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5n(2n+1) кратно, а 12 нет.№1091.(10n + 5)2 = 100n(n + 1) + 25. Чтобы возвести в квадрат натуральное число,оканчивающееся цифрой 5, нужно умножить количество n десятков данного числа на 100(n + 1) и прибавить 25.(10n+ 5)2 = 100n2 + 100n + 25 = 100n(n + 1) + 25,252 = 100 ⋅ 2(2 + 1) + 25 = 625, 452 = 100 ⋅ 4(4 + 1) + 25 = 2025,752 = 100 ⋅ 7(7 + 1) + 25 = 5625, 1152 = 100 ⋅ 11(11 + 1) + 25 = 13225.Доказали.139Глава VI. Системы линейных уравнений§ 15.
Линейные уравнения с двумя переменнымии их системы39. Линейное уравнение с двумя переменными№1092.Линейные: а), в).№1093.Линейные: а), в), г).№1094.x+ y = 6.52 52При x = 1 , y = 4 , 1 + 4 = 6 эти числа — корни уравнения.77 77№1095.а) 2x + y = –5 (–1; –3); (0; –5); (–4; –3) — корни;б) x + 3y = –5 (–5; 0); (4; –3) — корни.№1096.3x + y = 10 (3; 1); (0; 10); (2; 4) — корни.№1097.10x + y = 12 (0,1; 11); (1; 2) — корни.№1098.а) x = 2; y = 4,5, y – x = 2,5; б) x = –1; y = –2, x + y = –3.№1099.4x – 3y = 12;13а) 4x – 3y = 12, –3y = 12 – 4x, y = – 4 + 1 x;б) 4x – 3y = 12, 4x = 12 + 3y, x = 3 +0,75y.№1100.2u + v = 4;а) 2u + v = 4, v = 4 – 2u; б) 2u + v = 4, 2u = 4 – v, u = 2 –0,5v.№1101.а) 6x – y = 12, –y = 12 – 6x, y = 6x – 12;б) 10x + 7y = 0, 10x = –7y, x = –0,7y.№1102.а) x + y = 27, y = 27 – x, (0; 27); (1; 26); (2; 25).б) 2x – y= 4,5, y = 2x – 4,5, (0; –4,5); (4; 3,5); (8; 11,5).в) 3x + 2y = 12, 2y = 12 – 3x, y = 6 – 1,5x, (0; 6); (3; 1,5); (6; –3);г) 5y – 2x = 1, 5y = 1 + 2x, 4y = 0,2 + 0,4x, (0; 0,2); (3; 1,4); (6; 2,6).№1103.x – 6y = 4, x = 4 + 6y, (4; 0); (10; 1); (34; 5).№1104.а) 3x – y = 10, y = 3x – 10, (0; –10); (1; 7); (2; –4);б) 6x + 2y = 7, 2y = 7 – 6x, y = 3,5 – 3x, (0; 3,5); (2; –3,5); (1; 0,5).140№1105.x + 2x = 18, 3x = 18, x = 6.Ответ: (3; 3).№1106.Подставим x и y в уравнение: a⋅2 + 2 ⋅ 1 = 8, 2a + 2 = 8, 2a = 6, a = 3.Т.е.
3x + 2y = 8.№1107.а) 2c(c – 4)2 – c2(2c – 10) = 2c(c2 – 8c + 16) – 2c3 + 10c2 == 2c3 – 16c2 + 32c – 2c3 + 10c2 = –6c2 + 32c = 2c(–3c + 16);при c = 0,2, 2c(–3c + 16) = 2 ⋅ 0,2(–3 ⋅ 0,2 + 16) = 0,4 ⋅ 15,4 = 6,16б) (a – 4b)(4b + a) = a2 – 16b2;при a = 1,2; b = –0,6, a2 – 16b2 = 1,44 – 16 ⋅ 0,36 = 1,44 – 5,76 = –4,32.№1108.а) 1+a–a2 – a3 = (1 + a) – a2(1 + a) = (1 – a2)(1 + a)=(1–a)(1 + a)(1 + a);б) 8–b3+4b–2b2=(2–b)(4+2b + b2) + 2b(2 – b)=(2–b)(4 + 2b + b2 + 2b) == (2 – b)(4 + 4b + b2) = (2 – b)(2 + b)2 = (2 – b)(2 + b)(2 + b).40. График линейного уравнения с двумя переменными№1109.3x + 4y = 12;а) A(4; 1), т.к. 3 ⋅ 4 + 4 ⋅ 1 = 12 + 4 = 16 ≠ 12 , то A ∉ графику;б) B(1; 3), т.к.
