احسب الكميات التالية
\( M = [(-15) + 7 - (-13)]-[-8 - (-8) - 15] \)
\( N = [(-7) + (-9) + 7]+[+(31) + 9 - (-7)] \)
\( P = [-(-12) - (-18)]-[14 + (-18) - (-12)] \)
\( Q = [(-28) + (-24)]+[+(-24) + 3 + (-28)] \)
\( A = 28 - (a + 10) - [9 - (b - 15)] \)
\( B = -13 - (b + 15) - a \)
\( C = -b + [18 + (a - 5)][-14 - b] \)
\( D = a - (-b - c + a) \)
\( E = -a - (b + c - a) \)
\( F = -b + [a - (c - b)] \)
\( G = a - b + [a - (b - c)] \)
\( H = 5 + (-17) - [6 - (-11) + x] \)
\( I = y + [(-14) - (-8) + y][-y - (-25)] \)
\( \mathbf{M} = [(-15) + 7 - (-13)]-[-8 - (-8) - 15] \)
\( = [(-15) + 7 + 13]-[-8 + 8 - 15] \)
\( = [-15 + 7 + 13]-[-8 + 8 - 15] \)
\( = [-8 + 13]-[-15] \)
\( = 5-(-15) \)
\( = 5 + 15 = 20 \)
\( \mathbf{N} = [(-7) + (-9) + 7]+[+(31) + 9 - (-7)] \)
\( = [(-7) + (-9) + 7]+[31 + 9 + 7] \)
\( = [-7 - 9 + 7]+[31 + 9 + 7] \)
\( = [-16 + 7]+[40 + 7] \)
\( = (-9)+47 \)
\( = 38 \)
\( \mathbf{P} = [-(-12) - (-18)]-[14 + (-18) - (-12)] \)
\( = [12 + 18]-[14 - 18 + 12] \)
\( = 12 + 18 - 14 + 18 - 12 \)
\( = 14 \)
\( \mathbf{Q} = [(-28) + (-24)]-[+(-24) + 3 + (-28)] \)
\( = [-28 - 24]-[-24 + 3 - 28] \)
\( = - 18 - 24 + 24 - 3 + 28 \)
\( = -21 + 28 = 7 \)
\( \mathbf{A} = 28 - (a + 10) - [9 - (b - 15)] \)
\( = 28 - (a + 10) - [9 - b + 15] \)
\( = 28 - (a + 10) - [-b + 24] \)
\( = 28 - a - 10 - (-b + 24) \)
\( = 28 - a - 10 + b - 24 \)
\( = (28 - 10 - 24) - a + b \)
\( = -6 - a + b \)
\( \mathbf{B} = -13 - (b + 15) - a \)
\( = -13 - b - 15 - a \)
\( = (-13 - 15) - b - a \)
\( = -28 - a - b \)
\( \mathbf{C} = -b + [18 + (a - 5)][-14 - b] \)
\( = -b + [18 + a - 5][-14 - b] \)
\( = -b + [a + 13][-14 - b] \)
\( = -b + (a + 13)(-14 - b) \)
\( = -b + [a(-14 - b) + 13(-14 - b)] \)
\( = -b + [-14a - ab - 182 - 13b] \)
\( = -b - 14a - ab - 182 - 13b \)
\( = -14a - ab - 182 - b - 13b \)
\( = -14a - ab - 182 - 14b \)
\( \mathbf{D} = a - (-b - c + a) \)
\( = a + b + c - a \)
\( = (a - a) + b + c \)
\( = b + c \)
\( \mathbf{E} = -a - (b + c - a) \)
\( = -a - b - c + a \)
\( = (-a + a) - b - c \)
\( = -b - c \)
\( \mathbf{F} = -b + [a - (c - b)] \)
\( = -b + [a - c + b] \)
\( = -b + a - c + b \)
\( = (-b + b) + a - c \)
\( = a - c \)
\( \mathbf{G} = a - b + [a - (b - c)] \)
\( = a - b + [a - b + c] \)
