Based on the type of equations in the system, what is the greatest possible number of solutions?
$\left\{\begin{array}{l}
x^2+y^2=9 \\
9 x+2 y=16
\end{array} ?\right.$
1
2
3
4
Answer (2)
The greatest possible number of solutions for the system of equations is 2, as the quadratic formed has a positive discriminant, allowing for two distinct real solutions. This means the line intersects the circle at two points. Thus, the answer is 2.
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Substitute y from the linear equation into the circle equation.
Simplify the equation to a quadratic form: 85 x 2 − 288 x + 220 = 0 .
Calculate the discriminant: 0"> D = ( − 288 ) 2 − 4 × 85 × 220 = 8144 > 0 .
Since 0"> D > 0 , there are two distinct real solutions. 2
Explanation
Problem Analysis The problem asks for the greatest possible number of solutions for the system of equations:
x 2 + y 2 = 9 (Equation of a circle) 9 x + 2 y = 16 (Equation of a line)
Solving for y To find the number of solutions, we can solve the system of equations. We can solve the linear equation for y in terms of x and substitute it into the equation of the circle.
Expressing y in terms of x From the second equation, we have:
2 y = 16 − 9 x
y = 8 − 2 9 x
Substituting and simplifying Substitute this expression for y into the first equation:
x 2 + ( 8 − 2 9 x ) 2 = 9
x 2 + ( 64 − 72 x + 4 81 x 2 ) = 9
Multiply by 4 to eliminate the fraction:
4 x 2 + 256 − 288 x + 81 x 2 = 36
85 x 2 − 288 x + 220 = 0
Analyzing the Quadratic Equation This is a quadratic equation in x . The number of real solutions for x will determine the number of intersection points between the circle and the line. A quadratic equation can have at most two real solutions.
Calculating the Discriminant The discriminant of the quadratic equation a x 2 + b x + c = 0 is given by D = b 2 − 4 a c . In this case, a = 85 , b = − 288 , and c = 220 .
D = ( − 288 ) 2 − 4 ( 85 ) ( 220 ) = 82944 − 74800 = 8144
Determining the Number of Solutions Since the discriminant 0"> D = 8144 > 0 , the quadratic equation has two distinct real solutions for x . Each real solution for x will correspond to a unique value of y . Therefore, the system has two solutions.
Final Answer The greatest possible number of solutions to the system is 2.
Examples
Consider a scenario where you are tracking the path of a satellite orbiting the Earth. The orbit can be modeled by a circle, and a ground-based radar can be represented by a line. The points where the radar intersects the satellite's orbit are the solutions to the system of equations. Knowing the maximum number of possible intersection points helps in predicting the number of times the satellite will be detected by the radar during its orbit. This has practical applications in satellite tracking and communication systems.
