Equations

An equation is a statement that two expressions are equal. Solutions to an equation are the values of the variables in it that make it true. If the equation is true for all values of the variables, then we call it an identity; identities are often discovered by manipulating one expression into another. The solutions of an equation in one variable form a set of numbers; the solutions of an equation in two variables form a set of ordered pairs, which can be graphed in the plane. Equations can be combined into systems to be solved simultaneously. An equation can be solved by successively transforming it into one or more simpler equations. The process is governed by deductions based on the properties of equality. For example, one can add the same constant to both sides without changing the solutions, but squaring both sides might lead to extraneous solutions. Strategic competence in solving includes looking ahead for productive manipulations and anticipating the nature and number of solutions. Some equations have no solutions in a given number system, stimulating the formation of expanded number systems (integers, rational numbers, real numbers and complex numbers). A formula is a type of equation. The same solution techniques used to solve equations can be used to rearrange formulas. For example, the formula for the area of a trapezoid, //A// = ((b1 + b2)/2) //h//, can be solved for //h// using the same deductive process. Inequalities can be solved in much the same way as equations. Many, but not all, of the properties of equality extend to the solution of inequalities. //Connections to Functions, Coordinates, and Modeling.// Equations in two variables may define functions. Asking when two functions have the same value leads to an equation; graphing the two functions allows for the approximate solution of the equation. Equations of lines involve coordinates, and converting verbal descriptions to equations is an essential skill in modeling. Students understand that: > > > > Students can and do: > > > > > >
 * Core Concepts**
 * 1) An equation is a statement that two expressions are equal.
 * 1) The solutions of an equation are the values of the variables that make the resulting numerical statement true.
 * 1) The steps in solving an equation are guided by understanding and justified by logical reasoning.
 * 1) Equations not solvable in one number system may have solutions in a larger number system.
 * Core Skills**
 * 1) Understand a problem and formulate an equation to solve it.
 * 1) Solve equations in one variable using manipulations guided by the rules of arithmetic and the properties of equality.
 * 1) Rearrange formulas to isolate a quantity of interest.
 * 1) Solve systems of equations.
 * 1) Solve linear inequalities in one variable and graph the solution set on a number line.
 * 1) Graph the solution set of a linear inequality in two variables on the coordinate plane.