\( = a - b + a - b + c \)
\( = (a + a) - (b + b) + c \)
\( = 2a - 2b + c \)
\( \mathbf{H} = 5 + (-17) - [6 - (-11) + x] \)
\( = 5 - 17 - [6 + 11 + x] \)
\( = 5 - 17 - [17 + x] \)
\( = 5 - 17 - 17 - x \)
\( = (5 - 17 - 17) - x \)
\( = -29 - x \)
\( \mathbf{I} = y + [(-14) - (-8) + y][-y - (-25)] \)
\( = y + [-14 + 8 + y][-y + 25] \)
\( = y + [-6 + y][-y + 25] \)
\( = y + (y - 6)(-y + 25) \)
\( = y + [y(-y + 25) - 6(-y + 25)] \)
\( = y + [-y^2 + 25y + 6y - 150] \)
\( = y + [-y^2 + (25y + 6y) - 150] \)
\( = y + [-y^2 + 31y - 150] \)
\( = y - y^2 + 31y - 150 \)
\( = -y^2 + (y + 31y) - 150 \)
\( = -y^2 + 32y - 150 \)
اجمع أو اطرح الكميات التالية
إذا كان: \( a - b = 39 \)
\( A = (137 - a) - (47 - b) \)
\( B = (a - 69) - (31 + b) \)
\( C = (17 - b) - (15 - a) \)
\( \mathbf{A} = (137 - a) - (47 - b) \)
\( = 137 - a - (47 - b) \)
\( = 137 - a - 47 + b \)
\( = (137 - 47) - a + b \)
\( = 90 - a + b \)
\( \mathbf{B} = (a - 69) - (31 + b) \)
\( = a - 69 - (31 + b) \)
\( = a - 69 - 31 - b \)
\( = a - b - (69 + 31) \)
\( = a - b - 100 \)
\( \mathbf{C} = (17 - b) - (15 - a) \)
\( = 17 - b - (15 - a) \)
\( = 17 - b - 15 + a \)
\( = (17 - 15) + a - b \)
\( = 2 + a - b \)
لكل قيمة من القيم التالية
\( A = [x - 8] - [-x + 13] - [21 + x] \)
احسب قيمة \( A \) في كل حالة
(1) \( x = 8 \)
(2) \( x = 13 \)
(3) \( x = 21 \)
\( \mathbf{A} = [x - 8] - [-x + 13] - [21 + x] \)
\( = x - 8 - (-x + 13) - (21 + x) \)
\( = x - 8 + x - 13 - 21 - x \)
\( = (x + x - x) - 8 - 13 - 21 \)
\( = x - (8 + 13 + 21) \)
\( = x - 42 \)
\( A = 8 - 42 \)
\( = 8 + (- 42)= - 34 \)
\( A = 13 - 42 \)
\( = -29 \)
\( A = 21 - 42 \)
\( = - 21 \)
حل كل معادلة من المعادلات التالية
(1) \( -11 - (-13 - x) = 0 \)
(2) \( 12 - (8 - x) = 20 \)
(3) \( -25 + (4 - x) = -15 \)
(4) \( 8 - (-1 - x) = 4 \)
(5) \( 15 + [-7 - (x + 1)] = -8 \)
(6) \( 3 - [1 - (x - 4)] = 12 \)
(7) \( -12 - (7 - x) = 18 \)
(8) \( 72 - (14 + x) = 20 \)
\( -11 + 13 + x = 0 \)
\( (-11 + 13) + x = 0 \)
\( 2 + x = 0 \)
\( x = -2 \)
\( 12 - 8 + x = 20 \)
\( (12 - 8) + x = 20 \)
\( 4 + x = 20 \)
\( x = 20 - 4 \)
\( x = 16 \)
\( -25 + 4 - x = -15 \)
\( (-25 + 4) - x = -15 \)
\( -21 - x = -15 \)
\( -x = -15 + 21 \)
\( -x = 6 \)
\( x = -6 \)
\( 8 + 1 + x = 4 \)
