Skip to main content
Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 2 - Functions and GraphsProblema 67
Capitolo 3, Problema 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

Guida verificata passo dopo passo
1
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.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
4m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.
Video consigliato:
5:18
Circles in Standard Form

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.
Video consigliato:
06:00
Categorizing Linear Equations

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.
Video consigliato:
04:27
Finding Equations of Lines Given Two Points
Pratica correlata
Domanda del libro di testo

In Exercises 59-66, a. Rewrite the given equation in slope-intercept form. b. Give the slope and y-intercept. c. Use the slope and y-intercept to graph the linear function. 4y+ 28 = 0

107
views
Domanda del libro di testo

Use intercepts to graph each equation. 6x-9y-18 = 0

1212
views
Domanda del libro di testo

Begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = √x + 1

775
views
Domanda del libro di testo

Begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. h(x) = -2(x+2)²+1

711
views
Domanda del libro di testo

In Exercises 65–70, use the graph of f to find each indicated function value. f(4)

607
views
Domanda del libro di testo

In Exercises 67-74, find a. (fog) (x) b. the domain of f o g. f(x) = 2/(x+3), g(x) = 1/x

1678
views