Giải hệ phương trình

1

Sắp xếp lại các số hạng theo dạng chuẩn



x+y=124x8,9+y7=152 lines; Line 1: x plus y equals 124; Line 2: the fraction with numerator x and denominator 8 comma 9 end-fraction plus y over 7 end-fraction equals 15 end-lines;

𝑥+𝑦=124𝑥8,9+𝑦7=15



{x+y=1240,112x+17y=152 cases; Case 1: x positive y equals 124; Case 2: 0 comma 112 x positive 1 over 7 end-fraction y equals 15 end-cases;

𝑥+𝑦=1240,112𝑥+17𝑦=15

2

Phép khử biến 



xx

𝑥



{x+y=1240,112x+17y=152 cases; Case 1: x positive y equals 124; Case 2: 0 comma 112 x positive 1 over 7 end-fraction y equals 15 end-cases;

𝑥+𝑦=1240,112𝑥+17𝑦=15

Nhân (các) phương trình để các hệ số của biến cần khử là nghịch đảo phép cộng



{x+y=1240,112x+17y=152 cases; Case 1: x positive y equals 124; Case 2: 0 comma 112 x positive 1 over 7 end-fraction y equals 15 end-cases;

𝑥+𝑦=1240,112𝑥+17𝑦=15



{-10(x+y)=-10(124)89(0,112x+17y)=89(15)2 cases; Case 1: negative 10 open paren x positive y close paren equals negative 10 open paren 124 close paren; Case 2: 89 open paren 0 comma 112 x positive 1 over 7 end-fraction y close paren equals 89 open paren 15 close paren end-cases;

−10(𝑥+𝑦)=−10(124)89(0,112𝑥+17𝑦)=89(15)

Phân phối



{-10x−10y=-124010x+12,714y=13352 cases; Case 1: negative 10 x negative 10 y equals negative 1240; Case 2: 10 x positive 12 comma 714 y equals 1335 end-cases;

−10𝑥−10𝑦=−124010𝑥+12,714𝑦=1335

Thêm hai phương trình



{−10x−10y=−124010x+12,714y−10x−10y=1335−12402 cases; Case 1: negative 10 x negative 10 y equals negative 1240; Case 2: 10 x positive 12 comma 714 y negative 10 x negative 10 y equals 1335 negative 1240 end-cases;

−10𝑥−10𝑦=−124010𝑥+12,714𝑦−10𝑥−10𝑦=1335−1240

Kết hợp các số hạng đồng dạng



10x+12,714y−10x−10y=1335−124010 x plus 12 comma 714 y minus 10 x minus 10 y equals 1335 minus 1240

10𝑥+12,714𝑦−10𝑥−10𝑦=1335−1240



2,714y=952 comma 714 y equals 95

2,714𝑦=95



{x+y=1242,714y=952 cases; Case 1: x positive y equals 124; Case 2: 2 comma 714 y equals 95 end-cases;

𝑥+𝑦=1242,714𝑦=95

3

Nhân cả hai vế của phương trình với 



0.370.37

0.37



{x+y=1240.37(2,714y)=0.37(95)2 cases; Case 1: x positive y equals 124; Case 2: 0.37 open paren 2 comma 714 y close paren equals 0.37 open paren 95 close paren end-cases;

𝑥+𝑦=1240.37(2,714𝑦)=0.37(95)



{x+y=124y=352 cases; Case 1: x positive y equals 124; Case 2: y equals 35 end-cases;

𝑥+𝑦=124𝑦=35

4

Phép thế trong phương trình



{x+y=124y=352 cases; Case 1: x positive y equals 124; Case 2: y equals 35 end-cases;

𝑥+𝑦=124𝑦=35

Phép thế trong phương trình



{x+y=124y=352 cases; Case 1: x positive y equals 124; Case 2: y equals 35 end-cases;

𝑥+𝑦=124𝑦=35



{x+35=124y=352 cases; Case 1: x positive 35 equals 124; Case 2: y equals 35 end-cases;

𝑥+35=124𝑦=35

Trừ 



3535

35

 ở cả hai vế



{x=124−(35)y=352 cases; Case 1: x equals 124 negative open paren 35 close paren; Case 2: y equals 35 end-cases;

