Solve a linear equation and Quadratic equation (1)

Example 1:

Solve the following equations

x= 2 y , x² + y² = 45

the answer :

Substitute the expression 2 y for x into the second equation and solve for y

( 2y)² + y² = 45

4 y² + y² = 45

5 y² = 45

y² =  9

y = ± 3 

y = 3                              or           y = -3

x = 2 y =2 × 3                              x = 2 y =2 ×(- 3)

X= 6                                             x= - 6

the solution to this equations = {(6, 3) ,( - 6 , - 3) }

Example 2:

Solve the following equations

y  ½  x=

 x² + y² = 20

the answer :

Substitute the expression   ½ y for x into the second equation and solve for y

(  ½ y)² + y² = 20

 ¼ y² + y² = 20  ………… ×4

  y² +4 y² = 80 

5 y² = 80

Y² =  16

Y = ± 4 

Y = 4                              or           y = -4

X = ½  y =½ × 4                         X = ½ y =½ ×(- 4)

X= 2                                              x= - 2

the solution to this equations = {(2, 4) ,( - 2 , - 4) }

Example 3:

Solve the following equations

X= 3 y , x² - y² = 72

the answer :

Substitute the expression 3 y for x into the second equation and solve for y

( 3y)² -  y² = 72

9 y² - y² = 72

8 y² = 72

Y² =  9

Y = ± 3 

Y = 3                              or           y = -3

X = 3 y =3 × 3                              X = 3 y =3 ×(- 3)

X= 9                                            x= - 9

the solution to this equations = {(9, 3) ,( - 9 , - 3) }

Example 4:

Solve the following equations

X= y + 3 , x² + y² = 23

the answer :

Substitute the expression  y + 3  for x into the second equation and solve for y

( y + 3)² + y² = 23

  y² + 6 y + 3  + y² = 23

2 y² + 6 y + 3 – 23  = 0

2 y² + 6 y – 20 = 0   …….   ÷   2 

Y²  + 3 y – 10  =  0

( y – 2 )( y + 5 ) = 0

Y = 2                              or              y = -5

X = y + 3 =2 + 3                              X =  y + 3 =- 5 + 3

X= 5                                                x= - 2

the solution to this equations = {(5, 2) ,( - 2 , - 5) }

Example 5:

Solve the following equations

X= y - 2 , x² + y² = 74

the answer :

Substitute the expression  y - 2  for x into the second equation and solve for y

( y - 2)² + y² = 74

  y² - 4 y + 4  + y² = 74

2 y² -  4 y + 4 – 74  = 0

2 y² -  4 y – 70 = 0   …….  ÷  2 

Y²  -  2 y – 35  =  0

( y – 7 )( y + 5 ) = 0

Y = 2                              or              y = -5

X = y + 3 =2 + 3                              X =  y + 3 =- 5 + 3

X= 5                                                x= - 2

the solution to this equations = {(5, 2) ,( - 2 , - 5) }