In Exercises 29–42, solve each system by the method of your choice. x^2+y^2+3y=22, 2x+y=−1

Verified step by step guidance

1. Step 1: Start by solving the second equation for y in terms of x. The equation is 2x + y = -1. Rearrange it to get y = -1 - 2x.
2. Step 2: Substitute the expression for y from Step 1 into the first equation. The first equation is x^2 + y^2 + 3y = 22. Replace y with (-1 - 2x) to get x^2 + (-1 - 2x)^2 + 3(-1 - 2x) = 22.
3. Step 3: Expand the equation from Step 2. Calculate (-1 - 2x)^2 and 3(-1 - 2x), then substitute these into the equation.
4. Step 4: Simplify the equation obtained in Step 3. Combine like terms to form a quadratic equation in terms of x.
5. Step 5: Solve the quadratic equation for x. Use the quadratic formula or factorization, if possible, to find the values of x. Once x is found, substitute back into y = -1 - 2x to find the corresponding y values.

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 a nonlinear equation (a circle) and a linear equation (a line), which can be solved using various methods such as substitution or elimination.

Substitution Method

The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. This method is particularly useful when one equation is easily solvable for a variable, allowing for a straightforward substitution that simplifies the system into a single equation with one variable.

Quadratic Equations

Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. In the given system, the first equation represents a circle, which can be rearranged into a standard quadratic form. Understanding how to manipulate and solve quadratic equations is essential for finding the intersection points of the given system.