𝑥=124−(35)𝑦=35

Kết hợp các số hạng đồng dạng



{x=89y=352 cases; Case 1: x equals 89; Case 2: y equals 35 end-cases;

𝑥=89𝑦=35



{x=89y=352 cases; Case 1: x equals 89; Case 2: y equals 35 end-cases;

𝑥=89𝑦=35

1

Sắp xếp lại các số hạng theo dạng chuẩn



x+y=124x8,9+y7=152 lines; Line 1: x plus y equals 124; Line 2: the fraction with numerator x and denominator 8 comma 9 end-fraction plus y over 7 end-fraction equals 15 end-lines;

𝑥+𝑦=124𝑥8,9+𝑦7=15



{x+y=1240,112x+17y=152 cases; Case 1: x positive y equals 124; Case 2: 0 comma 112 x positive 1 over 7 end-fraction y equals 15 end-cases;

𝑥+𝑦=1240,112𝑥+17𝑦=15

2

Phép khử biến 



xx

𝑥



{x+y=1240,112x+17y=152 cases; Case 1: x positive y equals 124; Case 2: 0 comma 112 x positive 1 over 7 end-fraction y equals 15 end-cases;

𝑥+𝑦=1240,112𝑥+17𝑦=15

Nhân phương trình với một số rồi cộng vào một phương trình khác



{x+y=1240,112x+17y=152 cases; Case 1: x positive y equals 124; Case 2: 0 comma 112 x positive 1 over 7 end-fraction y equals 15 end-cases;

𝑥+𝑦=1240,112𝑥+17𝑦=15



{x+y=1240,112x+17y-0.1(x+y)=15-0.1(124)2 cases; Case 1: x positive y equals 124; Case 2: 0 comma 112 x positive 1 over 7 end-fraction y negative 0.1 open paren x positive y close paren equals 15 negative 0.1 open paren 124 close paren end-cases;

𝑥+𝑦=1240,112𝑥+17𝑦−0.1(𝑥+𝑦)=15−0.1(124)

Phân phối



{x+y=1240,112x+17y-0.1(x+y)=15-0.1(124)2 cases; Case 1: x positive y equals 124; Case 2: 0 comma 112 x positive 1 over 7 end-fraction y negative 0.1 open paren x positive y close paren equals 15 negative 0.1 open paren 124 close paren end-cases;

𝑥+𝑦=1240,112𝑥+17𝑦−0.1(𝑥+𝑦)=15−0.1(124)



{x+y=1240,112x+17y−0,112x−0,112y=15-13,9332 cases; Case 1: x positive y equals 124; Case 2: 0 comma 112 x positive 1 over 7 end-fraction y negative 0 comma 112 x negative 0 comma 112 y equals 15 negative 13 comma 933 end-cases;

𝑥+𝑦=1240,112𝑥+17𝑦−0,112𝑥−0,112𝑦=15−13,933



{x+y=1240,0305y=1,0672 cases; Case 1: x positive y equals 124; Case 2: 0 comma 0305 y equals 1 comma 067 end-cases;

𝑥+𝑦=1240,0305𝑦=1,067

3

Nhân cả hai vế của phương trình với 



32.832.8

32.8



{x+y=12432.8(0,0305y)=32.8(1,067)2 cases; Case 1: x positive y equals 124; Case 2: 32.8 open paren 0 comma 0305 y close paren equals 32.8 open paren 1 comma 067 close paren end-cases;

𝑥+𝑦=12432.8(0,0305𝑦)=32.8(1,067)



{x+y=124y=352 cases; Case 1: x positive y equals 124; Case 2: y equals 35 end-cases;

𝑥+𝑦=124𝑦=35

4

Phép khử biến 



yy

𝑦



{x+y=124y=352 cases; Case 1: x positive y equals 124; Case 2: y equals 35 end-cases;

𝑥+𝑦=124𝑦=35

Nhân phương trình với một số rồi cộng vào một phương trình khác



{x+y=124y=352 cases; Case 1: x positive y equals 124; Case 2: y equals 35 end-cases;

𝑥+𝑦=124𝑦=35



{x+y−(y)=124−(35)y=352 cases; Case 1: x positive y negative open paren y close paren equals 124 negative open paren 35 close paren; Case 2: y equals 35 end-cases;

