In Exercises 1–18, solve each system by the substitution method. x+y=2, y=x²−4

Verified step by step guidance

1. Start with the given system of equations: $x+y=2$ and $y=x^2-4$.
2. From the first equation, solve for $y$ in terms of $x$: $y=2-x$.
3. Substitute this expression for $y$ into the second equation: replace $y$ with $2-x$ in $y=x^2-4$ to get $2-x=x^2$.
4. Rearrange the resulting equation to set it equal to zero: $x^2+x-6=\; 0$. This is a quadratic equation in standard form.
5. Solve the quadratic equation using factoring, completing the square, or the quadratic formula to find the values of $x$. Then substitute each $x$ value back into $y=2-x$ to find the corresponding $y$ values.

Key Concepts

System of Equations

A system of equations consists of two or more equations with the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. In this problem, the system includes a linear and a quadratic equation.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, making it easier to solve.

Solving Quadratic Equations

After substitution, the resulting equation may be quadratic, requiring methods like factoring, completing the square, or the quadratic formula to find solutions. These solutions correspond to the variable values that satisfy both original equations.