3 ⋅ 1 + 4 ⋅ 3 = 3 + 12 = 15 ≠ 12 , то B ∉ графику;в) C(–6; –7,5), т.к. 3 ⋅ (–6) + 4 ⋅ (–7,5) = –48 ≠ 12 , то C ∉ графику;г) D(0; 3), т.к. 3 ⋅ 0 + 4 ⋅ 3 = 12, то D ∈ графику.№1110.x – 2y = 4;а) A(6; 1), т.к. 6 – 2 ⋅ 1 = 4, то A ∈ графику;б) B(–6; –5), т.к. –6 – 2 ⋅ (–5) = 4 , то B ∈ графику;в) C(0; –3), т.к.
0 – 2 ⋅ (–2) = 4, то C ∈ графику;г) D(–1; 3), т.к. –1 – 2 ⋅ 3 = –7 ≠ 4, то D ∉ графику.№1111.3x – y = –5, т.к. 3 ⋅ (–1) – 2 = –5, то P ∈ графику;–x + 10y = 21, т.к. –(–1) + 10 ⋅ 2 = 21, то P ∈ графику;11x + 21y = 31, т.к. 11 ⋅ (–1) + 21 ⋅ 2 = 31, то P ∈ графику.№1112.yа) 2x – y = 6y = 2x – 6y = 2x –6x03y–6003x–6141б) 1,5x + 2y = 3, y = –0,75x + 1,5;xy01,5yв) x + 6y = 0, 6y = –x, y = –20xy6–1y31x +142y=–001x;61,5602xг) 0,5y – x = –1, 0,5y = x – 1,y = 2x – 2;xy0–21y=– x6xд)1,2x = –4,8, x = –4;10yy0x–4y = 2x – 2x = –4е) 1,5 y = 6, y = 4.y401420–6y=4x0x1113.а) x + y = 5, y = 5 – x;xy05б) y – 4x = 0; y = 4x;50xyy0014y54110 1в) 1,6x = 4,8;0 1x5x = 3;xг) 0,5y = 1,5; y = 3.yy31130 1x0 1№1114.а) x – y – 1 = 0; y = x – 1;xy0–1б) 3x = y + 4; y = 3x – 4;10xy0–44/30yy110 1–1xx0 1 4x3–4143в) 2(x – y) + 3y = 4;2x – 2y + 3y = 4; y = 4 – 2x;x02y40г) (x + y) – (x – y) = 4;2y = 4; y = 2.yy441120 1x№1115.0 1x№1116.yy30–7,4416210x11x–8,8Ответ: y = –8,8.–20Ответ: y = –7,4.№1117.а) 12x – 8y = 25: y = 1,5x – 3 ⋅ 125.Так как k = 1,5 > 0, то график проходит в I и III координатных четвертях.Т.к.
