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Mt1nt_07 tran wk bk 7/22/03 2:37 PM 7.2 Page 137 Solving Linear Systems by Substitution Goals p Use substitution to solve a linear system. p Model a real-life situation using a linear system. SOLVING A LINEAR SYSTEM BY SUBSTITUTION Step 1 Solve one of the equations for one of its variables . Step 2 Substitute the expression from Step 1 into the other equation and solve for the other variable . Step 3 Substitute the value from Step 2 into the revised equation from Step 1 and solve. Step 4 Check the solution in each of the original equations. Example 1 Choosing an Equation to Solve Tell which equation you would use to isolate a variable. Explain your reasoning. a. 6x ⫹ y ⫽ ⫺5 2x ⫺ 3y ⫽ 10 Equation 1 Equation 2 b. 5c ⫺ 2d ⫽ 1 c ⫹ 3d ⫽ 0 Equation 1 Equation 2 Solution a. It would be easiest to isolate y in Equation 1 because the variable has a coefficient of 1 . b. It would be easiest to isolate c in Equation 2 because the variable has a coefficient of 1 . Lesson 7.2 • Algebra 1 Notetaking Guide 137 Mt1nt_07 tran wk bk 7/22/03 2:38 PM Page 138 The Substitution Method Example 2 Solve the linear system. 2x 5y 13 Equation 1 x 3y 1 Equation 2 Solution 1. Solve for x in Equation 2 because it is easy to isolate x. x 3y 1 Revised Equation 2 2. Substitute 3y 1 for x in Equation 1 and solve for y. 2x 5y 13 2( 3y 1 ) 5y 13 6y 2 5y 13 11y 2 13 11y 11 y 1 Write Equation 1. Substitute for x. Distribute. Simplify. Add 2 to each side. Solve for y. 3. To find the value of x, substitute 1 for y in the revised Equation 2 and solve for x. x 3y 1 Write revised Equation 2. x 3(1) 1 Substitute for y. x 4 Simplify. 4. Check that ( 4 , 1 ) is a solution by substituting 4 for x and 1 for y in each of the original equations. Equation 1 Equation 2 2x 5y 13 x 3y 1 2( 4 ) 5( 1 ) ⱨ 13 4 3( 1 ) ⱨ 1 8 5 ⱨ 13 4 3 ⱨ 1 13 13 Answer The solution is ( 4 , 1 ). 138 Algebra 1 Notetaking Guide • Chapter 7 1 1 Mt1nt_07 tran wk bk 7/22/03 2:38 PM Page 139 Checkpoint Tell which equation you would use to isolate a variable. Explain your reasoning. 1. x ⫺ 2y ⫽ 0 2. 4x ⫹ 2y ⫽ 10 x ⫺ 8y ⫽ ⫺5 7x ⫺ y ⫽ 12 I would isolate x in the first equation because it is easy to do so and the value is easy to substitute in the other equation. I would isolate y in the second equation because it is easier to do so than to isolate y in the first equation or x in either equation. Use the substitution method to solve the linear system. 3. y ⫽ x ⫺ 1 4. y ⫽ ⫺5x ⫹ 3 x ⫺ 5y ⫽ ⫺15 3x ⫹ 2y ⫽ ⫺8 (5, 4) (2, ⫺7) Lesson 7.2 • Algebra 1 Notetaking Guide 139