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)?
x² + y = 2 (1)
x - y = 0 (2)

Verified step by step guidance

1. Identify the system of equations: (1) \(x^2 + y = 2\) and (2) \(x - y = 0\).
2. Solve equation (2) for \(y\): since \(x - y = 0\), rearrange to get \(y = x\).
3. Substitute the expression for \(y\) from equation (2) into equation (1): replace \(y\) in \(x^2 + y = 2\) with \(x\).
4. After substitution, the resulting equation becomes \(x^2 + x = 2\).
5. This new equation in terms of \(x\) can now be solved to find the values of \(x\), which can then be used to find \(y\).

Key Concepts

Solving Systems of Equations by Substitution

Substitution involves solving one equation for one variable and then replacing that variable in the other equation. This method reduces the system to a single equation with one variable, making it easier to solve. It is especially useful when one equation is easily solved for one variable.

Isolating a Variable

Isolating a variable means rewriting an equation so that one variable is expressed explicitly in terms of the others. For example, solving equation (2) for y means rewriting it as y = x. This step is crucial for substitution because it provides the expression to replace the variable in the other equation.

Substituting Expressions into Equations

After isolating a variable, substitution involves replacing that variable in the other equation with the expression found. For instance, substituting y = x into equation (1) transforms it into an equation with only x, such as x^2 + x = 2, which can then be solved using algebraic methods.