聯立方程(三題)

1. {y=x^2-2x+3
y=3x-1
3x - 1 = x^2 - 2x + 3
x^2 - 2x - 3x + 1 + 3 = 0
x^2 - 5x + 4 = 0
(x - 1)(x - 4) = 0
x - 1 = 0, x - 4 = 0
x = 1, x = 4

2. {y=4x^2-3x+2
y=8x+5
8x + 5 = 4x^2 - 3x + 2
4x^2 - 3x - 8x - 5 + 2 = 0
4x^2 - 11x - 3 = 0
(4x + 1)(x - 3) = 0
4x + 1 = 0, x - 3 = 0
x = -0.25, x = 3

3. {y=-3x
xy-3=6x
x(-3x) - 3 = 6x
-3x^2 - 3 = 6x
-3x^2 - 6x - 3 = 0
3x^2 + 6x + 3 = 0
(3x + 3)(x + 1) = 0
3x + 3 = 0, x + 1 = 0
x = -1, x = -1

(Question 1)
y=x^2-2x+3 -----(i)
y=3x-1 ----(ii)

Sub(ii) into (i) :3x-1 = x^2-2x+3

x^2-5x+4=0
(x-4)(x-1)=0

Therefore, x=4 or x=1
y=3(4)-1 or y=3(1)-1
=11 or y=2

Solutions are(4,11) and (1,2)

(Question 2)

y=4x^2-3x+2
y=8x+5

8x+5 = 4x^2-3x+2
4x^2-11x-3
(4x+1)(x-3) = 0

Therefore x =-1/4 or x=3
y=8(-1/4)+5 or y=8(3)+5
=3 or y=29

Solutions are(-1/4,3) and (3,29)

(Question 3)

y=-3x
xy-3=6x

x(-3x)-3=6x
-3x^2-6x-3=0
x^2+2x+1=0
(x+1)^2=0
x=-1

y=-3(-1)
=3

The solution is(-1,3)

1.
{y=x²-2x+3 .......... (1)
y=3x-1 ............ (2)

從(1) = (2)，
x²-2x+3 = 3x-1
x²-5x+4 = 0
(x-4)(x-1) = 0
x-4 = 0 或 x-1 = 0
x=4 或 x=1

若 x=4,
y = 3(4)-1 = 11 ....... 從 (2)
若 y = 1
y = 3(1)-1 = 2 ........ 從 (2)

所以 x=4,y=11 或 x=1, y=2
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2.
{y=4x²-3x+2 .......... (1)
y=8x+5 ............ (2)

從(1) = (2)，

4x²-3x+2 = 8x+5
4x²-11x-3 = 0
(4x+1)(x-3) = 0
4x+1 = 0 或 x-3 = 0
x = -1/4 或 x = 3

若 x=-1/4,
y = 8(-1/4)+5 = 3 ........ 從 (2)
若 x=3,
y = 8(3)+5 = 29 ........ 從 (2)

所以 x=-1/4,y=3 或 x=3, y=29
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3.
{y=-3x .......... (1)
xy-3=6x ............ (2)

將 (1) 代入 (2)

x(-3x)-3 = 6x
-3x² - 3 = 6x
3x² + 6x + 3 = 0
x² + 2x + 1 = 0
(x+1)² = 0
x+1 = 0
x = -1

若 x = -1,
y = -3(-1) = 3

所以 x=-1, y=3