To solve the system of equations:

$y = x^2 + 4$ $y = 2x + 4$

we can use the substitution method by setting the two expressions for $y$ equal to each other since they are both equal to $y$.

Step 1: Set the Equations Equal to Each Other

$x^2 + 4 = 2x + 4$

Step 2: Simplify the Equation

Subtract $2x + 4$ from both sides: $x^2 - 2x = 0$

Step 3: Factor the Equation

We can factor out $x$: $x(x - 2) = 0$

Step 4: Solve for $x$

This gives two possible solutions for $x$:

1. $x = 0$
2. $x = 2$

Step 5: Substitute $x$ Back into Either Original Equation to Find $y$

1. For $x = 0$: $y = 2(0) + 4 = 4$ So, one solution is $(0, 4)$.

2. For $x = 2$: $y = 2(2) + 4 = 8$ So, another solution is $(2, 8)$.

Final Answer

The solutions to the system are: $(0, 4) \quad \text{and} \quad (2, 8)$

Would you like a detailed explanation of each step or have any further questions?

Here are five related questions that expand on this topic:

1. How do we verify if these points are indeed solutions to the system?
2. What method would we use if both equations were nonlinear?
3. Can this system have more than two solutions? Why or why not?
4. How would the solution process change if the equations were both quadratic?
5. How do we interpret the solution geometrically in terms of graph intersections?

Tip: Always check your solutions by substituting them back into the original equations to ensure they satisfy both.