5.3 Solving Systems of Linear Equations by the Addition Method

Solving Using Addition Method Write both eqns. in standard form (Ax + By = C). Get opposite coefficients for one of the variables. You may need to mult. one or both eqns. by a nonzero number to do this. Add the eqns., vertically. Solve the remaining eqn. Substitute the value for the variable from step 4 into one of the original eqns. and solve for the other variable. Check soln. in BOTH eqns., if necessary.

Ex. Solve by the addition method: x + y = 3 x – y = 5 Done x + y = 3 x – y = 5 3. x + y = 3 4. 2x + 0 = 8 2x = 8 2 2 x = 4 5. x + y = 3 4 + y = 3 sub 4 for x y + 4 – 4 = 3 – 4 y = -1 Soln: {(4, -1)} 6. Check: x + y = 3 x – y = 5 4 + (-1) = 3 4 – (-1) = 3 3 = 3 4 + 1 = 5 5 = 5 add

Ex. Solve by the addition method: x + y = 9 -x + y = -3 Done 2. x + y = 9 -x + y = -3 3. x + y = 9 4. 0 + 2y = 6 2y = 6 2 2 y = 3 5. x + y = 9 x + 3 = 9 sub 3 for y x + 3 – 3 = 9 – 3 x = 6 Soln: {(6, 3)} 6. Check: x + y = 9 -x + y = -3 6 + 3 = 9 -6 + 3 = -3 9 = 9 -3 = -3 add

Ex. Solve by the addition method: -5x + 2y = -6 10x + 7y = 34 Done 2. -5x + 2y = -6  2(-5x + 2y)=2(-6)  -10x + 4y = -12 10x + 7y = 34  10x + 7y = 34  10x + 7y = 34 3. -10x + 4y = -12 10x + 7y = 34 4. 0 + 11y = 22 11y = 22 11 11 y = 2 add

5. 10x + 7y = 34 10x + 7(2) = 34 sub 2 for y 10x + 14 = 34 10x + 14 – 14 = 34 – 14 10x = 20 10 10 x = 2 Soln: {(2, 2)} Check: -5x + 2y = -6 10x + 7y = 34 -5(2) + 2(2) = -6 10(2) + 7(2) = 34 -10 + 4 = -6 20 + 14 = 34 -6 = -6 34 = 34

Ex. Solve by the addition method: 3x + 2y = -1 -7y = -2x – 9 Rewrite 2nd eqn. in standard form (Ax + By = C) -7y = -2x – 9 -7y + 2x = -2x – 9 + 2x 2x – 7y = -9 2. 3x + 2y = -1  7(3x + 2y) =7(-1)  21x + 14y = -7 2x – 7y = -9  2(2x – 7y) = 2(-9)  4x – 14y = -18 3. 21x + 14y = -7 4x – 14y = -18 4. 25x + 0 = -25 25x = -25 25 25 x = -1 add

5. 3x + 2y = -1 3(-1) + 2y = -1 sub -1 for x -3 + 2y = -1 -3 + 2y + 3 = -1 + 3 2y = 2 2 2 y = 1 Soln: {(-1, 1)}

Ex. Solve by the addition method: -2x = 4y + 1 2x + 4y = -1 Rewrite 1st eqn. in standard form (Ax + By = C) -2x = 4y + 1 -2x – 4y = 4y + 1 – 4y -2x – 4y = 1 2x + 4y = -1 3. -2x – 4y = 1 4. 0 + 0 = 0 No variables remain and a TRUE stmt. lines coincide infinite number of solns. dependent eqns. Soln: {(x, y)|2x + 4y = -1} add

Ex. Solve by the addition method: -3x – 6y = 4 3(x + 2y + 7) = 0 Rewrite 2nd eqn. in standard form (Ax + By = C) 3(x + 2y + 7) = 0 3x + 6y + 21 = 0 3x + 6y + 21 – 21 = 0 – 21 3x + 6y = -21 -3x – 6y = 4 3. -3x – 6y = 4 4. 0 + 0 = -17 No variables remain and a FALSE stmt. lines are parallel no solution inconsistent system Answer: no soln. or ø (empty set) add

Groups Page 315 – 316: 27, 41, 59 Groups or class discussion 27 -> answer has fractions 41-> clear fractions first 59-> distribute first