при x = 0 y = –3 ⋅ 125 < 0, то график проходит в IV четверти.Ответ: I, III, IV.144б) 1,5y = 150: Так как график параллелен оси Оx и y > 0 для всех x, то график проходит в I и II координатных четвертях.Ответ: I, II.в) 0,2x = 43: Так как график параллелен оси Оy и x > 0 для всех y, то графикпроходит в I и IV координатных четвертях.Ответ: I, IV.№1118.16 − x 18 − xа)−= 0 , 3(16 – x) – 2(18 – x) = 0, 48–3x – 36 + 2x = 0, x = 12;812x − 15 2 x + 1б)−+ 1 = 0 , 4(x – 15) – (2x + 1) + 8 = 0,284x – 60 – 2x + 1 – 8 = 0, 2x = 67, x = 33,5.№1119.а) a(a – 4) – (a + 4)2 = a2 – 4a –a2 – 8a – 16 = –12a – 16,1⎛ 1⎞при a = – 1 : –12a – 16 = –12 ⎜ −1 ⎟ – 16 = 15 – 16 = –1;4⎝ 4⎠б) (2a – 5)2 –4(a – 1)(3 + a) = 4a2 – 20a + 25 – 4(a2 + 3a – a– 3) == 4a2 – 20a + 25 – 4a2 –8a + 12 = –28a + 37,11−7 + 3 ⋅ 37 104при a = : –28a + 37 = –28 + 37 ==.12123341.
Системы линейных уравнений с двумя переменными№1120.3 +1 = 44=4не решение;б)а)2⋅3 –1= 25≠2№1121.3 ⋅ 3 + ( – 1) = 88=8а)решение;б)7 ⋅ 3 − 2 ⋅ ( − 1) = 23 23 = 23№1122.x = y−7−3 = 4 − 7−3 = − 3а):3x + 4 y = 0 3 ⋅ ( − 3) + 4 ⋅ 4 = 0 7 ≠ 0{{{{{{{{22 ⋅+22−=24= 2 {42 == 42решение.{3−1++22⋅ (⋅ 3−1)= 5=1 {15==15решение.(–3; 4) не решение{3−2⋅ (=−−2)6 +− 74 ⋅ ( − 6) = 0 {−−230≠≠−013 (–2; –6) не решение{3−4⋅ (=−34)−+7 4 ⋅ 3 = 0 {0−4==0−4 (–4; 3) решение3x − y = 03 ⋅ ( − 3) − 4 = 0 −13 ≠ 0б) {(–3; 4) не решение:5 x − y = −4 {5 ⋅ ( − 3) − 4 = −4 {−19 ≠ −4{35 ⋅⋅ (( −− 2)2) −− (( −− 6)6) == 0−4 {0−4==0 −4 (–2; –6) решение{35 ⋅⋅ (( −− 4)4) −− 33 == 0−4 {−−1523 ≠≠ −04 (–4; 3) не решение145№1123.x − 4y = 0x+ y=3а)б)x+ y=5x − 2 y = −6№1124.⎧⎪ y = x − 1x − y =11⇒ ⎨а)x + 3y = 9y = 3− x⎪⎩3{{{1y = 3− x3y=x–1xy0–110xy0390y3290–11Ответ: (3; 2).x + 2y = 4y = 2 − 0 ,5 xб)⇒−2 x + 5 y = 10y = 2 + 0,4 xу = 2 – 0,5х у = 2 + 0,4хx04xy20y{x3{02–50y2–5Ответ: (2; 2)146104xв){−x3+xy+=40y = 14 {yy == 0−,x75x + 3,5у = –хxyу = 0,75х + 3,5010–1xy03,5–14/30y3,52–14/3–20Ответ: (– 2; 2)3x − 2 y = 6y = 1,5 x − 3г)3x + 10 y = −12y = −1, 2 − 0,3 xу = 1,5х – 3 у = –1,2 – 0,3хx02xy–30y{x1{0–1/2–40y–4–1,21102x–1,5–3Ответ: (1; –1,5)147№1125.x − 2y = 6y = 0 ,5 x − 3а)3 x + 2 y = −6y = −3 − 1,5 xу = 0,5х – 3 у = –3 – 1,5хx06xy–30y{{0–3–20y1–206x–3Ответ: (0; 3)б){2x x−+y3=y0= −5у=хxy⎧⎪ y = x⎨y = − 5 − 2 x⎪⎩3 35 2y=− − x3 301x01y0–5/3–2,50y1–1–2,50–5/3Ответ: (–1; –1)148–1x№1126.а){34yy +− xx == 12−3⎧⎪ y = 3 + 0 ,25⎨ y = −1 − x⎪⎩31, то графики пересекаются, а, значит, решение одно.3⎧ y = 3xy − 3x = 0 ⎪1б)3y − x = 6 ⎨ y = 2 + x⎪⎩31Т.к.
