SOLUTION: Please help:
1. Solve for X and Y in the following problems. Make sure you show all your work.
a. X + Y=6 , 2X + Y = 8
b. 7X + 3Y = 14 , 5X + 9Y = 10
c. 4X + Y = 16
Algebra.Com
Question 174508: Please help:
1. Solve for X and Y in the following problems. Make sure you show all your work.
a. X + Y=6 , 2X + Y = 8
b. 7X + 3Y = 14 , 5X + 9Y = 10
c. 4X + Y = 16 , 2X + 3Y = 24
d. 12X + Y = 25 , 8X - 2Y = 14
for "a" I have x=3, y=2...I just did it in my head but I'm having trouble with what formula to use and how to arrive at the answer step by step
Found 2 solutions by checkley77, solver91311:
Answer by checkley77(12844) (Show Source): You can put this solution on YOUR website!
a. X + Y=6 OR X=6-Y
NOW REPLACE X IN THE SECOND EQUATION BY (6-Y) & SOLVE FOR Y.
2X + Y = 8
2(6-Y)+Y=8
12-2Y+Y=8
-Y=8-12
-Y=-4
Y=4 ans.
X=6-4
X=2 ans.
--------------------------
b.7X + 3Y = 14 MULTIPLY BY -3 & ADD TO THE SECOND EQUATION
5X + 9Y = 10
-21X-9Y=-42
------------------
-16X=-32
X=-32/-16
X=2 ans.
5*2+9Y=10
10+9Y=10
9Y=10-10
9Y=0
Y=0 ans.
-------------------------------
c. 4X + Y = 16 OR Y=16-4X
NOW REPLACE Y WITH (16-4X) IN THE SECOND EQUATION & SOLVE FOR X.
2X + 3Y = 24
2X+3(16-4X)=24
2X+48-12X=24
-10X=24-48
-10X=-24
X=-24/-10
X=2.4 ans.
4*2.4+Y=16
9.6+Y=16
Y=16-9.6
Y=6.4 ans.
d. 12X + Y = 25 OR Y=25-12X NOW REPLACE Y WITH (25-12X) IN THE SECOND EQUATION & SOLVE FOR X.
8X - 2Y = 14
8X-2(25-12X)=14
8X-50+24X=14
32X=14+50
32X=64
X=64/32
X=2 ans.
12*2+Y=25
24+Y=25
Y=25-24
Y=1 ans.
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
First thing to realize is that any solution must make BOTH equations true, and since 3 + 2 = 5, not 6, X=3 and Y=2 is NOT a solution to this system. Sorry, but your 'in my head' method failed.
Two ways to do these:
Substitution Method
Eq. 1:
Eq. 2:
Solve one of the equations for one of the variables in terms of the other. Eq. 1 is most convenient in this case, but that is simply a matter of personal taste. Properly done, it doesn't matter which.
Eq. 1:
Add -Y to both sides:
Now you have an expression for X in terms of Y, so Substitute (hence the name of the method) this expression for X into Eq. 2 in place of the X, thus:
Eq. 2: →
Now you have an equation in a single variable that can be solved by ordinary methods:
Now that we know that , we can substitute that value into either of the equations, let's use Eq. 1, and solve for the value of X.
→ →
Check:
and . Both equations are true statements for the values and so the solution set is the ordered pair (2,4).
Elimination Method (We'll use your problem b. to illustrate)
Eq. 1:
Eq. 2:
Step 1 is to multiply either or both of the equations by a constant or constants such that the coefficent on one of the variables in one of the equations becomes the additive inverse of the coefficient on the same variable in the other equation. Note that in this case if we multiply Eq. 1 by -3, the coefficient on Y in Eq. 1 becomes -9, which is the additive inverse of the coefficient (9) on Y in Eq. 2.
Eq. 3: (result of multiplying Eq. 1 by -3)
Eq. 2:
Step 2 is to add the resulting equations term by term:
Eq. 4: → . Notice that we have Eliminated the variable Y (hence the name of the method).
Step 3: Solve for X by ordinary means:
→
Step 4: You can proceed one of two ways. In this case the easiest thing to do is to substitute this determined value for X into either of the original equations and solve for Y. Alternatively, you could multiply Eq. 1 by -5 and Eq. 2 by 7 and then add the new equations term by term to eliminate X. Either way works, but substituting is going to require easier arithmetic in this case.
Eq. 2: → → →
So your solution set is the ordered pair (2,0).
Check:
→ Check
→ Check
Apply the given methods to your other problems and you should do fine. Remember: You MUST check your solution each time to make sure you haven't made an algebra or arithmetic error in the process of applying the solution method.
As to which method is the best, all I can say is that you will learn this by experience. As with anything else in life, there are three keys to success: Practice, Practice, and Practice. Good luck.
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