Solving a System by Substitution In Exercises 7-14, solve...

Solving a System by Substitution In Exercises
$7-14,$ solve the system by the method of substitution.
Check your solution(s) graphically.
$$\left\{\begin{array}{r}{x^{2}+y=0} \\ {x^{2}-4 x-y=0}\end{array}\right.$$

Key Concepts

System of Equations

A system of equations consists of two or more equations that are solved together, meaning that solutions must satisfy every equation in the system. In algebra, these systems often represent the intersection points of curves or lines, and understanding their structure is fundamental to finding common solutions.

Substitution Method

The substitution method is an algebraic technique in which one equation is solved for one variable, and this expression is substituted into the other equation(s). This process reduces the system to a single equation with one variable, making it easier to solve step-by-step before back-substituting to find the remaining variable(s).

Quadratic Equations

Quadratic equations are second-degree polynomial equations, typically expressed in the form ax² + bx + c = 0. Solving quadratic equations is a key aspect of algebra, and methods such as factoring, the quadratic formula, or completing the square are applied. They often appear in systems where one or more equations describe parabolic curves.

Graphical Verification

Graphical verification involves plotting the equations on a coordinate plane to visually identify their points of intersection, which correspond to the solutions of the system. This method serves as an additional check to ensure that the algebraic solutions are accurate and consistent with the original equations.

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