The problem involves solving the simultaneous equations:

1. $y = x^2$
2. $y - 5x + 4 = 0$

Step-by-Step Solution:

Step 1: Substitution

From the first equation, $y = x^2$. Substitute this into the second equation:

$x^2 - 5x + 4 = 0$

Step 2: Solve the quadratic equation

The quadratic equation is:

$x^2 - 5x + 4 = 0$

Factorize this equation:

$(x - 4)(x - 1) = 0$

So, $x = 4$ or $x = 1$.

Step 3: Solve for $y$

Using $y = x^2$:

• When $x = 4$: $y = 4^2 = 16$

• When $x = 1$: $y = 1^2 = 1$

Final Solution:

The solutions to the simultaneous equations are:

1. $(x, y) = (4, 16)$
2. $(x, y) = (1, 1)$

Would you like further details or a graphical representation?

Here are 5 related questions:

1. How do we graphically interpret the solution to simultaneous equations involving a parabola and a line?
2. How do we check if a solution satisfies both equations?
3. What if the parabola and line did not intersect? How would that affect the solution?
4. How can the discriminant be used to determine the number of solutions for quadratic equations?
5. Can this problem be solved using matrices or numerical methods?

Tip: Always substitute your solutions back into the original equations to verify correctness.