations and functions we study has an inverse. As a rule, in any "solve for ..." situation, you want to "undo" the operations and functions performed on the variable of interest. You can use the Order of Operations to determine the order those operations are performed, then <em>undo them in reverse order</em>.
<u>Trivial example</u>:
Solve x + 3 = 5 for x. We observe that the variable x has had 3 added to it. We "undo" the addition of 3 by adding its opposite, -3. The properties of equality require that we add this value to both sides of the equation:
x +3 -3 = 5 -3
x = 2 . . . . . after simplification
__
Rearranging also involves combining terms, finding a common denominator, rationalizing denominators, and similar algebraic operations. These sorts of operations are performed on variables and expressions in virtually the same way they are performed on numbers.
<u>Less trivial example</u>:
Suppose we have the equation of a circle, and we want to rearrange it to find an expression for y.
(x -h)^2 +(y -k)^2 = r^2
We observe that y has k subtracted, the result is squared, and an x-term is added. Undoing these in reverse order means we first subtract the x-term, then take the square root, and finally add k.
(y -k)^2 = r^2 -(x -h)^2 . . . . . . we subtracted the x-term from both sides
y -k = √(r^2 -(x -h)^2) . . . . . . and took the square root. Notice we have retained only the positive square root, so the bottom half of the circle will go missing from our final equation.
y = k + √(r^2 -(x -h)^2) . . . . . finally, we added k
This is the formula for a circle rearranged to find y in terms of x. The value of y is that associated with the top half of the circle. If we want the bottom half, we must negate the square root:
y = k - √(r^2 -(x -h)^2) . . . . . . formula rearranged to give the bottom half
Note that the root function is used to "undo" the operation of raising to a power. As we said, each of the operations we study in algebra has its "undo". It is useful to pay attention to the inverse operation associated with any new operation or function you learn about. (trig functions, log functions, exponentiation come to mind)
