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In algebra, a system of equations is a set of two or more equations involving the same set of variables. The goal is to find the values of these variables that satisfy all equations in the system simultaneously. There are several methods to solve systems of equations, including graphing, substitution, elimination, and matrix methods. The substitution method, which we focus on here, involves expressing one variable in terms of the others and then substituting this expression into another equation.

The key to using the substitution method effectively involves isolating one variable in one equation and substituting it into another, which reduces the system of equations to a single equation in one variable. This approach can be applied to linear and nonlinear systems. For instance, in our example we replaced the variable y in one of the linear equations with an expression derived from a quadratic equation, combining the two to find a common solution.

A quadratic equation is a second-degree polynomial equation in a single variable x, with a general form of ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to a quadratic equation are called the roots of the equation and can be found in various ways, such as factoring, completing the square, using the quadratic formula, or graphing.

In the given exercise, the quadratic equation appears as part of a system of equations. After substituting and simplifying, we were able to find factors of the quadratic expression, which allowed us to find the variable x values that satisfy both the linear and quadratic equations in the system. Understanding how to manipulate and solve quadratic equations is essential for solving many algebraic problems.

When solving equations step by step, the approach should be systematic to ensure accuracy and understanding of each stage.

Steps typically involve:

• Identifying the type of equation: Is it linear, quadratic, exponential, etc.?
• Isolating the variable: Aim to get the unknown on one side of the equation, often using operations that reverse the operation affecting the variable.
• Simplifying the equation: Combine like terms and perform operations to consolidate the equation into the simplest form.
• Solving for the variable: Use the appropriate method based on the type of equation.
• Checking solutions: Substitute the solution back into the original equation to verify its correctness.

For the problem at hand, substitution turned our system of equations into a simpler quadratic equation, which we could factor and solve for x, followed by substituting the found values into the original equation to solve for y. This methodical process makes complex problems more manageable and helps develop problem-solving skills.