3 ≠ , то графики пересекаются, а, значит, решение одно.32⎧⎪1,5 x = 1x=в)3−3 x + 2 y = −2 ⎨⎩⎪ y = −2 + 1,5 xГрафики этих функций пересекаются, а, значит, решение одно.x + 2 y = 3 y = 1,5 − 0,5 xг)y = −0 ,5 x y = −0 ,5 xТ.к. –0,5 ≠ –0,5; 1,5 ≠ 0, то графики пересекаются, а, значит, решений нет.2 x = 11 − 3 y 2 x = 11 − 3 yд)6 y = 22 − 4 x 2 x = 11 − 3 yТ.к.
два эти уравнения совпадают, то решений бесконечно много.− x + 2 y = 8 y = 4 + 0 ,5 xе)x + 4 y = 10 y = 2,5 − 0 , 25 xТ.к. 0,5≠–0,25, то графики этих функций пересекаются, а, значит, решение одно.№1127.x = 6y −1x = 6y −1а)2 x − 10 y = 3 x = 5 y + 1,5Т.к. 6 ≠ 5, то графики этих функций пересекаются, а, значит, решение одно.5x + y = 4y = 4 − 5xб)x+ y−6=0 y =6− xТ.к. –5≠–1, то графики этих функций пересекаются, а, значит, решение одно.12 x − 3 y = 524 x − 6 y = 10в)6 y − 24 x = −10 −6 y + 24 x = 10Т.к.
два эти уравнения совпадают, то решений бесконечно много.№1128.x − 3y = 5а), x – 3y = 5; (5; 0); (11; 2); (8; 1) — решения системы.3x − 9 y = 15Т.к. 0,25 ≠ –{{{{{{{{{{{{{{{1,5 y + x = −0,5б) {, 2x + 3y = –1;2 x + 3 y = −11⎞⎛⎛ 1 ⎞⎜ 0; − ⎟ ; (1; –1); ⎜ − ; 0 ⎟ — решения системы.3⎝⎠⎝ 2 ⎠149№1129.2x − 3x +1а), 2x – 3 – 12x = 2(x + 1), 2x – 12x – 2x = 2 + 3,− 3x =42512x = –5, x = – ;123x − 1 xб) 6 =− , 6 ⋅ 15 = 5(3x – 1) – 3x, 90 = 15x – 5 – 3x,35951112x = 95, x ==7 .1212№1130.а) (5c2 – c + 8)(2x – 3) – 16 = 10c3 – 15c2 – 2c2 + 6c + 16c – 24 – 16 == 10c3 – 17c2 + 24c – 40;б) 18m3–(3m–4)(6m2+m–2)=18m3–(18m3 + 3m2 – 6m – 24m2 – 4m + 8) == –3m2 + 6m + 24m2 + 4m – 8 = 21m2 + 10m – 8.№1131.а) a3+a2–x2a–x2=a2(a + 1) – x2(a + 1) = (a2 – x2)(a+1)=(a–x)(a+x)(a + 1);б) b3+b2c – 9b–9c=b2(b + c)–9(b + c)=(b2 – 9)(b+c)=(b – 3)(b+3)(b + c).§ 16. Решение систем линейных уравнений42. Способ подстановки№1132.y = x −1а), 5x + 2(x – 1) = 16, 5x + 2x – 2 = 16,5 x + 2 y = 16{18444= 2 , y = 2 −1 = 1 .7777⎛ 4 4⎞Ответ: ⎜ 2 ; 1 ⎟ .