7)
\Given equation
\a(x + 1) + 5a(x - 1) = 2(3b - 2a)
\Distribute terms using distributive property: a( b + c) = ab + ac.
\a*x + a*1 + 5a*x - 5a*1 = 2*3b - 2*2a
\ax + a + 5ax - 5a = 6b - 4a
\6ax - 4a = 6b - 4a
\6ax = 6b - 4a + 4a
\Seperate variables and constant terms.
\6ax = 6b
\x = 6b / 6a
\x = b / a
\8)
\Given equation
\x(a + 1) - x(a - 1) = 2a + 4
\x(a + 1) - x(a - 1) = 2a + 2*2
\Take common term using distributive property: ab + ac = a( b + c)
\x [ (a + 1) - (a - 1) ] = 2(a + 2)
\x [ a + 1 - a + 1 ] = 2(a + 2)
\x [ 1 + 1 ] = 2(a + 2)
\x [ 2 ] = 2(a + 2)
\2x = 2(a + 2)
\x = 2(a + 2) / 2
\x = a + 2
\9)
\Given equation
\(x + 2)(x - 5) = (x - 1)(x - 6)
\Distribute terms using distributive property: a( b - c) = ab - ac.
\x(x - 5) + 2(x - 5) = x(x - 6) - 1(x - 6)
\(x*x - x*5 + 2*x - 2*5) = (x*x - x*6 - 1*x + 1*6)
\(x² - 5x + 2x - 10) = (x² - 6x - x + 6)
\(x² - 5x + 2x - 10) = (x² - 6x - x + 6)
\(x² - 3x - 10) = (x² - 7x + 6)
\Seperate variables and constant terms.
\x² - 3x + 7x - x²= 6 + 10
\4x = 16
\x = 16 / 4
\x = 4
\10)
\Given equation
\(x - 8)(x + 1) = (x + 5)(x - 3)
\Distribute terms using distributive property: a( b - c) = ab - ac.
\x(x + 1) - 8(x + 1) = x(x + 5) - 3(x + 5)
\(x*x + x*1 - 8*x - 8*1) = (x*x + x*5 - 3*x - 3*5)
\(x² + x - 8x - 8) = (x² + 5x - 3x - 15)
\(x² - 7x - 8) = (x² + 2x - 15)
\Seperate variables and constant terms.
\x² - 7x - 2x - x²= -15 + 8
\- 9x = -7
\x = -7 / (-9)
\x = 7/9
\12)
\Given equation
\2(x - 2)(x + 3) - (2x + 4)(x - 2) = 0
\Take out term (x - 2) as common term.
\(x - 2) [ 2(x + 3) - (2x + 4) ] = 0
\(x - 2) [ 2*x + 2*3 - 2x - 4 ] = 0
\(x - 2) [ 2x + 6 - 2x - 4 ] = 0
\(x - 2) [ 6 - 4 ] = 0
\2(x - 2) = 0
\x - 2 = 0
\x = 2
\11)
\Given equation
\(x + 1)(6x - 2) = (2x + 4)(3x + 2)
\Distribute terms using distributive property : a( b - c) = ab - ac.
\6x(x + 1) - 2(x + 1) = 3x(2x + 4) + 2(2x + 4)
\6x*x + 6x*1 - 2*x - 2*1 = 3x*2x + 3x*4 + 2*2x + 2*4
\6x² + 6x - 2x - 2 = 6x² + 12x + 4x + 8
\6x² + 4x - 2 = 6x² + 16x + 8
\Seperate variables and constant terms.
\6x² + 4x - 6x² - 16x = + 8 + 2
\4x - 16x = + 8 + 2
\-12x = 10
\12x = - 10
\x = - 10 / 12
\x = -5/6
\