College Algebra

[ryan_kalle]

Assignment 4

Find a formula to solve the system:

ax + by = c

dx + ey = f

where a,b,c,d,e and f are real numbers

Im a little confused on this one.
Our assignments are usually an application of previous material on our last test...sometimes.
We dealt with classifying and solving systems of equations using the addition/subraction and substitution method and finding unknowns when the product or sum is known. I'm not sure on how to solve this system with the current data.

 

[Ryan Rigdon]

Assignment 4

Find a formula to solve the system:

ax + by = c

dx + ey = f

where a,b,c,d,e and f are real numbers

Im a little confused on this one.
Our assignments are usually an application of previous material on our last test...sometimes.
We dealt with classifying and solving systems of equations using the addition/subraction and substitution method and finding unknowns when the product or sum is known. I'm not sure on how to solve this system with the current data.

well its been a long time since i have tutored college algebra. anyway i took the given information and found formula's for x and y only. i hope that this helps you in some way or at least points you in the right direction. and if what i have shown is not correct way i apologize i am sorry. lets see what everyone else thinks. here is what i came up with. peace out.

 

[ryan_kalle]

Wouldn't it be x((ae-bd)/e) = (ce-bf)/d rather than x((ae+bf)/e) = (ce-bf)/d?

 

[Deleted member 4993]

Assignment 4

Find a formula to solve the system:

ax + by = c

dx + ey = f

where a,b,c,d,e and f are real numbers

Im a little confused on this one.
Our assignments are usually an application of previous material on our last test...sometimes.
We dealt with classifying and solving systems of equations using the addition/subraction and substitution method and finding unknowns when the product or sum is known. I'm not sure on how to solve this system with the current data.

For a quick review - go to

http://www.purplemath.com/modules/systlin1.htm

 

[Denis]

ax + by = c : x = (c - by) / a [1]
dx + ey = f : x = (f - ey) / d [2]

So [1] = [2]:
(c - by) / a = (f - ey) / d
Simplify:
y = (af - cd) / (ae - bd) [3]

Get y using [3]
Get x using [1] or [2]

Hope that helps...

 

[ryan_kalle]

ax+by=c
-Subtract "by" from both sides
ax=c-by
-Divide boths sides by "a"
x=(c-by)/a

dx+cy=f
-subreact "ey" from both sides
dx=f-ey
-Divide both sides by "d"
x=(f-ey)/d
-Both equations now equal "x". Set them equal to one another
(c-by)/a = (f-ey)/d
-Solve for "y"
d(c-by)=a(f-ey)
cd-bdy=af-aey
aey-bdy=af-cd
y(ae-bd)=af-cd
y=(af-cd)/(ae-bd)

Solution
x=(c-by)/a or (f-ey)/d
y=(af-cd)/(ae-bd)

I wanted to test it and I dont know if I went about it correctly.
I set values to a-f
a=1
b=2
c=3
d=4
e=5
f=6

Then pluged them into the equations
y=[1(6)-3(4)]/[1(5)-2(4)]=2
x=[3-2(2)]/1=-1
x=[6-5(2)]/4=-1

Then I pluged in the values for x and y in the original equations and got this

1(-1)+2(2)=3
-1+4+3
3=3
True

4(-1)+5(2)=6
-4+10=6
6=6
True

Is that the proper way to prove them?

 

[mmm4444bot]

Our assignments are usually an application of previous material

We dealt with classifying and solving systems of equations using the addition/subraction and substitution method

Aha! This exercise IS an application of your previous material because it is a system of equations that can be solved using the Substitution Method.

One of the hardest things for algebra students is learning how to think symbolically (i.e., abstractly). It takes a lot of mental practice.

 

[mmm4444bot]

x = (c-by)/a or (f-ey)/d

y = (af-cd)/(ae-bd)

I wanted to test [these results]

I set values

a=1
b=2
c=3
d=4
e=5
f=6

Is that the proper way to prove them?

If you used the pronoun "them" to mean those specific values that you assigned, then the answer to your question is yes.

The confirmation that you got only applies to those specific values, however.

If you want a confirmation for all possibilities, then you need to work symbolically.



If you're interested in the solution itself (versus a "formula" to find the solution), then the expression for x should not be a variable expression; it should be a constant. In other words, the symbol y should not appear in the solution for x.

If we substitute Denis' result for y into each of the "formula" results for x, and simplify, we find that both results are exactly the same expression.

The solution to the posted system is:

x = (ce - bf)/(ae - bd)

y = (af - cd)/(ae - bd)



Now, if you would like to confirm this solution for all possible values of the parameters a through f, then substitute the expressions for x and y into the given equations, and simplify. (That will be good practice, for you.)

The first equation simplifies to c = c.

The second equation simplifies to f = f.