⎝ 7 7⎠7x = 18, x =б){3xx=−22−y y− 11 = 0 ,3(2 – y) – 2y – 11 = 0, 5y = –5, y = –1, x = 2 – (–1) = 3.Ответ: (3; –1).№1133.y − 2x = 1а), y = 2x + 1, 6x – (2x + 1) = 7,6x − y = 7{6x – 2x – 1 = 7, 4x = 8, x = 2, тогда y = 2 ⋅ 2 + 1 = 5.Ответ: (2; 5).7 x − 3 y = 13, x = 2y + 5, 7(2y + 5) – 3y = 13,б)x − 2y = 5{14y + 35 – 3y = 13, 11y = –22, y = –2, значит, x = 2 ⋅ (–2) + 5 = 1;Ответ: (1; –2).150в){3xx+−y5=y 6= 2 ,y = 6 – x, 3x – 5(6 – x) = 2, 3x – 30 + 5x = 2,8x = 32, x = 4, y = 6 – 4 = 2;Ответ: (4; 2).4 x − y = 11, y = 4y – 11, 6x – 2(4x – 11) = 13,г)6 x − 2 y = 13{6x – 8x + 22 = 13, 2x = 9, x = 4,5, y = 4 ⋅ 4,5 – 11 = 7.Ответ: (4,5; 7).y − x = 20д), y = x + 20, 2x – 15(x + 20) = –1,2 x − 15 y = −12x – 15x – 300 = –1, 13x = –299, x = –23, y = –23 + 20 = –3.Ответ: (–23; –3).25 − x = −4 y, x = 4y + 25, 3(4y + 25) – 2y = 30,е)3x − 2 y = 30{{12y + 75 – 2y = 30, 10y = –45, y = –4,5, x = 4 ⋅ (–4,5) + 25 = 7.Ответ: (7; –4,5).№1134.2 x + y = 12а), y = 12 – 2x, 7x – 2(12 – 2x) = 31,7 x − 2 y = 31{7x – 24 + 4x = 31, 11x = 55, x = 5, y = 12 – 2 ⋅ 5 = 2.Ответ: (5; 2).y − 2x = 4, y = 2x + 4, 7x – (2x + 4) = 1,б)7x − y = 1{7x – 2x – 4 = 1, 5x = 5, x = 1, y = 2 ⋅ 1 + 4 = 6.Ответ: (1; 6).8y − x = 4, x = 8y – 4, 2(8y – 4) – 21y= 2, 16y – 8 – 21y = 2,в)2 x − 21 y = 2{5y = –10, y = –2, x = 8 ⋅ (–2) – 4= –20.Ответ: (–20; –2).2 x = y + 0 ,5, y = 2x – 0,5, 3x – 5 ⋅ (2x – 0,5) = 13,г)3x − 5 y = 13{3x – 10x + 2,5 = 13, 7x –10,5, x = –1,5, y = 2 ⋅ (–1,5) – 0,5 = –3,5.Ответ: (–1,5; –3,5).№1135.2u + 5v = 0u = −2,5vа),, −8 ⋅ ( −2,5 ) v + 15v = 7 ,−8u + 15v = 7−8u + 15v = 7{{35v = 7 , v = 0,2 , u = –2,5 ⋅ 0,2 = –0,5.Ответ: (–0,5; 0,2).5 p − 3q = 0, p = 0,6q , 3 ⋅ 0,6q + 4q = 2q , 5,8q = 29 , q = 5 , p=3.б)3 p + 4q = 29Ответ: (3; 5).