Algebra II Day 1
An equation is a mathematical statement that asserts the equality of two expressions.
Equations often express relationships between given quantities, the knowns, and quantities yet to be determined, the unknowns. By convention, unknowns are denoted by letters at the end of the alphabet, x, y, z, w… while knowns are denoted by letters at the beginning, a, b, c, d… The process of expressing the unknowns in terms of the knowns is called solving the equation.
In an equation with a single unknown, a value of that unknown for which the equation is true is called a solution or root of the equation. In a set simultaneous equations, or system of equations, multiple equations are given with multiple unknowns. A solution to the system is an assignment of values to all the unknowns so that all of the equations are true.
One use of equations is in mathematical identities, assertions that are true independent of the values of any variables contained within them. For example, for any given value of x it is true
However, equations can also be correct for only certain values of the variables. In this case, they can be solved to find the values that satisfy the equality. For example, consider the following
The equation is true only for two values of x, the solutions of the equation. In this case, the solutions are x = 0 and x = 1.
If an equation in algebra is known to be true, the following operations may be used to produce another true equation:
- Any real number can be added to both sides.
- Any real number can be subtracted from both sides.
- Any real number can be multiplied to both sides.
- Any non-zero real number can divide both sides.
- Some functions can be applied to both sides. Caution must be exercised to ensure that the operation does not cause missing or extraneoussolutions. For example, the equation y*x=x has 2 solutions: y=1 and x=0. Dividing both sides by x “simplifies” the equation to y=1, but the second solution is lost.