How do u solve this problem?

Question

First transform the equations so that either the x-coefficients or the y-coefficients are opposite. Then solve by the linear combination method.



1) 7u+8v=23
3u-2v=-1


2) 3x-5y=-29
2x-10y=-42

Please tell me what to do.
Thanks.

Answer 1

"Transform the equations so that the y-coefficients are opposite"; that should probably be "Transform the equations so that the y-coefficients are EQUAL AND opposite". Then when you "combine" (add) the equations, the "y" terms will drop out, leaving only the x, and you can solve for x. Example using the second pair:

Multiply the first eq by -2 to get -6x+10y=58; notice the coefficient of y is now +10, while in the second eq. the y coefficient is -10, so these are now equal and opposite. If we add the equations now, get -4x = 16, so x = -4. Plug this x into either eq. to get y.

Answer 2

1) You have two equations and two unknowns (i.e., u and v). It is almost always solvable.

2) consider the general case:
au + bv = c
du + ev = f
where a, b, c, d, e, f can be almost any constants. In your case they are (7, 8, 23, 3, -2, -1)

3) There are many ways to solve this system of equations. A simple way is to solve the first for u in terms of v [i.e., . u = (c-bv)/a]. Then, substitute this into the second and solve the second for u. Then, substitute u back into the first and solve for v; that's it; really. Do it carefully and the answer will follow.

4) I'll do a little more. If you follow the algorithm outlined in 3), you will get

4.1) u = ( ce-bf ) / ( ae-bd )

4.2) v = ( dc-af ) / ( bd-ae )

5) In your case: (a,b,c,d,e,f) =(7, 8, 23, 3, -2, -1)
5.1) u = ( 23 * (-2) - 8 * (-1) ) / ( 7 * (-2) - 8 * (3) ) = 1
5.2) v = ( 3 * 23 - 7 * (-1) ) / ( 8 * 3 - 7 * (-2) ) = 2

6) So, the solution is : u = 1 and v = 2;

7) Test it on the first equation: 7u+8v = 7(1)+8(2) = 23 as we hoped it would; how about that!!

8) Test it on the second equation: 3u-2v = 3(1)-2(2) = -1; wow! Ain't algebra beautiful

Answer 3

Multiply the second equation in (1) by 4 to get

12u - 8v = -4

7u + 8v = 23

Then write the equations with u terms first, v terms next and constant terms to the right of the equal sign. Add the two equations and you get:

19u = 19

Which means u = 1. And then you can pluq 1 into one of the equations and solve for v.

3(1) -2v = -1
-2v = -4
v = 2


Now, you try it, with the second set of equations. (Hint, multiply the first equation by -2.)

Answer 4

I'll explain in a moment !!

1) lets take v for the opposites .. so as the above one is 8v .. the bottom one shud be -8v .. what to multiply by ?? 4 !!

so

7u+8v=23
12u-8v=-4

add them now!

so

19u = 19

so u = 1

now u can solve for v !!

and the second equation .. multiply the top one by -2 .. and u will get the answer !!

Answer 5

1)
7u+8v=23..................(1)
3u-2v=-1 ..................(2)

equation (2)x4
12u - 8v = -4...............(3)

equation (3) + (1)
12u + 7u + (-8v) + 8v = -4 +23
19u = 19
so,
u = 19/19 = 1.............ans

substitute value of u = 1 into equation (2)
3(1) - 2v = -1
3 + 1 = 2v
so,
v= 4/2 = 2................ans.

check;
subs. both value u & v into equation (3);
12u - 8v = -4...............(3)
12(1) - 8 (2) = -4
12 - 16 = -4
-4 = -4.................both side is same, so this value is right.

2).
3x-5y=-29.................(1)
2x-10y=-42...............(2)

equation (1) x 2
6x - 10y = - 58...........(3)

equation (3) - (2)
6x - 2x -10y - (-10y) = -58 - (-42)
4x = -16
so,
x = -16 / 4 = -4...............ans

subs value of x into equation (1)
3x-5y=-29
3(-4) - 5y = -29
-12 + 29 = 5y
17 = 5y
so;
y = 17/ 5.......ans

check;
subs both value into equation (30
6x - 10y = - 58...........(3)
6(-4) - 10(17/5) = -58
-24 - 34 = -58
-58 = -58.............both side is same, so this value is right



gut luck... math is easy...remember that..