In Exercises 67–70, graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. x² + y² = 16, x-y = 4

Verified step by step guidance

1. <Step 1: Identify the equations.> The first equation is a circle: $x^2 + y^2 = 16$. The second equation is a line: $x - y = 4$.
2. <Step 2: Graph the circle.> The equation $x^2 + y^2 = 16$ represents a circle centered at the origin (0,0) with a radius of 4.
3. <Step 3: Graph the line.> The equation $x - y = 4$ can be rewritten as $y = x - 4$. This is a line with a slope of 1 and a y-intercept of -4.
4. <Step 4: Find points of intersection.> Substitute $y = x - 4$ into the circle's equation $x^2 + y^2 = 16$ to find the points where the line intersects the circle.
5. <Step 5: Verify the solutions.> Solve the resulting system of equations to find the intersection points, and check that these points satisfy both the original equations.

Key Concepts

Graphing Equations

Graphing equations involves plotting points on a coordinate system to visualize the relationship between variables. For the given equations, x² + y² = 16 represents a circle centered at the origin with a radius of 4, while x - y = 4 is a linear equation representing a straight line. Understanding how to graph these shapes is essential for identifying their points of intersection.

Points of Intersection

Points of intersection are the coordinates where two graphs meet on the coordinate plane. To find these points, one must solve the system of equations simultaneously. This involves substituting one equation into the other or using methods such as substitution or elimination to determine the values of x and y that satisfy both equations.

Verification of Solutions

Verification of solutions entails substituting the found points of intersection back into the original equations to confirm they satisfy both. This step is crucial to ensure that the identified points are indeed valid solutions to the equations. It reinforces the accuracy of the graphing and solving process, ensuring that the results are reliable.