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CHAPTER OUTLINE 4 Linear Equations and Inequalities in Two Variables 2 4.1Solving Systems by Graphing 4.2Solving Systems by the Substitution Method 4.3Solving Systems by the Elimination Method 4.4Applications of Systems of Two Equations 4.5Linear Inequalities in Two Variables

Objectives 4.2Solving Systems by the Substitution Method 3 1Solve a Linear System by Substitution 2Solve a System Containing Fractions or Decimals 3Solve a System by Substitution: Special Cases

4.2Solving Systems by the Substitution Method 1Solve a Linear System by Substitution 4 Another way to solve a system of equations is to use the substitution method. When we use the substitution method, we solve one of the equations for one of the variables in terms of the other. The substitution method is especially good when one of the variables has a coefficient of 1 or –1.

EXAMPLE 1 5 Solution

EXAMPLE 1 6 Solution We write the solution of the system as an ordered pair, (1, 1).

4.2Solving Systems by the Substitution Method 1Solve a Linear System by Substitution 7 Procedure Solving a System by the Substitution Method Step 1: Solve one of the equations for one of the variables. If possible, solve for a variable that has a coefficient of 1 or –1. Step 2: Substitute the expression found in Step 1 into the other equation. The equation you obtain should contain only one variable.

4.2Solving Systems by the Substitution Method 1Solve a Linear System by Substitution 8 Procedure Solving a System by the Substitution Method Step 3: Solve the equation you obtained in Step 2. Step 4: Substitute the value found in Step 3 into either of the equations to obtain the value of the other variable. Step 5: Check the values in each of the original equations, and write the solution as an ordered pair.

EXAMPLE 2 9

2 10 Solution Step 1: Step 2:

EXAMPLE 2 11 Solution Step 3:

EXAMPLE 2 12 Solution Step 4: Step 5: The check is left to the reader. The solution of the system is (-3, -5).

4.2Solving Systems by the Substitution Method 2Solve a System Containing Fractions or Decimals 13 If a system contains an equation with fractions, first multiply the equation by the least common denominator to eliminate the fractions. Likewise, if an equation in the system contains decimals, begin by multiplying the equation by the lowest power of 10 that will eliminate the decimals.

EXAMPLE 3 14

EXAMPLE 3 15 Solution

EXAMPLE 3 16 Solution Step 1: Step 2:

EXAMPLE 3 17 Solution Step 3: Step 4:

EXAMPLE 3 18 Solution Step 5: Check x = 6 and y = 4 in the original equations. The solution of the system is (6, 4).

4.2Solving Systems by the Substitution Method 3Solve a System by Substitution: Special Cases 19 When we solve a system by substitution, how do we know whether the system is inconsistent or dependent?

EXAMPLE 4 20

EXAMPLE 4 21 Solution Step 1: Step 2:

EXAMPLE 4 22 Solution Step 3: Because the variables drop out and we get a false statement, there is no solution to the system. The system is inconsistent, so the solution set is .

EXAMPLE 4 23 Solution Step 3:

EXAMPLE 5 24

EXAMPLE 5 25 Solution Step 1: Step 2:

EXAMPLE 5 26 Solution Step 3: Because the variables drop out and we get a true statement, the system has an infinite number of solutions. The equations are dependent, and the solution set is {(x, y)|x – 3y = 5}.

EXAMPLE 5 27

4.2Solving Systems by the Substitution Method 3Solve a System by Substitution: Special Cases 28 Note When you are solving a system of equations and the variables drop out: 1)If you get a false statement, like 3 = 5, then the system has no solution and is inconsistent. 2)If you get a true statement, like –4 = –4, then the system has an infinite number of solutions. The equations are dependent.