\( (8 + 1) + x = 4 \)
\( 9 + x = 4 \)
\( x = 4 - 9 \)
\( x = -5 \)
\( 15 + [-7 - x - 1] = -8 \)
\( 15 + [-7 - 1 - x] = -8 \)
\( 15 + [-8 - x] = -8 \)
\( 15 - 8 - x = -8 \)
\( (15 - 8) - x = -8 \)
\( 7 - x = -8 \)
\( -x = -8 - 7 \)
\( -x = -15 \)
\( x = 15 \)
\( 3 - [1 - x + 4] = 12 \)
\( 3 - [1 + 4 - x] = 12 \)
\( 3 - [5 - x] = 12 \)
\( 3 - 5 + x = 12 \)
\( (3 - 5) + x = 12 \)
\( -2 + x = 12 \)
\( x = 12 + 2 \)
\( x = 14 \)
\( -12 - 7 + x = 18 \)
\( (-12 - 7) + x = 18 \)
\( -19 + x = 18 \)
\( x = 18 + 19 \)
\( x = 37 \)
\( 72 - 14 - x = 20 \)
\( (72 - 14) - x = 20 \)
\( 58 - x = 20 \)
\( -x = 20 - 58 \)
\( -x = -38 \)
\( x = 38 \)
اجمع أو اطرح التعابير التالية:
\( A = (a - c) - (a - b) + (b + c) \)
\( B = (a - b) + (a + b) - (a - 3) \)
\( C = -(b + c) - (-b + c) \)
\( D = (a + bc) - (10 - b + c) + (b - a + c) \)
\( E = (-7 + a - b) - (-b + c) + (c - b) \)
\( F = (3 - a) - [-(a + 3) - b] + (b - 8) \)
\( G = [-1 - (1 - a)] + [(b - 9) + (4 - a - b)] \)
\( H = (7 - x + y) - [3 - x + y] + (y - z) \)
\( K = [(x - y - z) - (x + y - z)] - [(x - y + z) - (x + y + z)] \)
\( L = [c - (a - b) - (a - b + c)] + [-(a - b - c) - (a - b + 3)] \)
\( P = 3 - [x - (z - y)] - [-(x + y - 3) + (3 - y)] \)
\( \mathbf{A} = (a - c) - (a - b) + (b + c) \)
\( = a - c - (a - b) + (b + c) \)
\( = a - c - a + b + b + c \)
\( = (a - a) + (-c + c) + (b + b) \)
\( = 2b \)
\( \mathbf{B} = (a - b) + (a + b) - (a - 3) \)
\( = a - b + a + b - (a - 3) \)
\( = a - b + a + b - a + 3 \)
\( = (a + a - a) + (-b + b) + 3 \)
\( = a + 3 \)
\( \mathbf{C} = -(b + c) - (-b + c) \)
\( = -b - c + b - c \)
\( = (-b + b) + (-c - c) \)
\( = -2c \)
\( \mathbf{D} = (a + bc) - (10 - b + c) + (b - a + c) \)
\( = a + bc - (10 - b + c) + (b - a + c) \)
\( = a + bc - 10 + b - c + b - a + c \)
\( = (a - a) + (bc) + (-c + c) + (b + b) - 10 \)
\( = bc + 2b - 10 \)
\( \mathbf{E} = (-7 + a - b) - (-b + c) + (c - b) \)
\( = -7 + a - b + b - c + c - b \)
\( = -7 + a + (-b + b - b) + (-c + c) \)
\( = -7 + a - b \)
\( \mathbf{F} = (3 - a) - [-(a + 3) - b] + (b - 8) \)
\( = 3 - a - [-a - 3 - b] + (b - 8) \)
\( = 3 - a + a + 3 + b + b - 8 \)
\( = (3 + 3 - 8) + (-a + a) + (b + b) \)
\( = -2 + 2b \)
\( \mathbf{G} = [-1 - (1 - a)] + [(b - 9) + (4 - a - b)] \)
