Copyright © 2011 Pearson Education, Inc. Equations and Graphs in Two Variables Section 1.3 Equations, Inequalities, and Modeling

1.3 Copyright © 2011 Pearson Education, Inc. Slide 1-3 If x and y are real numbers, then (x, y) is called an ordered pair of real numbers. The numbers x and y are the coordinates of the ordered pair, with x being the first coordinate or abscissa, and y being the second coordinate or ordinate. To picture ordered pairs of real numbers we use the rectangular coordinate system or Cartesian coordinate system, named after the French mathematician René Descartes (1596–1650). The Cartesian Coordinate System

1.3 Copyright © 2011 Pearson Education, Inc. Slide 1-4 The Cartesian coordinate system consists of two number lines drawn perpendicular to one another, intersecting at zero on each number line. The point of intersection of the number lines is called the origin. The horizontal number line is the x-axis and its positive numbers are to the right of the origin. The vertical number line is the y-axis and its positive numbers are above the origin. The two number lines divide the plane into four regions called quadrants. The quadrants do not include any points on the axes. The Cartesian Coordinate System

1.3 Copyright © 2011 Pearson Education, Inc. Slide 1-5 We call a plane with a rectangular coordinate system the coordinate plane or the xy-plane. Just as every real number corresponds to a point on the number line, every ordered pair of real numbers (a, b) corresponds to a point P in the xy-plane. For this reason, ordered pairs of numbers are often called points. So, a and b are the coordinates of (a, b) or the coordinates of the point P. Locating the point P that corresponds to (a, b) in the xy-plane is referred to as plotting or graphing the point, and P is called the graph of (a, b). In general, a graph is a set of points in the rectangular coordinate system. The Cartesian Coordinate System

1.3 Copyright © 2011 Pearson Education, Inc. Slide 1-6 The Distance Formula The distance d between the points (x 1, y 1 ) and (x 2, y 2 ) is given by the formula Theorem: The Midpoint Formula The midpoint of the line segment with endpoints (x 1, y 1 ) and (x 2, y 2 ) is The Distance Formula and The Midpoint Formula

1.3 Copyright © 2011 Pearson Education, Inc. Slide 1-7 An ordered pair is a solution to or satisfies an equation in two variables if the equation is correct when the variables are replaced by the coordinates of the ordered pair. The solution set to an equation in two variables is the set of all ordered pairs that satisfy the equation. The graph of (the solution set to) an equation is a geometric object that gives us a visual image of an algebraic object. Circles provide a nice example of this relationship between algebra and geometry. A circle is the set of all points in a plane that lie a fixed distance from a given point in the plane. The fixed distance is called the radius, and the given point is the center. The Circle

1.3 Copyright © 2011 Pearson Education, Inc. Slide 1-8 The distance formula can be used to write an equation for a circle with center (h, k) and radius r for r > 0. A point (x, y) is on the circle if and only if it satisfies the equation Since both sides of the equation are positive, we can square each side to get the standard form for the equation of a circle. The Circle

1.3 Copyright © 2011 Pearson Education, Inc. Slide 1-9 Theorem: Equation for a Circle in Standard Form The equation for a circle with center (h, k) and radius r for r > 0 is A circle centered at the origin has equation x 2 + y 2 = r 2. The Circle

1.3 Copyright © 2011 Pearson Education, Inc. Slide 1-10 Rule for Completing the Square x 2 + bx + ? The last term of a perfect square trinomial (with a = 1) is the square of one-half of the coefficient of the middle term. In symbols, the perfect square trinomial whose first two terms are x 2 + bx is Completing the Square

1.3 Copyright © 2011 Pearson Education, Inc. Slide 1-11 Theorem: Equation of a Line in Standard Form If A, B, and C are real numbers, then the graph of the equation Ax + By = C is a straight line, provided that A and B are not both zero. Every straight line in the coordinate plane has an equation in the form Ax + By = C, the standard form for the equation of a line. The Line