Answer:
Therefore, the solutions to the system of equations are:
x = 3 + √13 and y = -3 + √13
x = 3 - √13 and y = -3 - √13
These are the simplest surd forms of the solutions to the given system of equations.
Explanation:
To solve the system of equations x - y = 6 and xy = 4, we can use the method of substitution.
From the first equation, we can isolate x by adding y to both sides:
x = y + 6
Substituting this expression for x in the second equation, we get:
(y + 6)y = 4
Expanding the equation, we have:
y^2 + 6y = 4
Rearranging the equation, we get:
y^2 + 6y - 4 = 0
To solve this quadratic equation, we can use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = 6, and c = -4. Substituting these values into the quadratic formula, we get:
y = (-6 ± √(6^2 - 4(1)(-4))) / (2(1))
Simplifying the expression, we have:
y = (-6 ± √(36 + 16)) / 2
y = (-6 ± √52) / 2
y = (-6 ± 2√13) / 2
y = -3 ± √13
Now, we can substitute the values of y back into the first equation to find the corresponding values of x:
For y = -3 + √13, x = (-3 + √13) + 6 = 3 + √13
For y = -3 - √13, x = (-3 - √13) + 6 = 3 - √13