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8th Grade Chapter 8 – Solving Systems of Equations 8.3 By “The Addition Method”: Page 367 8.3 Solve by the “Addition Method”: This technique for solving a system of equations is useful when they are written in the Standard Form of Ax + By = C. x + y = 5 and x–y=1 The Addition Property of Equality. We can add the same number to either side and the Equations are still “balanced” or equal. X + y = 5 +x – y = 1 2x + 0y = 6 2x = 6 x=3 Substitute back into either of the equations: Check in the 2nd equation: x – y = 1: Try This: page 367 a) x + y = 5 and 2x - y = 4 x + y = 5: 3–2=1 3+y=5 1 = 1 Valid ∴y=2 b) 3x - 3y = 6 and 3x + 3y = 0 The Multiplication Property of Equality is we can multiply each side of an equation by the same number and still have a balanced or equal equation. 2x + 3y = 8 and x + 3y = 7 In the above example, no variables will be eliminated by just adding the terms, BUT, if we multiple one of the equations by (-1) we create a Linear Combination and additive inverse. 2x + 3y = 8 2x + 3y = 8 and (-1)(x + 3y) = (-1)(7) + -x – 3y = -7 x + 0y = 1 Substitute back into the equation: x + 3y = 7: 1 + 3y = 7: 3y = 6 : y=2 Check in the other equation: 2x + 3y = 8: 2(1) + 3(2) = 8: 2+6=8 Try This: page 368: c) 5x + 3y =17 and d) 8x + 11y = 37 and 5x – 2y = -3 -2x + 11y = 7 Sometimes you can find a Multiplicative Inverse to eliminate a term: 3x + 6y = -6 and 5x – 2y = 14 The “Y” term of -2 can be multiplied by 3. So: 3(5x – 2y) = (3)(14) 15x – 6y = 42 3x + 6y = -6 + 15x – 6y = 42 18x + 0y = 36 18x = 36 x=2 8th Grade Chapter 8 – Solving Systems of Equations 8.3 By “The Addition Method”: Page 367 Try This page: 369 e) 4a + 7b = 11 f) 7x – 5y = 76 g) 5b + 10c – 15 and and and 4a + 6b = 10 4x + y = 55 3b – 2c = -7 The Multiplication Property can be used multiple times to FORCE the elimination of a term by creating the additive inverse: 3x + 5y = 30 and 5x + 8y = 49 5(3x + 5y) = (5)(30) 15x + 25y = 150 and and (-3)(5x + 8y) = (-3)(49) -15x – 24y = -147 15x + 25y = 150 + -15x – 24y = -147 0x + y = 3 Substitute back into the equation: 5x + 8y = 49 5(x) + 8(3) = 49 5(x) + 24 = 49 5x = 25 x=5 Try This page 369 h) 5x + 3y = 2 i) 6x + 2y = 4 and and 3x + 5y = -2 10x + 7y = -8 The Multiplication Property can be used eliminated fractional coefficients to create additive inverse terms: 𝟏 𝟏 x + y = 56 and 𝒙 + 𝟒 𝒚 = 𝟏𝟔 𝟑 𝟏 𝟏 12(𝟑 𝒙 + 𝟒 𝒚) = (𝟏𝟐)𝟏𝟔 -4(x _+ y) = (-4)(56) so…… 4x + 3y = 192 - 4x – 4y = - 224 0x – y = -32 y = 32 8th Grade Chapter 8 – Solving Systems of Equations 8.3 By “The Addition Method”: Page 367 Exercises 8-3 page 371 1) x + y = 10 and x–y=8 2) x – y = 7 and x+y=3 3) x + y = 8 and – x + 2y = 7 4) x + y = 6 and –x + 3y = -2 5)3x – y = 9 and 2x + y = 6 6) 4x – y = 1 and 3x + y = 13 7) 4a + 3b = 7 and -4a + b = 5 8) 7c + 5d = 18 and c – 5d = -2 9) 8x – 5y = -9 and 3x + 5y = -2 10) 3a – 3b = -15 and -3a – 3b = -3 11) 4x – 5y = 7 and -4x + 5y = 7 12) 2x + 3y = 4 and -2x – 3y = -4 13) –x – y =8 and 2x – y = -1 14) x + y = -7 and 3x + y = -9 15) x + 3y =19 and x – y = -1 16) 3x – y =8 and x + 2y =5 17) x + y = 5 5x – 3y = 17 18) x – y = 7 4x – 5y -25 19) 2w + 3z =17 and 3w + 4z = 24 20) 7p + 5q =2 and 8p +-9q =17 21) 2a + 3b = -1 and 3a + 5b = -2 22) 3x – 4y =16 and 5x + 6y = 14 23) x – 3y = 0 and 5x – y = -14 24) 5a – 2b =0 and 2a – 3b = -11 25) 3x – 2y = 10 and 5x + 3y = 4 26) 2p + 5q = 9 and 3p – 2q = 4 27) 3x – 8y = 11 and x + 6y -8 = 0 28) m – n = 32 and 3m = 8n – 6 = 0 29) a + b =12 and ½ a + ¼b = 4 30) 2p – q = 8 and 1/3p + 1/4q = 3 and and Translate: 31) The sum of two numbers is 115 and the difference is 21. 32) The sum of two numbers is 26.4. One is five times the other. 33) The sum of the length and width of a rectangle is 19 inches. The length is one less than twice the width. 34) The perimeter of a rectangle is 48m. The width of the rectangle is 2 more than half the length 35) Two angles are complementary. Their difference is 34o. 36) Two angles are complementary. One angle is 420 more than ½ the other. 38) 3(x – y) = 9 and x+y=7 39) 5(a – b) = 10 and a + b = 2 40) 2(x – y) = 3 + x and x = 3y + 4 41) 2(5a – 5b) =10 and -5(6a + 2b) =10 42) 1.5x + .85y = 1637.5 and .01(x + y) = 15.25 44) y = ax + b and y = x + c 45) ax + by + c = 0 and ax + cy + b = 0 46) 2(7-a) – 2(1 + 2b) + 5 = 0 and 2a + 2b -18 = 0 47) 2/x– 3/y = - ½ and 1/x + 2/y = 11/12

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