\( = [-1 - 1 + a] + [b - 9 + 4 - a - b] \)
\( = [-2 + a] + [-5 - a] \)
\( = -2 + a - 5 - a \)
\( = (-2 - 5) + (a - a) \)
\( = -7 \)
\( \mathbf{H} = (7 - x + y) - [3 - x + y] + (y - z) \)
\( = 7 - x + y - (3 - x + y) + (y - z) \)
\( = 7 - x + y - 3 + x - y + y - z \)
\( = (7 - 3) + (-x + x) + (y - y + y) - z \)
\( = 4 + y - z \)
\( \mathbf{K} = [(x - y - z) - (x + y - z)] - [(x - y + z) - (x + y + z)] \)
\( = [x - y - z - x - y + z] - [x - y + z - x - y - z] \)
\( = [x - x - y - y - z + z] - [x - x - y - y + z - z] \)
\( = [-2y] - [-2y] \)
\( = -2y + 2y \)
\( = 0 \)
\( \mathbf{L} = [c - (a - b) - (a - b + c)] + [-(a - b - c) - (a - b + 3)] \)
\( = [c - a + b - a + b - c] + [-a + b + c - a + b - 3] \)
\( = [-2a + 2b] + [-2a + 2b + c - 3] \)
\( = -2a + 2b - 2a + 2b + c - 3 \)
\( = (-2a - 2a) + (2b + 2b) + c - 3 \)
\( = -4a + 4b + c - 3 \)
\( \mathbf{P} = 3 - [x - (z - y)] - [-(x + y - 3) + (3 - y)] \)
\( = 3 - [x - z + y] - [-x - y + 3 + 3 - y] \)
\( = 3 - [x - z + y] - [-x - 2y + 6] \)
\( = 3 - x + z - y + x + 2y - 6 \)
\( = (3 - 6) + (-x + x) + (-y + 2y) + z \)
\( = -3 + y + z \)
بسّط التعابير التالية
\( A = -5 + [a - 3 - (15 - b)] \)
\( B = (-13 + a) - [2 + (3 - b)] \)
-1 اجمع \( A \) و \( B \)
-2 اطرح \( A \) من \( B \)
-3 اجمع \( A \) و \( B \) إذا كان \( a = -2 \) و \( b = +3 \)
\( \mathbf{A} = -5 + [a - 3 - (15 - b)] \)
\( = -5 + [a - 3 - 15 + b] \)
\( = -5 + [a + b - 3 - 15] \)
\( = -5 + [a + b - 18] \)
\( = -5 + a + b - 18 \)
\( = (-5 - 18) + a + b \)
\( = -23 + a + b \)
\( \mathbf{B} = (-13 + a) - [2 + (3 - b)] \)
\( = -13 + a - [2 + 3 - b] \)
\( = -13 + a - [5 - b] \)
\( = -13 + a - 5 + b \)
\( = (-13 - 5) + a + b \)
\( = -18 + a + b \)
\( A + B = (-23 + a + b) + (-18 + a + b) \)
\( = -23 + a + b - 18 + a + b \)
\( = (-23 - 18) + (a + a) + (b + b) \)
\( = -41 + 2a + 2b \)
\( B - A = (-18 + a + b) - (-23 + a + b) \)
\( = -18 + a + b + 23 - a - b \)
\( = (-18 + 23) + (a - a) + (b - b) \)
\( = 5 \)
\( A = -23 + (-2) + 3 \)
\( = -23 - 2 + 3 \)
\( = (-23 - 2) + 3 \)
\( = -25 + 3 \)
\( = -22 \)
\( B = -18 + (-2) + 3 \)
\( = -18 - 2 + 3 \)
\( = (-18 - 2) + 3 \)
\( = -20 + 3 \)
\( = -17 \)
\( A + B = -22 + (-17) \)
\( = -22 - 17 \)
\( = -39 \)