Simultaneous Equations?

1) X + Y = 3
2) 6X - Y = 7

addition of both sides:
7x = 10
x = 10/7

x in 1') y = 3 - x = 21/7 - 10/7 = 11/7


OR
1') y = 3 - x
2') y = 6x - 7

3 - x = 6x - 7
10 = 7x
x = 10/7

x in 1') y = 3 - 10/7 = 21/7 - 10/7 = 11/7


OR
1') y = 3 - x
y in 2) 6x -(3 - x) = 7
6x - 3 + x = 7
7x = 10
x = 10/7

x in 1') y = 3 - 10/7 = 21/7 - 10/7 = 11/7

Value of x:
3 - x = 6x - 7
7x = 10
x = 10/7

Value of y:
= 3 - 10/7
= 21/7 - 10/7
= 11/7

Answer: x = 10/7, y = 11/7

Check (1st equation):
10/7 + 11/7 = 3
21/7 = 3
3 = 3

Check (2nd equation):
6(10/7) - 11/7 = 7
60/7 - 11/7 = 7
49/7 = 7
7 = 7

1: x+y = 3...........................................Multiply by- 6
2: 6x+y =7

-6x-6y =-18
6x + y = 7
--------------------------------
-5y = -11

y=11/5

x= 3-y -----------------------------as per equation 1

x = 3 -11/5
x= 4/5

x + y = 3
x = 3 - y
6x = 6(3 - y)

6x - y = 7
6(3 - y) - y = 7
18 - 6y - y = 7
-7y = 7 - 18
-7y = -11
y = -11/-7
y = 11/7

x + y = 3
x + 11/7 = 3
x = 3 - 11/7
x = 21/7 - 11/7
x = 10/7

∴ x = 10/7 , y = 11/7

I agree with sirpetethegreat..=)

Perhaps the easiest way to do this is to use substitution, that is, solve one equation for one variable and substitute that into the other equation. The easier equation is obviously the top one. So solving for Y, we get:

Y = 3 - X

Since that is true, we can use that and substitute "3-X" for "Y" in the second equation, yielding:

6X - (3-X) = 7

From here, we solve for X and get:

6X -3 + X = 7
7X = 10
X = 10/7

We can now substitute "10/7" for "X" in either equation to find out Y. Use the first equation, it's easier.

10/7 + Y = 3
10/7 + Y = 21/7
Y = 11/7

And there you go. Not too much different from solving and getting whole number answers.

EDIT: Ok, after seeing the first answer, my way is not the easiest way, but you get the idea.

in your specific example I like to use the substitution method.

Arrange the first equation in terms of x: x = 3 - y
Then plug that in place of x in your other equation, multiply out, combine the terms, and solve for x. Now that we know the value of x, go back to the first equation and solve for y.

7X = 10
X = 10/7

10/7 + Y = 3
Y = 21/7 - 10/7
Y = 11/7

X = 10/7 , Y = 11/7

X + Y = 3
6X - Y = 7

Step 1: Transform X + Y = 3 into a new equation by isolating Y at the left side of the equation

X + Y = 3 (into)
Y = 3 - X

Step 2: Substitute X = 3 - Y in the equation, 6X - Y = 7

6X - Y = 7
6x - (3 - X) = 7
6x - 3 + X = 7
7x = 7 + 3
7X = 10
X = 10 / 7

Step 3: Substitute X = 10 / 7 in the equation, X + Y = 3

X + Y = 3
(10/7) + Y = 3
Y = 3 - 10/7
Y = 21/7 - 10/7
Y = (21 - 10) / 7
Y = 11 / 7

Note: 3 is just = 21 / 7


Final Answers:
X = 10 / 7
Y = 11 / 7