A system of equations is shown.
$\left\{\begin{array}{c}
y^2+x^2=20 \\
y=-x^2
\end{array}\right.$
Plot the real solutions to the system of equations.
Plot each point on the coordinate grid.
Answer (1)
Substitute y = − x 2 into y 2 + x 2 = 20 to get x 4 + x 2 − 20 = 0 .
Let u = x 2 , then solve the quadratic equation u 2 + u − 20 = 0 , which factors to ( u + 5 ) ( u − 4 ) = 0 . Thus, u = − 5 or u = 4 .
Since x is real, x 2 cannot be negative, so x 2 = 4 , which gives x = − 2 or x = 2 .
Substitute x values into y = − x 2 to find corresponding y values: y = − ( − 2 ) 2 = − 4 and y = − ( 2 ) 2 = − 4 . The solutions are ( − 2 , − 4 ) , ( 2 , − 4 ) .
Explanation
Understanding the Problem We are given the system of equations:
{ y 2 + x 2 = 20 y = − x 2
Our objective is to find the real solutions (x, y) to this system and plot them on a coordinate grid.
Substitution Substitute the second equation, y = − x 2 , into the first equation to eliminate y. This gives us:
( − x 2 ) 2 + x 2 = 20
Simplifying, we get:
x 4 + x 2 = 20
x 4 + x 2 − 20 = 0
Solving for u Let u = x 2 . Then the equation becomes:
u 2 + u − 20 = 0
This is a quadratic equation in u. We can solve it by factoring:
( u + 5 ) ( u − 4 ) = 0
So, u = − 5 or u = 4 .
Solving for x Since u = x 2 , we have x 2 = − 5 or x 2 = 4 .
For x 2 = − 5 , there are no real solutions for x, because the square of a real number cannot be negative.
For x 2 = 4 , we have x = − 2 or x = 2 .
Solving for y Now, substitute the values of x back into the equation y = − x 2 to find the corresponding y values.
If x = − 2 , then y = − ( − 2 ) 2 = − 4 .
If x = 2 , then y = − ( 2 ) 2 = − 4 .
So, the real solutions are ( − 2 , − 4 ) and ( 2 , − 4 ) .
Final Answer The real solutions to the system of equations are ( − 2 , − 4 ) and ( 2 , − 4 ) . These are the points we need to plot on the coordinate grid.
Examples
Systems of equations are used in various real-world applications, such as modeling supply and demand in economics, determining the trajectory of objects in physics, and designing structures in engineering. For example, if you want to optimize the dimensions of a rectangular garden with a fixed perimeter and a constraint on the area, you can set up a system of equations to find the optimal length and width. Similarly, in electrical engineering, systems of equations are used to analyze circuits and determine the currents and voltages at different points.
