In Exercises 19–28, solve each system by the addition method. $\(\begin{cases}\)y^2 - x = 4 \(\x\)^2 + y^2 = 4\(\end{cases}\)$

Verified step by step guidance

1. Rewrite the first equation \(y^2 - x = 4\) to express \(x\) in terms of \(y\): add \(x\) to both sides and subtract 4 from both sides to get \(x = y^2 - 4\).
2. Substitute the expression for \(x\) from the first equation into the second equation \(x^2 + y^2 = 4\) to eliminate \(x\) and have an equation in terms of \(y\) only.
3. After substitution, the second equation becomes \((y^2 - 4)^2 + y^2 = 4\). Expand the squared term \((y^2 - 4)^2\) carefully using the formula \((a - b)^2 = a^2 - 2ab + b^2\).
4. Simplify the resulting equation to form a polynomial equation in \(y\), combining like terms and setting the equation equal to zero.
5. Solve the polynomial equation for \(y\) (this may involve factoring or using the quadratic formula), then substitute each \(y\) value back into \(x = y^2 - 4\) to find the corresponding \(x\) values, giving the solution pairs \((x, y)\).

Key Concepts

Systems of Equations

A system of equations consists of two or more equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to interpret and manipulate these systems is essential for finding their solutions.

Addition Method (Elimination Method)

The addition method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the remaining variable. This technique requires aligning terms and sometimes multiplying equations to facilitate elimination.

Handling Nonlinear Equations

Nonlinear equations include variables raised to powers other than one, such as squares. Solving systems with nonlinear equations often requires substitution or elimination combined with algebraic manipulation to isolate variables and find all possible solutions.