Find all points of intersection of each of the following

Question

Find all points of intersection of each of the following:

a) 2𝑥−3𝑦=13 and 5𝑥+3𝑦=1 

b) 𝑥^2+𝑦^2 =25 and 2𝑥+𝑦=10 

c) 𝑦=𝑥^3−4𝑥 and 𝑦=−𝑥 

d) 𝑦=𝑥^4−2𝑥^2+1 and 𝑦=1−𝑥2

Please show works

Answer 1

(a)

adds them together: 7x = 14 -->x=2

when x=2, y = -3

(b)

per second equation y = 10-2x

x^2 + (10-2x)^2 = 25

x^2 + 100 - 40x + 4x^2 = 25

5x^2 - 40x + 75 = 0

x^2 - 8x + 15 = 0

(x - 3 )(x - 5) = 0

x=3 --> y=4 so (3,4)

x=5 ---> y=0 so (5,0)

(c) -x = x^3 - 4x

3x = x^3

0 = x^3 - 3x

0 = x(x^2-3)

x = 0, x = +or- sqrt(3)

(0,0) ( sqrt(3), -sqrt(3)) (-sqrt(3), sqrt(3))

(d) y = (x^2 - 1)^2 and y = 1 - x^2

(x^2 - 1)^2 = 1 - x^2

(x^2 - 1)^2 + (x^2-1) = 0

(x^2-1) [ (x^2 - 1) + 1] = 0

(x^2 - 1) x^2 = 0

x=1,-1,0,0

(1,0) (-1,0) (0,1)