Question

Solve the simultaneous equation x+2y=5x²-y²=-3​

Answer 1

Answer:

(1, 2)

(-13/3, 14/3).

Step-by-step explanation:

x + 2y = 5

x² - y² = -3​

From the first equation x = 5 - 2y

Substitute for y in the second equation:

(5 - 2y)^2 - y^2 = -3

25 + 4y^2 - 20y - y^2 = -3

3y^2 - 20y + 28 = 0

Factoring:-

(3y - 14(y  -  2)   = 0

y = 2, 14/3.

So when y = 2, x = 5 - 2(2) = 1.

When y = 14/3, x = 5 - 2(14/3) = -13/3.

Checking the results in the second equation:

(1, 2):-

1^2 - 2^2 = -3    Correct.

(-13/3, 14/3):-

(-13/3)^2 - (14/3)^2 = -3   Correct.

Answer 2

Answer:

y = 14/3 and 2

x = -13/3 and 1

(-13/3, 14/3) and (1, 2)

Step-by-step explanation:

$x + 2y = 5 - - - (a) \\ {x}^{2} - {y}^{2} = - 3 - - - (b)$

for equation (a)

make x the subject of the formular:

$x = 5 - 2y - - - (c)$

for equation (b)

${x}^{2} - {y}^{2} = - 3$

substitute for x as 5 - 2y in equation (b):

${(5 - 2y)}^{2} - {y}^{2} = - 3 \\ (25 - 20y + 4 {y}^{2} ) - {y}^{2} = - 3 \\ 25 - 20y + 3 {y}^{2} = - 3 \\ {3y}^{2} - 20y + 28 = 0 \\ (3y - 14)(y - 2) = 0$

therefore,

$y = \frac{14}{3} \: \: and \: \: 2$

substitute for all values of y in equation (c):

for y = 14/3:

$x = 5 - 2( \frac{14}{3} ) \\ \\ x = 5 - \frac{28}{3} \\ \\ x = - \frac{13}{3}$

for y = 2:

$x = 5 - 2(2) \\ x = 5 - 4 \\ x = 1$