x+y=4
xy=16
(system of equations, variables, algebra)
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Step-by-Step Solution
Step 1: Express one variable in terms of the other
We are given two equations: x+y=4 and xy=16. To solve this system, we can use the substitution method. From the first equation, x+y=4, we can express y in terms of x by subtracting x from both sides.
Step 2: Substitute into the second equation
Now, substitute the expression for y from the first step into the second equation, xy=16. This will give us an equation with only one variable, x.
Step 3: Form a quadratic equation
Distribute x in the equation x(4−x)=16 to get 4x−x2=16. Rearrange this into the standard quadratic form ax2+bx+c=0 by moving all terms to one side, resulting in x2−4x+16=0.
Step 4: Calculate the discriminant
To find the solutions for x, we can use the quadratic formula. First, we calculate the discriminant, Δ=b2−4ac. For the equation x2−4x+16=0, we have a=1, b=−4, and c=16. Substituting these values, we get Δ=(−4)2−4(1)(16)=16−64=−48.
Step 5: Determine the nature of the roots
Since the discriminant Δ=−48 is less than zero, the quadratic equation x2−4x+16=0 has no real roots. This means there are no real values of x and y that satisfy both given equations simultaneously.