Remember, there are infinitely many ordered pairs that would satisfy the equation.Д®xample 1 Sketch the graph of 2x + y = 3.

A sketch can be described as the 'curve of best fit.' In other words, it is necessary to locate enough points to give a reasonably accurate picture of the equation. Of course we could never find all numbers x and y such that x + y = 7, so we must be content with a sketch of the graph. If we represent these answers as ordered pairs (x,y), then we have (5,2) and (3,4) as two points on the plane that represent answers to the equation x + y = 7.ДЄll possible answers to this equation, located as points on the plane, will give us the graph (or picture) of the equation. Also, if x = 3 then y = 4, since 3 + 4 = 7. For instance, if x = 5 then y - 2, since 5 + 2 = 7. In other words, we will sketch a picture of an equation in two variables.Д¬onsider the equation x + y - 7 and note that we can easily find many solutions. In this section we will discuss the method of graphing an equation in two variables. Graphs are used because a picture usually makes the number facts more easily understood. You can usually find examples of these graphs in the financial section of a newspaper. There are many types of graphs, such as bar graphs, circular graphs, line graphs, and so on. Draw a straight line through those points that represent the graph of this equation.ДЄ graph is a pictorial representation of numbered facts.

Locate these points on the Cartesian coordinate system.

Find several ordered pairs that make a given linear equation true.

Check each one to determine how they are located. The first number of the ordered pair always refers to the horizontal direction and the second number always refers to the vertical direction.Д®xample 1 On the following Cartesian coordinate system the points A (3,4), B (0,5), C (-2,7), D (-4,1), E (-3,-4), F (4,-2), G (0,-5), and H (-6,0) are designated. The numbers represented by x and y are called the coordinates of the point (x,y). Ordered pairs are always written with x first and then y, (x,y). Second, from the point on the x-axis given by the first number count up or down the number of spaces designated by the second number of the ordered pair. Points are located on the plane in the following manner.ДЇirst, start at the origin and count left or right the number of spaces designated by the first number of the ordered pair. The ordered pair (5,7) is not the same as the ordered pair (7,5). This is called an ordered pair because the order in which the numbers are written is important. Points on the plane are designated by ordered pairs of numbers written in parentheses with a comma between them, such as (5,7). These are numbered in a counterclockwise direction starting at the upper right. The plane is divided into four parts called quadrants. Thus the plane extends indefinitely in all directions. The arrows indicate the number lines extend indefinitely. Positive is to the right and up negative is to the left and down. The zero point at which they are perpendicular is called the origin. The horizontal line is the x-axis and the vertical is the y-axis. Study the diagram carefully as you note each of the following facts. Perpendicular means that two lines are at right angles to each other. This system is composed of two number lines that are perpendicular at their zero points. This scheme is called the Cartesian coordinate system (for Descartes) and is sometimes referred to as the rectangular coordinate system. Rene Descartes (1596-1650) devised a method of relating points on a plane to algebraic numbers. Note that this concept contains elements from two fields of mathematics, the line from geometry and the numbers from algebra. We have already used the number line on which we have represented numbers as points on a line.

Given a point on the Cartesian coordinate system, state the ordered pair associated with it.

Given an ordered pair, locate that point on the Cartesian coordinate system.

Represent the Cartesian coordinate system and identify the origin and axes.

Upon completing this section you should be able to: We will now study methods of solving systems of equations consisting of two equations and two variables. In previous chapters we solved equations with one unknown or variable.

Grapher 12