Solving Linear System of Equations: Elimination

[ilovecake]

Homework Statement

A general system of linear equations is
ax+by=e
cx+dy=f

where a,b,c,d,e,f are constant values.

a)Use elimination to solve for x and y in terms of a,b,c,d,e,f.

b)Are there any values that a,b,c,d,e,f cannot have?

The Attempt at a Solution

For part A, I got x=(e-f)/(a-c), assuming using subtraction for elimination, but in the answers it says x=(de-bf)/(ad-bc). Why are the coefficients of y there?

For part B, I think ad=/=bc because then both equations would cancel out, and it is either the exact same equation or has the same slope but parallel and never intersecting.

for example, 15x+9y & 5x+3y

 

[Ray Vickson]

Homework Statement

A general system of linear equations is
ax+by=e
cx+dy=f

where a,b,c,d,e,f are constant values.

a)Use elimination to solve for x and y in terms of a,b,c,d,e,f.

b)Are there any values that a,b,c,d,e,f cannot have?

The Attempt at a Solution

For part A I got x=(e-f)/(a-c), assuming using subtraction for elimination, but in the answers it says x=(de-bf)/(ad-bc). Why is the coefficient of y there?

For part B I believe ad=/=bc because if ad=bc, then the x and y of both equations would have the same coefficients meaning it's the exact same equation or same slope but parallel and never intersecting.

You need to show your work---how you got the answer, not just the answer itself.

RGV

 

[ilovecake]

using elimination by subtraction, solving for x:
ax+by=e
cx+dy=f

(ax+by)-(cx+dy)=(e-f)
ax-cx +by-dy=(e-f)
(a-c)x=(e-f)
x=(e-f)/(a-c)

Nevermind, I just realized that it's to make the coefficients of y the same.

(ad)x+[STRIKE](bd)y[/STRIKE]=(de)
(cb)x+[STRIKE](bd)y[/STRIKE]=(bf)