Problem Breakdown:

We are tasked with finding the points of intersection between the curve and the line for $k = 8$. Let's break it down step by step.

Given Equations:

1. The equation of the curve is: $y^2 + 2x = 13$
2. The equation of the line is: $2y + x = k$ For $k = 8$, the line becomes: $2y + x = 8 \quad \text{or equivalently,} \quad x = 8 - 2y.$

Step 1: Substitute $x = 8 - 2y$ into the curve equation

From the curve equation: $y^2 + 2x = 13,$ substitute $x = 8 - 2y$: $y^2 + 2(8 - 2y) = 13.$ Simplify: $y^2 + 16 - 4y = 13.$ Rearrange: $y^2 - 4y + 3 = 0.$

Step 2: Solve the quadratic equation for $y$

The equation is: $y^2 - 4y + 3 = 0.$ Factorize: $(y - 3)(y - 1) = 0.$ Thus: $y = 3 \quad \text{or} \quad y = 1.$

Step 3: Find corresponding $x$-coordinates

Using $x = 8 - 2y$:

• For $y = 3$: $x = 8 - 2(3) = 8 - 6 = 2.$ So, one intersection point is $(2, 3)$.
• For $y = 1$: $x = 8 - 2(1) = 8 - 2 = 6.$ So, the other intersection point is $(6, 1)$.

Final Answer:

The points of intersection are: $(2, 3) \quad \text{and} \quad (6, 1).$

Would you like me to explain any step in more detail, or do you have any questions?

Related Questions:

1. What happens to the points of intersection if $k \neq 8$?
2. How do you determine whether the curve and line intersect for all values of $k$?
3. How can the quadratic equation be solved using the quadratic formula instead of factoring?
4. What is the geometric interpretation of these intersection points?
5. How can we confirm the solutions by substituting the points back into the original equations?

Tip:

When solving problems involving intersections of curves and lines, always substitute one equation into the other to eliminate a variable and simplify systematically.