Question

10ac-5bc+2ax-bx = 6a²-6ab-7a+7b = 4x³-4x²y+x-y = x⁴y-x³y-x+1=

Answer 1

10ac-5bc+2ax-bx =5c(2a-b)+x(2a-b)=(5c+x)(2a-b)
6a²-6ab-7a+7b =6a(a-b)-7(a-b)=(6a-7)(a-b)
4x³-4x²y+x-y =4x^2(x-y)+(x-y)=(4x^2+1)(x-y)
x⁴y-x³y-x+1=x^3y(x-1)-(x-1)=(x^3y-1)(x-1)

Answer 2

We can simplify each equation separately as follows:

1. 10ac - 5bc + 2ax - bx = 6a² - 6ab - 7a + 7b

We can factor out a common factor of 5 from the first two terms on the left side and factor out a common factor of a from the last two terms on the left side to get:

5c(2a - b) + a(2x - b) = (6a - 7)(a - b)

We can further simplify this equation by dividing both sides by (a - b) and simplifying:

5c(2a - b)/(a - b) + a(2x - b)/(a - b) = 6a - 7

2. 4x³ - 4x²y + x - y = 6a² - 6ab - 7a + 7b

We can rearrange terms and group like terms to get:

4x³ - 4x²y - x + 6a² - 6ab - 7a + y - 7b = 0

3. x⁴y - x³y - x + 1 = 6a² - 6ab - 7a + 7b

We can rearrange terms and group like terms to get:

x⁴y - x³y - 6a² + 6ab + 7a - 7b - x + 1 = 0

Overall, we have three equations in four variables (a, b, x, y). We can use the first equation to eliminate one variable, say b, in terms of the other three variables (a, c, x):

b = 2a(5c + x)/(5c + 2x)

Substituting this expression for b into the second and third equations yields two equations in three variables (a, c, x):

4x³ - 4x²y - x + 6a² - 6a(2a(5c + x)/(5c + 2x)) - 7a + y - 7(2a(5c + x)/(5c + 2x)) = 0

x⁴y - x³y - 6a² + 6a(2a(5c + x)/(5c + 2x)) + 7a - 7(2a(5c + x)/(5c + 2x)) - x + 1 = 0

We can simplify each of these equations further by multiplying through by (5c + 2x) and expanding the products:

4x⁴ + 2x³y - x²(5c + 2x) + 6a²(5c + 2x) - 12a³(5c + x) - 7a(5c + 2x) + y(5c + 2x) - 14ac - 7x = 0

x⁴y - x³y - 6a²(5c + 2x) + 12a³(5c + x) + 7a(5c + 2x) - 14a(5c + x) - x(5c + 2x) + (5c + 2x) = 0

These two equations can be solved simultaneously for a, c, and y in terms of x, giving:

a = (-x³ + 3x² - 2x)/(12x³ - 35x² +