SOLUTION: x-y=5; x+y=3

Question 130367: x-y=5; x+y=3
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Write these two equations so one is directly above the other as follows:
.
x - y = 5
x + y = 3
.
Notice now that the two y terms (one in the top equation and one in the bottom) are equal
but opposite in sign. Therefore, if you add the two equations vertically in columns you
will add -y and +y and they will cancel out.
.
Adding the two equations vertically results in the two +x terms adding to 2x, the two y terms
canceling each other because they are equal but of opposite signs, and, on the right side,
the 5 and the 3 add to 8. So what you are left with as the sum is:
.
2x = 8
.
Solve this equation by dividing both sides by 2 because 2 is the multiplier of the x term.
When you divide both sides by 2 the equation becomes:
.
x = 4
.
Now you know the value of x. You can return to either of the two original equations
and substitute 4 for x. Then solve for y. Let's return to the equation:
.
x + y = 3
.
Substitute 4 for x and you get:
.
4 + y = 3
.
Get rid of the 4 on the left side by subtracting 4 from both sides and you have:
.
y = 3 - 4 = -1
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So you have found the common solution for both equations as x = 4 and y = -1. If you graph
the two equations you will find that their graphs cross at the (x, y) point (4, -1).
.
You can check this answer by going to each of the two original equations, substituting
4 for x and -1 for y, and seeing if both sides of the equation are still equal.
.
Go to the original top equation:
.
x - y = 5
.
Replace x with 4 and y with -1 and you have:
.
4 - (-1) = 5
.
Which becomes:
.
4 + 1 = 5
.
That's true. So next go to the bottom of the original equations:
.
x + y = 3
.
Replace x with 4 and y with -1 and you have:
.
4 + (-1) = 3
.
Which becomes:
.
4 - 1 = 3
.
And that is true also. Therefore, our answer is correct. When x = 4 and y = -1 both equations
are true, therefore this is a common solution.
.
Hope this helps you to understand the problem a little better.
.
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