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5.3 Solving Systems of Linear Equations by the Addition Method

Learn how to solve systems of linear equations using the addition method. Write the equations in standard form, find opposite coefficients, add the equations vertically, solve for the remaining equation, substitute values, and check solutions.

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5.3 Solving Systems of Linear Equations by the Addition Method

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  1. 5.3 Solving Systems of Linear Equations by the Addition Method

  2. 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.

  3. Ex. Solve by the addition method: x + y = 3 x – y = 5 • Done • x + y = 3 x – y = 5 3. x + y = 3 x – y = 5 4. 2x + 0 = 8 2x = 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

  4. Ex. Solve by the addition method: x + y = 9 -x + y = -3 • Done 2. x + y = 9 -x + y = -3 3. x+ y = 9 -x + y = -3 4. 0 + 2y = 6 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

  5. 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 11y = 22 11 11 y = 2 add

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

  7. 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 25x = -25 25 25 x = -1 add

  8. 5. 3x + 2y = -1 3(-1) + 2y = -1 sub -1 for x -3 + 2y = -1 -3 + 2y + 3 = -1 + 3 2y = 2 2y = 2 2 2 y = 1 Soln: {(-1, 1)}

  9. 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 2x + 4y = -1 3. -2x – 4y = 1 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

  10. 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 3x + 6y = -21 3. -3x – 6y = 4 3x+ 6y = -21 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

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

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