Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 67

Chapter 3, Problem 67

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

Identify the two given equations: the first is a circle equation \(x^{2} + y^{2} = 16\), and the second is a linear equation \(x - y = 4\).

From the linear equation \(x - y = 4\), solve for one variable in terms of the other. For example, express \(x\) as \(x = y + 4\).

Substitute the expression for \(x\) from the linear equation into the circle equation. This gives \( (y + 4)^{2} + y^{2} = 16 \).

Expand and simplify the resulting equation to form a quadratic equation in terms of \(y\). Then solve this quadratic equation to find the possible \(y\)-values.

Use the \(y\)-values found to calculate the corresponding \(x\)-values using \(x = y + 4\). These \((x, y)\) pairs are the points of intersection. Finally, verify each point by substituting back into both original equations.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Graphing Circles

The equation x² + y² = 16 represents a circle centered at the origin with radius 4. Understanding how to graph this circle involves plotting all points (x, y) that satisfy the equation, which lie exactly 4 units from the origin in all directions.

Graphing Linear Equations

The equation x - y = 4 is a linear equation representing a straight line. To graph it, rewrite it in slope-intercept form (y = x - 4) and plot points accordingly. This line will intersect the circle at points where both equations hold true.

Finding Points of Intersection

Points of intersection satisfy both equations simultaneously. To find them, substitute the linear equation into the circle's equation and solve for x and y. These solutions correspond to the coordinates where the circle and line cross on the graph.

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