Answer each of the following. When appropriate, fill in the blank to correctly complete the sentence. If we want to solve the following nonlinear system by substitution and we decide to solve equation (2) for y, what will be the resulting equation when the substitution is made into equation (1)? x^2 + y = 2 (1) x - y = 0 (2)

Verified step by step guidance

1. Solve equation (2) for y in terms of x. Since x - y = 0, you can rearrange this to express y as y = x.
2. Substitute the expression for y from step 1 into equation (1). Replace y in equation (1) with x, resulting in x^2 + x = 2.
3. Rearrange the substituted equation to set it to zero, which is a standard form for solving quadratic equations. This results in x^2 + x - 2 = 0.
4. Factorize the quadratic equation obtained in step 3. Look for two numbers that multiply to -2 and add up to 1.
5. Solve the factorized equation for x by setting each factor equal to zero and solving for x.

Key Concepts

Nonlinear Systems of Equations

A nonlinear system of equations consists of two or more equations where at least one equation is not linear. In this case, the first equation, x^2 + y = 2, is nonlinear due to the x^2 term. Understanding how to manipulate these equations is crucial for finding solutions that satisfy all equations in the system.

Substitution Method

The substitution method is a technique used to solve systems of equations by expressing one variable in terms of another and substituting it into the other equation. In this problem, we are instructed to solve equation (2) for y, which allows us to replace y in equation (1) with an equivalent expression, simplifying the system.

Solving for a Variable

Solving for a variable involves isolating that variable on one side of the equation. In this case, solving equation (2), x - y = 0, for y gives us y = x. This step is essential as it provides a direct relationship between x and y, which can then be substituted into the other equation to find the values of both variables.