A linear system is consistentwhen it has at least one solution. A linear system is
The coeffi cient of −1 in Equation 3 makes ya convenient variable to eliminate.
x+y5z= −12 Equation 2 4xEquation 3− 2y+z= 10 A solutionof such a system is an ordered triple(x, y, z) whose coordinates make each equation true. The graph of a linear equation in three variables is a plane in three-dimensional space. The graphs of three such equations that form a system are three planes whose intersection determines the number of solutions of the system, as shown in the diagrams below.
To be profi cient in math, you need to look for entry points to the solution of a problem.
Because you obtain a false equation, the original system has no solution.
The algebraic methods you used to solve systems of linear equations in two variables can be extended to solve a system of linear equations in three variables.
An extraneous solution is an apparent solution that must be rejected because it does not satisfy the original equation.
An absolute value equationis an equation that contains an absolute value expression. You can solve these types of equations by solving two related linear equations. Property of Absolute Value
extraneous. An extraneous solutionis an apparent solution that must be rejected because it does not satisfy the original equation.
If the absolute values of two algebraic expressions are equal, then they must either be equal to each other or be opposites of each other.
Always check your solutions in the original equation to make sure they are not extraneous.
To solve an equation, isolate the variable so that it has a coefficient of 1 on one side of the equation. If a number is being added to or subtracted from the variable in the original equation, use the inverse function to isolate the variable. Use the Addition or Subtraction Property of Equality to preserve equality. These properties stress the importance of performing the same operation on each side of the equation to result in an equivalent equation.
You also solve equations in which the variable is multiplied or divided by a rational number by using the inverse operation. The Multiplication and Division Properties of Equality state that you can multiply or divide each side of an equation by the same number and preserve equality.
The Key Vocabulary list introduces students to some of the main vocabulary terms included in this chapter. For a more thorough vocabulary list with pronunciations of new words, give students the Vocabulary Builder worksheets found on pages vii and viii of the Chapter 3 Resource Masters. Encourage them to complete the definition of each term as they progress through the chapter. You may suggest that they add these sheets to their study notebooks for future reference when studying for the Chapter 3 test.
Writing equations from verbal sentences is an essential tool for solving real-world problems. Variables are used to represent unspecified amounts. There are key words to assist in writing the mathematical equa- tions, such as equals, is, times, and, sum, difference, less, more, and so on. Use the Four-Step Problem-Solving Plan to solve problems. Always examine the solution to make sure the answer is reasonable. Translating equa- tions to verbal sentences can help give meaning to the equations.
Some equations contain more than one vari- able. The process for solving one-step or multi-step equations is applied to solve these equations for one of the variables in terms of the other terms. Formulas are written as equations with multiple variables. They can be solved for one of the variables to make computation easier.