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304 Chapter 3 Systems of Linear Equations and Inequalities Solving Systems of Linear Equations by the Substitution Method Examples Example 1 2y 6x 14 Isolate a variable. 2x y 5 y 2x 5 212x 52 6x 14 4x 10 6x 14 Now solve for y. 2x 10 14 2x 4 y 2x 5 y 2122 5 The ordered pair (2, 1) checks in both equations. The solution set is . Example 2 4x 212x 32 4x 2y 1 1 4x 4x 6 1 Contradiction. The system is inconsistent. There is no solution, . Example 3 2x 6y 2 213y 12 6y 2 6y 2 6y 2 Identity. The equations are dependent. There are infinitely many solutions. 51x, y2 0 x 3y 16 2 2 x 3y 1 5 6 6 1 y 2x 3 512, 126 y 1 x 2 Key Concepts Substitution Method 1. Isolate one of the variables. 2. Substitute the quantity found in step 1 into the other equation. 3. Solve the resulting equation. 4. Substitute the value from step 3 back into the equation from step 1 to solve for the remaining variable. 5. Check the ordered pair in both equations, and write the answer as an ordered pair within set notation. A system is consistent if there is at least one solution. A system is inconsistent if there is no solution. An inconsistent system is detected by a contradiction (such as 0 52. Two linear equations are independent if the equations represent different lines. The equations are dependent if they represent the same line. This produces infinitely many solutions. Two equations in a system of equations are dependent if the system reduces to an identity (such as 0 0). Substitute v Section 3.2


miller_intermediate_algebra_4e_ch1_3
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