{151в){54uu −+ 33vv == 1425 ,3v = 14 − 4u , 5u + 4u − 14 = 25 , 9u = 39 ,114 16 41u = 4 , u = − − = −1 .33 3 991⎞⎛ 1Ответ: ⎜ 4 ; −1 ⎟ .9⎠⎝ 310 p + 7 q = −2, p = 2,5q + 11 , 25q + 110 + 7 q = −2 ,г)2 p − 22 = 5q32q = −112 , q = –3,5, p = 2,25.Ответ: (2,25; –3,5).№1136.3x + 4 y = 0а), x = 0 ,5 − 1,5 y , 1,5 − 4 ,5 y + 4 y = 0 , 1,5 = 0 ,5 y ,2x + 3y = 1y = 3, x = 0,5 – 1,5 ⋅ 3 = –4.Ответ: (–4; 3).7x + 2 y = 0, 2 y = −7 x , −14 x + 9 x = 10 , −5 x = 10 , x=–2, y=7.б)4 y + 9 x = 10Ответ: (–2; 7).5 x + 6 y = −20, x = 12 ,5 − 4 ,5 y , 5 ⋅ 12 ,5 − 5 ⋅ 4 ,5 y + 6 y = −20 ,в)9 y + 2 x = 25−16 ,5 y = −82 ,5 , y = 5, x = 12,5 – 4,5 ⋅ 5 = –10.Ответ: (–10; 5).3x + 1 = 8 y, 3x = 8 y − 1 , 11y − 8 y + 1 = −11 , 3 y = −12 , y = –4.г)11 y − 3x = −11Ответ: (–11; –4).№1137.7 x + 4 y = 23а), 2 y = 11,5 − 3,5 x , 8 x − 5,75 + 17 ,5 = 19 ,8 x − 10 y = 1925,5 x = 76 ,5 , x = 3, y = 0,5.Ответ: A(3; 0,5).11x − 6 y = 2б), y = 0 ,6 + 1,6 x , 11x − 3,6 − 9 ,6 x = 2 ,−8 x + 5 y = 31,4 x = 5,6 , x = 4, y = 7.Ответ: B(4; 7).№1138.5 x − 4 y = 16а), x = 2y + 6, 5(2y + 6) – 4y = 16,x − 2y = 6{{{{{{{{10y + 30 – 4y = 16, 6y = –14, y = –1⎞⎛ 1Ответ: ⎜1 ; −2 ⎟ .3⎠⎝ 31527171= −2 , x = –2 ⋅ + 6 = 1 .3333б){320x x−−y15= y6 = 100 ,y = 3x – 6, 20x – 15(3x – 6) = 100,4x – 9x + 18 = 20, 5x = –2, x = –0,4, y = 3 ⋅ (–0,4) – 6 = –7,2.Ответ: (–0,4; –7,2).№1139.3(x − 5) − 1 = 6 − 2 x3x − 16 = 6 − 2 x5 x = 22а),,,3(x − y ) − 7 y = −43 x − 3 y − 7 y = −43x − 10 y = −4{{{x = 4,4, 3 ⋅ 4.4 – 10y = –4, 10y = 17,2, y = 1,72.Ответ: (4,4; 1,72).6(x + y ) − y = −16 x + 5 y = −16 x + 5 y = −1б)7(y + 4) − (y + 2) = 0 7 y + 28 − y − 2 = 0 6 y = −26{{{13113625, x =3 .= −4 , 6x – 5 ⋅ = –1, 6x =333391⎞⎛ 5Ответ: ⎜ 3 ; −4 ⎟ .3⎠⎝ 9№1140.2(3 x − 2 y ) + 1 = 7 x6x − 4 y + 1 = 7 xа)12(x + y ) − 15 = 7 x + 12 y 12 x + 12 y − 15 = 7 x + 12 yy=–{x = 3 , y = −0 ,5 .