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Equations and Graphs in Two Variables

Section 1.3

Equations, Inequalities, and Modeling

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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

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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

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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

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The Distance FormulaThe distance d between the points (x1, y1) and (x2, y2) is given

by the formula

Theorem: The Midpoint FormulaThe midpoint of the line segment with endpoints (x1, y1) and

(x2, y2) is

.212

2

12 yyxxd

.2

,2

2121

yyxx

The Distance Formulaand The Midpoint Formula

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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

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.22 rkyhx

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

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Theorem: Equation for a Circle in Standard FormThe equation for a circle with center (h, k) and radius r forr > 0 is

A circle centered at the origin has equation x2 + y2 = r2.

.222 rkyhx

The Circle

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Rule for Completing the Square x2 + bx + ?The last term of a perfect square trinomial (with a = 1) is thesquare of one-half of the coefficient of the middle term. Insymbols, the perfect square trinomial whose first two termsare x2 + bx is

.2

2

2

b

bxx

Completing the Square

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Theorem: Equation of a Line in Standard FormIf 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