Title: How to Solve a System Involving a Circle and a Line – Substitution Method

This post demonstrates how to find the points of intersection between a line and a circle using algebra, specifically the substitution method. These types of systems appear often in Algebra 2 and Precalculus, especially in coordinate geometry units.

Given System:

{x−y=−2x2+y2−4y=4\begin{cases} x - y = -2 \\ x^2 + y^2 - 4y = 4 \end{cases}

Step 1: Isolate a Variable

From the first equation:
x – y = –2
Solve for y:
y = x + 2

Step 2: Substitute into the Second Equation

Plug y = x + 2 into the circle equation:

x2+(x+2)2−4(x+2)=4x^2 + (x + 2)^2 - 4(x + 2) = 4 x2+x2+4x+4−4x−8=4x^2 + x^2 + 4x + 4 - 4x - 8 = 4 2x2−4=4→2x2=8→x2=4→x=±22x^2 - 4 = 4 → 2x^2 = 8 → x^2 = 4 → x = ±2

Step 3: Plug Back to Find y

Use y = x + 2:

• If x = 2 → y = 4

• If x = –2 → y = 0

Final Answer:

Points of intersection:
P₁(–2, 0), P₂(2, 4)

This is an example from Chapter 6: Systems of Equations and Inequalities in the Ultimate Crash Course for STEM Majors, where you’ll learn both algebraic and graphical techniques for solving systems accurately on exams.

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