Ответ: (3; –0,5).3(x + y ) − 7 = 12 x + yб)6(y − 2 x) − 1 = −45 x{{{5−xx =−154 y + 1 = 0{36xy +− 312yx−−71==12−45x +x y {33−9xx ++ 62yy ==171y = 4,5x + 3,5, 33x + 6(4,5x + 3,5) = 1, 60x = –20, x = – ,31y = –4,5 ⋅ + 3,5 = 2.3⎛ 1 ⎞Ответ: ⎜ − ; 2 ⎟ .⎝ 3 ⎠в){5(4(xx +− 32yy)) −− 503 ==3−x 33+ 5y {54xx +−1012 yy −− 350==3x−33+ 5y {24xx ++ 1021yy == 850x = 4 – 5y, 4(4 – 5y) + 21y = 50, 16 – 20y + 21y = 50, y = 34,x = 4 – 5 ⋅ 34 = –166.Ответ: (–166; 34).4 x + 1 = 5(x − 3 y ) − 64 x + 1 = 5 x − 15 y − 6− x + 15 y = −7г)3(x + 6 y ) + 4 = 9 y + 193x + 18 y + 4 = 9 y + 193x + 9 y = 15{{x = 15y + 7, (15y + 7) + 3y = 5, 18y = –2, y = –{11⎛ 1⎞, x = –15 ⋅ ⎜ − ⎟ + 7 = 5 .993⎝⎠1⎞⎛ 1Ответ: ⎜ 5 ; − ⎟ .9⎠⎝ 3153№1141.5 y + 8(x − 3 y ) = 7 x − 125 y + 8 x − 24 y = 7 x − 12x − 19 y = −12а)9 x + 3(x − 9 y ) = 11 y + 46 12 x − 27 y = 11 y + 4612 x − 38 y = 46x = 19y – 12, 12(19y – 12) – 38y = 46, 190y = 190, y = 1,x = 19 ⋅ 1 – 12 = 7.Ответ: (7; 1).−2(a − b) + 16 = 3(b + 7)−2a + 2b + 16 = 3b + 21 −2a − b = 4б)6a − (a − 5) = −8 − (b + 1)5a + 5 = − 8 − b − 15a + b = −14{{{{{{b = –5a – 14, –2a – (–5a – 14) = 5, 3a = 5 – 14, a=–3, b = –5 ⋅ (–3) – 14 = 1.Ответ: (–3; 1).№1142.⎧x y⎪ − = −42 x − 3 y = −24а) ⎨ 3 2y = 2x + 8,2 x − y = −8x y⎪ + = −2⎩2 42x – 3(2x + 8) = –24, –4x = 0, x = 0, y = 8.Ответ: (0; 8).⎧a⎪ − 2b = 6a − 12b = 36б) ⎨ 6a = 12b + 36, –6(12b + 36) + b = –74,−6a + b = −74b⎪−3a + = −37⎩2–71b = 142, b = –2, a = 12 ⋅ (–2) + 36 = 12.Ответ: (12; –2).⎧ 2m n+ =1⎪6m + 5n− = 15в) ⎨ 5 3n = 3 – 1,2m, 3m – 35(3 – 1,2m) = 120,m 7n3m − 35n = 120⎪ −=4⎩10 645m = 120 + 105, m = 5, n = 3 – 1,2 ⋅ 5 = –3.Ответ: (5; –3).3y⎧⎪ 7 x − 5 = −435 x − 3 y = −20y = –2,5x – 7,5,г) ⎨2y5 x + 2 y = −15⎪x += −35⎩35x – 3(–2,5x – 7,5) = –20, 42,5x = –42,5, x = –1, y = –2.5 ⋅ (–1) – 7,5 = –5.Ответ: (–1; –5).№1143.