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x + y = 3 x² + y? = 5

x + y = 3 x² + y? = 5

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Solve conic equations

The given image shows a system of equations involving two variables, \( x \) and \( y \). The equations are as follows: 1. \( x + y = 3 \) 2. \( x^2 + y^2 = 5 \) ### Explanation: 1. **Linear Equation: \( x + y = 3 \)** - This is a linear equation representing a straight line in a 2-dimensional coordinate system. Every pair \((x, y)\) that satisfies this equation lies on this line. 2. **Quadratic Equation: \( x^2 + y^2 = 5 \)** - This is an equation of a circle centered at the origin \((0,0)\) with a radius of \(\sqrt{5}\). ### Graphical Representation: - If plotted on a Cartesian plane, the line \( x + y = 3 \) will intersect the circle \( x^2 + y^2 = 5 \). The points of intersection are the solutions that satisfy both equations simultaneously. - These intersections represent the pairs \((x, y)\) that make both equations true at the same time. ### Solving the System of Equations: To find the exact points of intersection, one can solve the system algebraically or graphically by substituting and solving for the variables \( x \) and \( y \). Analyzing systems like this is fundamental in understanding how different types of equations can be used together to find common solutions, which is an essential concept in algebra and coordinate geometry.
Transcribed Image Text:The given image shows a system of equations involving two variables, \( x \) and \( y \). The equations are as follows: 1. \( x + y = 3 \) 2. \( x^2 + y^2 = 5 \) ### Explanation: 1. **Linear Equation: \( x + y = 3 \)** - This is a linear equation representing a straight line in a 2-dimensional coordinate system. Every pair \((x, y)\) that satisfies this equation lies on this line. 2. **Quadratic Equation: \( x^2 + y^2 = 5 \)** - This is an equation of a circle centered at the origin \((0,0)\) with a radius of \(\sqrt{5}\). ### Graphical Representation: - If plotted on a Cartesian plane, the line \( x + y = 3 \) will intersect the circle \( x^2 + y^2 = 5 \). The points of intersection are the solutions that satisfy both equations simultaneously. - These intersections represent the pairs \((x, y)\) that make both equations true at the same time. ### Solving the System of Equations: To find the exact points of intersection, one can solve the system algebraically or graphically by substituting and solving for the variables \( x \) and \( y \). Analyzing systems like this is fundamental in understanding how different types of equations can be used together to find common solutions, which is an essential concept in algebra and coordinate geometry.
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