In Exercises 29–42, solve each system by the method of your choice. x^2+(y−2)^2=4, x^2−2y=0

Verified step by step guidance

1. Identify the equations given in the system: Equation 1 is a circle equation, $x^2+(y-2{)}^2=4$, and Equation 2 is a parabolic equation, $x^2-2y=0$.
2. Express $x^2$ from Equation 2: Since $x^2-2y=0$, you can rearrange it to $x^2=2y$.
3. Substitute $x^2$ from Equation 2 into Equation 1: Replace $x^2$ in the circle equation with $2y$ from the parabolic equation, resulting in $(2y)+(y-2{)}^2=4$.
4. Simplify and solve the resulting equation for $y$: Expand and combine like terms in the equation $2y+(y-2{)}^2=4$ to form a quadratic equation in terms of $y$.
5. Substitute the values of $y$ back into $x^2=2y$ to find corresponding $x$ values: For each solution for $y$, substitute back into $x^2=2y$ to solve for $x$.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations that share the same variables. The goal is to find the values of these variables that satisfy all equations simultaneously. In this case, we have one nonlinear equation (a circle) and one linear equation, which can be solved using various methods such as substitution or elimination.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically in the form ax^2 + bx + c = 0. They can represent parabolas when graphed. In the given system, the first equation represents a circle, which can be rewritten in standard form to identify its properties, while the second equation is linear, making it easier to analyze intersections.

Graphical Interpretation

Graphical interpretation involves visualizing equations on a coordinate plane to understand their relationships. The first equation describes a circle centered at (0, 2) with a radius of 2, while the second equation is a straight line. Solving the system graphically can provide insights into the number of solutions, which correspond to the points where the graphs intersect.