Linear Equations in One Variable MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012
Objectives In this section we will learn how to: Recognize and combine like terms. Solve linear equations of the form: ax + b = c. Solve absolute value equations of the form: ax + b = c.
Recognizing Like Terms A term in an equation is an expression combining variables and constants using the operations of multiplication and division. A term containing only a real number is called a constant term or just a constant. The numerical factor of a non-constant term is called a numerical coefficient or just the coefficient. If no explicit coefficient is written, it is assumed to be 1. If a negative sign precedes a non-constant term without an explicit numerical coefficient, the coefficient is assumed to be 1. If an equation contains more than one term, like terms have the same variables raised to the same powers.
Combining Like Terms To combine like terms, add or subtract the coefficients and keep the common variable expression. Unlike terms cannot be combined.
Linear Equations: ax + b = c Terminology: An algebraic expression is a combination of variables and numbers using the operations of addition, subtraction, multiplication, and division. An equation is a statement that two algebraic expressions are equal. If an equation contains a variable, any number which makes the equation true when substituted for the variable is called a solution. The set of all numbers making an equation true is called a solution set. Equations with the same solution sets are said to be equivalent. An equation of the form ax + b = c with a 0 is called a linear equation in x.
Solving Linear Equations (1 of 2) Equations are solved by using the following properties of equations. Addition/Subtraction Property: if the same expression is added to (or subtracted from) both sides of an equation, the two equations are equivalent. A = B A + C = B + C A C = B C Multiplication/Division Property: if both sides of an equation are multiplied (or divided) by the same non-zero expression, the two equations are equivalent. A = B AC = BC A C = B C if C 0.
Solving Linear Equations (2 of 2) To solve a linear equation: 1 Simplify each side of the equation by removing grouping symbols and combining like terms. 2 Use the Addition/Subtraction Property to add the opposites of constants or variable expressions so that the variable expressions are on one side of the equation and the constants are on the other. 3 Use the Multiplication/Division Property to multiply both sides of the equation by the reciprocal of the coefficient of the variable, so that the new coefficient is 1. 4 Check your answer by substituting it into the original equation.
Types of Equations Equations can be classified depending on the number of solution they possess. If an equation has a finite number of solutions, it is called a conditional equation. If an equation is solved by every real number in R then the equation is called an identity. If an equation has no solutions (the solution set is the empty set ) the equation is called a contradiction.
Absolute Value Definition For any real number x, x = { x if x 0, x if x < 0. Solving equations involving absolute value. If x = c > 0 then x = c or x = c. If ax + b = c > 0 then ax + b = c or ax + b = c. If a = b then a = b or a = b. If ax + b = cx + d then ax + b = cx + d or ax + b = (cx + d).