Final answer:
To solve the system of equations y = x + 4 and y = x² + 2, we can use substitution method or elimination method. Let's use the substitution method. Rearrange the equation, solve the quadratic equation, and substitute the values of y back into the equation to solve for x. We get (x, y) = (3, 7) or (-2, 2).
Step-by-step explanation:
To solve the system of equations y = x + 4 and y = x² + 2, we can use substitution method or elimination method. Let's use the substitution method.
Step 1: Rewrite one of the equations in terms of one variable. We can rewrite the first equation as x = y - 4.
Step 2: Substitute the value of x from Step 1 into the other equation. We get y = (y - 4)² + 2.
Step 3: Simplify the equation and solve for y. Expanding and simplifying the equation, we get y = y² - 8y + 14.
Step 4: Solve the quadratic equation. Rearrange the equation as y² - 9y + 14 = 0. Taking factors of the quadratic, we have (y - 7)(y - 2) = 0.
Step 5: Solve for y. Set each factor equal to zero, so y - 7 = 0 or y - 2 = 0. Solving for y, we find y = 7 or y = 2.
Step 6: Substitute the values of y back into the equation x = y - 4 to solve for x. When y = 7, x = 7 - 4 = 3. When y = 2, x = 2 - 4 = -2.
Therefore, the solution to the system of equations is (x, y) = (3, 7) or (-2, 2).