Solve this maths equation?

X + Y = 9

SQUARE BOTH SIDES

X^2 + Y^2 + 2XY = 81

53 + 2XY = 81

2XY = 81-53 = 28


X^2 + Y^2 - 2XY = 53 -28

( X - Y)^2 = 25

X- Y = 5 oR -5

X + Y = 5

X+Y = 9

ADD 2X = 14

X = 7 ANSWER

Y = 2 ANSWER

X - Y = -5
X+Y = 9

ADD 2X = 4

X = 2 ANSWER

Y = 7 ANSWER


ANSWER 7,2 or 2 , 7

x + y = 9 --- eqn 1
x² + y² = 53 --- eqn 2
=> From eqn 1, make y the subject of formula by subtracting x from both sides, then
y = 9 - x --- eqn 3
=> Substitute 9 - x for y in eqn 2, i.e.
x² + (9 - x)² = 53
x² + 81 - 18x + x² = 53
x² + x² - 18x + 81 - 53 = 0
2x² - 18x + 28 = 0
=> Divide all through by 2, then
x² - 9x + 14 = 0
=> Solving quadratically, you'll have
x² - 2x - 7x + 14 = 0
x(x - 2) - 7(x - 2) = 0
(x - 2)(x - 7) = 0
=> From the principle of zero products: (a)(b) = 0, where a = 0 or b = 0, then
x - 2 = 0 or x - 7 = 0
x = 2 or 7.
=> Substitute 2 for x in eqn 3, i.e.
y = 9 - 2
y = 7.
=> Substitute 7 for x in eqn 3, i.e.
y = 9 - 7
y = 2.
Hence (x, y) = (2, 7), (7, 2) ...Ans.

x + y = 9 ← this is a straight line

x² + y² = 53 ← this is circle


There is no solution if the straight line don't cross the circle.

There is only one solution if the straight line is tangent to the circle.

There are 2 solutions if the straight line crosses the circle.


x² + y² = 53 → you know that: x + y = 9 → y = 9 - x

x² + (9 - x)² = 53

x² + 81 - 18x + x² = 53

2x² - 18x + 28 = 0 → you can simplify by 2

x² - 9x + 14 = 0

x² - (2x + 7x) + 14 = 0

x² - 2x - 7x + 14 = 0

(x² - 2x) - (7x - 14) = 0

x(x - 2) - 7(x - 2) = 0

(x - 2)(x - 7) = 0


First case: (x - 2) = 0 → x = 2 → recall: y = 9 - x → y = 7 → first point (2 ; 7)

Second case: (x - 7) = 0 → x = 7 → recall: y = 9 - x → y = 2 → first point (7 ; 2)

x + y = 9 eq1
x^2 + y^2 = 53 eq2

from eq1

x + y = 9
y = 9 - x

plug in eq1 to eq2

x^2 + (9 - x)^2 = 53
x^2 + x^2 - 18x + 81 = 53
2x^2 - 18x + 81 - 53 = 0
2x^2 - 18x + 28 = 0
x^2 - 9x + 14 = 0
(x - 7)(x - 2) = 0
x - 7 = 0, x - 2 = 0
x = 7, x = 2

x = 7, 2 answer//

solving for y from eq1

y = 9 - x

when x = 7

y = 9 - 7 = 2 answer//

when x = 2

y = 9 - 2 = 7 answer//

1)...X + Y = 9
2)...X^2 + Y^2 = 53
From 1) X = 9 - Y
Substitute for X in 2)
(9 - Y)^2 + Y^2 = 53
81 - 18Y + 2Y^2 = 53
2Y^2 - 18Y + 28 = 0
Y^2 - 9Y + 14 = 0
(Y - 7)(Y - 2) = 0

y = 9 - x....Substitute this in the second equation

x^2 + 81 - 18x + x^2 = 53
2x^2 - 18x + 28 = 0
x^2 - 9x + 14 = 0
(x -7)(x - 2) = 0
x = 7, 2

x + y = 9
so...x = (9 - y)

substitute, then solve...

(9 - y)^2 + y^2 = 53
81 - 18y + y^2 + y^2 = 53

2y^2 - 18y + 28 = 0
y^2 - 9y + 14 = 0
(y - 2)(y - 7) = 0
y = 2 or y = 7
x = 7 or x = 2

the solution sets are: (2, 7) and (7, 2)

check

MAD

7^2 + 2^2 = 53

49 + 4 = 53