Question 143797: Hi, can someone help me with this problem?
Solve for x and y: 2x + 3y = 1
5x + 3y = 16
Thanks. Answer by nabla(475) (Show Source): You can put this solution on YOUR website! Let's have a little fun with this problem. Let's generalize a formula for 2 equations in 2 variables!
consider:
ax+by=c
dx+ey=f
let's solve each for y!
y=(c-ax)/b
y=(f-dx)/e
Now, we need to find where they are equal.
(f-dx)/e=(c-ax)/b
This gives:
b(f-dx)=e(c-ax)
and thus
bf-bdx=ec-eax
eax-bdx=ec-bf
x=(ec-bf)/(ea-bd)
moreover y=(c-a(ec-bf)/(ea-bd))/b.
It is very constructive to do exercises like this to build understanding.
In any case, let's apply the formulae to your equations:
x=(3*1-3*16)/(3*2-3*5)=(3-48)/(6-15)=-45/-9=5 !!!
Thus y=(c-ax)/b=(1-2(5))/3=-3.
Giving us solution set (5,-3).
Now, something more in keeping with what your instructor will want:
2x + 3y = 1
5x + 3y = 16
subtract the second equation from the first
-3x=-15
gives x=5
put x=5 into the first equation
2(5)+3y=1
3y=-9
y=-3
Another way would be through substitution:
Solve: We'll use substitution. After moving 3*y to the right, we get: , or . Substitute that
into another equation: and simplify: So, we know that y=-3. Since , x=5.
Answer: .
If you have any questions, now or in the future, send me an E-mail at enabla@gmail.com.