The textbook wasn\'t very clear on how all the steps happen, it just showed the

Question

The textbook wasn't very clear on how all the steps happen, it just showed the final factored and simplified version of each step. Can you please explain how all of the numbers combine and multiply together and then how they get simplified?

5x-2y = 13 4x-3y = 9 1. Use the elimination met hod to solve the system of equations: X2-64 8-? x2 +6x-16 x+4 2 Perform the division and simplify: , Y-3 ??? +13?-10 Perform the addition and simplify: . ??-2 +5 4 Simplify X- 1 5 x-1 5 3 12 5 Solve the equation:

Explanation / Answer

1. To eliminate y, let us multiply the 1st equation by 3 and the 2nd equation by 2. Then we have 15x-6y = 39…(3) and 8x-6y = 18…(4). Now, on subtracting the 4th equation from the 3rd equation, we get 15x-6y –(8x-6y) = 39-18 or, 7x = 21 so that x = 21/7 = 3. Now, on substituting x = 3 in the 1st equation, we get 5*3-2y = 13 or, 2y = 15-13 = 2 so that y = 1. Thus, x = 3 and y = 1.

2. We have (x2-64) = (x2-82) = (x+8)(x-8). Also, x2+6x-16 = x2+8x-2x-16 = x(x+8)-2(x+8) = (x+8)(x-2).

Therefore, (x2-64)/( x2+6x-16) = (x+8)(x-8)/ (x+8)(x-2) = (x-8)/(x-2). Then the given expression is equal to (x-8)/(x-2) ÷ (8-x)/(x+4) =[ (x-8)/(x-2) ]* [(x+4)/(8-x) ] = - [(x-8)/(x-8)]*[(x+4)/(x+2)] = - (x+4)/(x+2).

3. We have [(y-3)/(3y2+13y-10 )]+ 2/(3y-2) = [(y-3)(3y-2)+ 2(3y2+13y-10 ) ]/ (3y-2) (3y2+13y-10 ) = (3y2-11y+6 +6y2 +26y-20)/ (3y-2) (3y2+13y-10 ) = (9y2+15y-14)/(3y-2)(3y2+13y-10 ).

Now (9y2+15y-14) = (3y+7)(3y-2) ( we can use the quadratic formula to find roots of the expression9y2+15y-14, and then factorize), so that [(y-3)/(3y2+13y-10 )]+ 2/(3y-2) = (3y+7)(3y-2)/( 3y-2)(3y2+13y-10 ) = (3y+7)/ (3y2+13y-10 ).

4. We have 5/(x-1) +5 = 5[1/(x-1)+1] = 5(1+x-1)/(x-1) = 5x/(x-1). Also, 5/(x-1)-5 = 5[1/(x-1)-1] = 5(1-x+1)/(x-1) = 5(2-x)/(x-1). Hence the given expression is equal to [5x/(x-1)]* [(x-1)/5(2-x)] = x/(2-x).

5. 5/(y+2)+3/(y-2) = 12/(y2-4) or, [5(y-2)+3(y+2)]/(y+2)(y-2) = 12/(y+2)(y-2) or, (8y-4)/ (y+2)(y-2) = 12/(y+2)(y-2) so that 8y-4 = 12 or, 8y = 12+4 = 16. Hence y = 16/8 = 2.

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