⎧y x⎪ − =65 y − 4 x = 120а) ⎨ 4 54x = –3y, 5y – (–3y) = 120,xy4x + 3y = 0⎪ +=0⎩15 128y = 120, y = 15, 4x = –45, x = –10,8.Ответ: (–10,8; 15).⎧ 6x y⎪ + = 2 ,3 18 x + y = 34,5б) ⎨ 5 15y = 34,5 – 18x, 3x – 20(34,5 – 18x) = 36,x 2y3x − 20 y = 36⎪ −= 1, 2⎩10 3363x = 726, x = 2, y = 34,5 – 18 ⋅ 2 = –1,5.Ответ: (2; –1,5).{{{{{{154№1144.а) (2x–3y)2+(2x+3y)2 = 4x2 – 12xy + 9y2 + 4x2 + 12xy + 9y2 = 8x2 + 18y2;б) (2x + 3y)2 – (2x – 3y)2 = 4x2 + 12xy + 9y2 – (4x2 – 12xy + 9y2) == 4x2 + 12xy + 9y2 – 4x2 + 12xy – 9y2 = 24xy;2⎛ x2 xy y 2 ⎞⎛x y⎞22в) 2 ⎜ + ⎟ + (2 x − y )2 = 2 ⎜ ++⎟ + 4x – 4xy + y =4 16 ⎠⎝2 4⎠⎝ 4=x 2 xy y 2111+++ 4x2 – 4xy + y2 = 4 x 2 − 3 xy + 1 y 2 ;4282282⎛ x 2 2 xy y 2 ⎞⎛x y⎞22г) 3 ⎜ + ⎟ − (3x − y )2 = 3 ⎜ ++⎟ – (9x – 6xy + y ) =27 81 ⎠⎝3 9⎠⎝ 9x 2 2 xy y 22226 2++– 9x2 + 6xy – y2 = −8 x 2 + 6 xy −y ;39273927д) (x + 2)3+(x – 2)3 = (x + 2 + x – 2)((x + 2)2 – (x + 2)(x – 2) + (x – 2)2) == 2x(x2 + 4x + 4 – x2 + 4 + x2 – 4x + 4) = 2x(x2 + 8);е) (x + 2)3–(x – 2)3 = (x + 2 – x + 2)((x + 2)2 + (x + 2)(x – 2) + (x – 2)2) == 4(x2 + 4x + 4 + x2 – 4 + x2 – 4x + 4) = 4(3x2 + 4).№1145.а) x5 + 4a2x3 – 4ax4 = x3(x2 – 4ax + 4a2) = x3(x – 2a)2 = x3(x – 2a)(x – 2a);б) 4a6–12a5b+9a4b2=a4(4a2–12ab+9b2)=a4(2a–3b)2=a4(2a–3b)(2a – 3b);=в)2⎛ y2 yc c2 ⎞1 4 1 2 2 1 3⎛ y c⎞⎛ 7 c ⎞⎛ 7 c ⎞y + y c – y c = y2 ⎜ − + ⎟ = y2 ⎜ − ⎟ = y2 ⎜ − ⎟⎜ − ⎟ ;43923493⎝⎠⎝ 2 3 ⎠⎝ 2 3 ⎠⎝⎠24 5⎛4⎞⎛2⎞b + 4b3c + 9bc2 = b ⎜ b4 + 4b2c + 9c 2 ⎟ = b ⎜ b2 + 3c ⎟ =9⎝9⎠⎝3⎠22⎛⎞⎛⎞= b ⎜ b2 + 3c ⎟⎜ b2 + 3c ⎟ ;⎝3⎠⎝ 3⎠г)2⎞y⎟ =⎠221 2 2 ⎛1⎞ ⎛1⎞⎛ 1⎞ ⎛1x – y + ⎜ x + y ⎟ = ⎜ x − y ⎟⎜ x + y ⎟ + ⎜ x +4⎝2⎠ ⎝2⎠⎝ 2⎠ ⎝21111⎛⎞⎛⎞⎛⎞= ⎜ x − y + x + y ⎟⎜ x + y ⎟ = x ⎜ x + y ⎟ ;2⎝2⎠⎝ 2⎠⎝2⎠д)21 2 2 ⎛1⎞ ⎛1⎞⎛ 1⎞ ⎛1⎞x – y – ⎜ x − y ⎟ = ⎜ x − y ⎟⎜ x + y ⎟ − ⎜ x − y ⎟ =4⎝2⎠ ⎝2⎠⎝ 2⎠ ⎝2⎠1111⎛⎞⎛⎞⎛⎞= ⎜ x + y − x + y ⎟⎜ x − y ⎟ = 2 y ⎜ x − y ⎟ .2⎝2⎠⎝ 2⎠⎝2⎠№1146.y = x2 – 4x + 5 = (x2 – 4x + 4) + 1 = (x – 2)2 + 1 > 0,т.е.