Linear Systems question - solve this system:
1/x + 3/y = 3/4
3/x - 2/y = 5/12
Polynomials question:
Divide 2x – 1 into 4x4 – 5x2 + 2x + 4.
For the first question, multiply the first equation by -3 and then add it to the second equation:
-3(1/x + 3/y = 3/4)
+ 3/x - 2/y = 5/12
-3/x - 9/y = -9/4
+3/x - 2/y = 5/12 Find a common denominator for the right hand sides of the equations:
-3/x - 9/y = -27/12
+3/x - 2/y = 5/12 Multiplied the right hand side of the top equation by 3/3, which is essentially multiplying by 1.
Now add them:
-11/y = -22/12
Cancel out the negatives and cross multiply to get:
12*11 = 22*y
132 = 22y
y = 6
Substitute this value back into this original equation:
3/x - 2/y = 5/12 to find x.
3/x - 2/6 = 5/12
3/x = 5/12 + 4/12 got a common denominator for the constants
3/x = 9/12
Cross Multiply:
3*12 = 9*x
36 = 9x
x = 4
For practice, Substitute the x and y values back into both of the original equations to show that they check out.
For the polynomials question, you start by asking yourself how many times 2x can go into 4x^4. Well, it would be 2x^3, because 2x*2x^3 = 4x^4.
So you multiply (2x-1) by 2x^3 and subtract it from the entire above polynomial: Looks like:
2x^3(2x-1) = 4x^4 - 2x^3
Now, do: (4x^4 - 5x^2 + 2x + 4) - (4x^4 - 2x^3)
The polynomial now looks like:
ii) 2x^3 - 5x^2 + 2x + 4
We do the same thing as above, ask ourselves how many times 2x can go into 2x^3. it would be x^2 times, since 2x*x^2 = 2x^3
So you multiply (2x-1) by x^2 and subtract it from the second stage polynomial (ii) above:
(x^2)(2x-1) = 2x^3 - x^2
Now, do (2x^3 - 5x^2 + 2x + 4) - (2x^3 - x^2)
The polynomial now looks like:
(iii) -4x^2 + 2x + 4
We do the same, and se that 2x can go into -4x^2 a number of -2x times because 2x*-2x = -4x^2.
So, you multiply (2x-1) by -2x and subtract it from the third stage polynomial (iii)
-2x(2x - 1) = -4x^2 + 2x
Now, do (-4x^2 + 2x + 4) - (-4x^2 + 2x)
The polynomial now looks like:
(iv) 4
You can't divide 2x into 4 any further, so we are done. The ending form looks like:
(2x-1)(2x^3 + x^2 -2x) + 4
Expand this to check to see if you get the original polynomial at the beginning. Hope this helps