數學問題(中二)(20點)

解答條題主項變換,詳細列明步驟


1.

2a(y-x)=3b(x+y)



【變換做a】

2a(y-x)=3b(x+y)

a = [3b(x + y)]/[2(y - x)]



【變換做b】

2a(y-x)=3b(x+y)

b = [2a(y - x)]/[3(x + y)]



【變換做x】

2a(y-x)=3b(x+y)

2ay - 2ax = 3bx + 3by

2ay - 3by = 3bx + 2ax

(2a - 3b)y = (3b + 2a)x

x = (2a - 3b)y/(3b + 2a)



【變換做y】

2a(y-x)=3b(x+y)

2ay - 2ax = 3bx + 3by

2ay - 3by = 3bx + 2ax

(2a - 3b)y = (3b + 2a)x

y = (3b + 2a)x/(2a - 3b)





2.

1/x+1/y=1/m



【變換做x】

1/x + 1/y = 1/m

1/x = 1/m - 1/y

1/x = (y - m)/(my)

x = my/(y - m)



【變換做y】

1/x + 1/y = 1/m

1/y = 1/m - 1/x

1/y = (x - m)/(mx)

y = (mx)/(x - m)



【變換做m】

1/x + 1/y = 1/m

(y + x)/(xy) = 1/m

m = xy/(y + x)





3.

(3 - a)/(b - c) = (a - 2)/(c + 1)



【變換做a】

(3 - a)(c + 1) = (a - 2)(b - c)【交叉相乘】

3c + 3 - ac - a = ab - ac -2b + 2c

3c + 3 + 2b - 2c = ab - ac + ac - a

c + 2b + 3 = ab - a

c + 2b + 3 = a(b - 1)

a = (c + 2b + 3)/(b - 1)



【變換做b】

(3 - a)(c + 1) = (a - 2)(b - c)【交叉相乘】

3c + 3 - ac - a = ab - ac -2b + 2c

3c + 3 - ac - a + ac - 2c = ab - 2b

c - a + 3 = b(a - 2)

b = (c - a + 3)/(a - 2)



【變換做c】

(3 - a)(c + 1) = (a - 2)(b - c)【交叉相乘】

3c + 3 - ac - a = ab - ac -2b + 2c

3c - ac + ac - 2c = ab - 2b - 3 + a

c = (a - 2)b + a - 3





4.

vt = (v + 3) + (t - 4)



【變換做v】

vt = (v + 3) + (t - 4)

vt = v + t - 1

vt - v = t - 1

v(t - 1) = t - 1

v = (t - 1)/(t - 1) = 1



【變換做t】

vt = (v + 3) + (t - 4)

vt = v + t - 1

vt - t = v - 1

t(v - 1) = v - 1

t = (v - 1)/(v - 1) = 1


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解答條題化簡,詳細列明步驟


1.

(x - 3)/(x + 3) + (x + 3)/(x - 3)

= (x - 3)(x - 3)/[(x + 3)(x - 3)] + (x + 3)(x + 3)/[(x - 3)(x + 3)]【通分母】

= [(x - 3x - 3x + 9) + (x + 3x + 3x + 9)]/[(x - 3)(x + 3)]

= [(x - 6x + 9) + (x + 6x + 9)]/[(x - 3)(x + 3)]

= (2x + 18)/[(x - 3)(x + 3)]

= 2(x + 9)/[(x - 3)(x + 3)]


2.

(x + 3)/(x - 2) + (x + 2)/(x - 3)

= (x + 3)(x - 3)/[(x - 2)(x - 3)] + (x + 2)(x - 2)/[(x - 3)(x - 2)]【通分母】

= [(x - 3x + 3x - 9) + (x - 2x + 2x - 4)]/[(x - 2)(x - 3)]

= [(x - 9) + (x - 4)]/[(x - 2)(x - 3)]

= (2x - 13)/[(x - 2)(x - 3)]