Solve the following system of equations:
{x+y=61+6xy=9
To solve this problem, we will follow these steps:
- Step 1: Identify the equations and express one variable in terms of the other.
- Step 2: Substitute into the other equation and simplify.
- Step 3: Perform calculations to solve for the variable.
- Step 4: Use the solution to find the second variable.
Let's work through the solution together:
Step 1: Given xy=9, express y as x9.
Step 2: Substitute into the first equation:
x+x9=61+6.
Step 3: Simplify this equation. Let a=x and b=y.
Then, a+b=61+6 and ab=9=3.
Squaring both sides of the linear equation:
(a+b)2=61+6.
a2+2ab+b2=61+6.
Using ab=3, we get 2ab=6.
This leads to a2+b2=61.
Replacing a=x and b=y:
Let a2=x and b2=y and use the identity (a−b)2=a2+b2−2ab=61−6.
So, a−b=61−6.
Now let S=a+b and P=ab from previous steps.
From S=61+6 and P=3, solve: t2−St+P=0.
This quadratic in t gives solutions t=2S±S2−4P.
The quadratic roots are a=261+6±25 and b=261+6∓25.
Thus, x=a2=(261+2.5)2 or (261−2.5)2.
Similarly for y.
Therefore, the solutions are:
x=261−2.5, y=261+2.5
or
x=261+2.5, y=261−2.5.
x=261−2.5
y=261+2.5
or
x=261+2.5
y=261−2.5