𝑥+𝑦−(𝑦)=124−(35)𝑦=35



{x=89y=352 cases; Case 1: x equals 89; Case 2: y equals 35 end-cases;

𝑥=89𝑦=35

1

Sắp xếp lại các số hạng theo dạng chuẩn



x+y=124x8,9+y7=152 lines; Line 1: x plus y equals 124; Line 2: the fraction with numerator x and denominator 8 comma 9 end-fraction plus y over 7 end-fraction equals 15 end-lines;

𝑥+𝑦=124𝑥8,9+𝑦7=15



{x+y=1240,112x+17y=152 cases; Case 1: x positive y equals 124; Case 2: 0 comma 112 x positive 1 over 7 end-fraction y equals 15 end-cases;

𝑥+𝑦=1240,112𝑥+17𝑦=15

2

Chuyển đổi hệ phương trình tuyến tính thành ma trận tăng cường



x+y=1240,112x+17y=152 lines; Line 1: x plus y equals 124; Line 2: 0 comma 112 x plus 1 over 7 end-fraction y equals 15 end-lines;

𝑥+𝑦=1240,112𝑥+17𝑦=15



[11|1240,11217|15]the 2 by 4 matrix; Row 1: Column 1: 1, Column 2: 1, Column 3: vertical line, Column 4: 124; Row 2: Column 1: 0 comma 112, Column 2: 1 over 7 end-fraction, Column 3: vertical line, Column 4: 15 end-matrix;

11|1240,11217|15

3

Nhân hàng (

1

) với



-0.1negative 0.1

−0.1

 rồi cộng vào hàng (

2

)



[11|1240,11217|15]the 2 by 4 matrix; Row 1: Column 1: 1, Column 2: 1, Column 3: vertical line, Column 4: 124; Row 2: Column 1: 0 comma 112, Column 2: 1 over 7 end-fraction, Column 3: vertical line, Column 4: 15 end-matrix;

11|1240,11217|15



[11|12400,0305|1,067]the 2 by 4 matrix; Row 1: Column 1: 1, Column 2: 1, Column 3: vertical line, Column 4: 124; Row 2: Column 1: 0, Column 2: 0 comma 0305, Column 3: vertical line, Column 4: 1 comma 067 end-matrix;

11|12400,0305|1,067

4

Nhân hàng (

2

) của ma trận với 



32.832.8

32.8



[11|12400,0305|1,067]the 2 by 4 matrix; Row 1: Column 1: 1, Column 2: 1, Column 3: vertical line, Column 4: 124; Row 2: Column 1: 0, Column 2: 0 comma 0305, Column 3: vertical line, Column 4: 1 comma 067 end-matrix;

11|12400,0305|1,067



[11|12401|35]the 2 by 4 matrix; Row 1: Column 1: 1, Column 2: 1, Column 3: vertical line, Column 4: 124; Row 2: Column 1: 0, Column 2: 1, Column 3: vertical line, Column 4: 35 end-matrix;

11|12401|35

5

Nhân hàng (

2

) với



-1negative 1

−1

 rồi cộng vào hàng (

1

)



[11|12401|35]the 2 by 4 matrix; Row 1: Column 1: 1, Column 2: 1, Column 3: vertical line, Column 4: 124; Row 2: Column 1: 0, Column 2: 1, Column 3: vertical line, Column 4: 35 end-matrix;

11|12401|35



[10|8901|35]the 2 by 4 matrix; Row 1: Column 1: 1, Column 2: 0, Column 3: vertical line, Column 4: 89; Row 2: Column 1: 0, Column 2: 1, Column 3: vertical line, Column 4: 35 end-matrix;

10|8901|35

6

Chuyển đổi ma trận tăng cường thành hệ phương trình tuyến tính



[10|8901|35]the 2 by 4 matrix; Row 1: Column 1: 1, Column 2: 0, Column 3: vertical line, Column 4: 89; Row 2: Column 1: 0, Column 2: 1, Column 3: vertical line, Column 4: 35 end-matrix;

10|8901|35



x=89y=352 lines; Line 1: x equals 89; Line 2: y equals 35 end-lines;

𝑥=89𝑦=35

Đáp án