Ordered Pairs Often, to get an idea of the behavior of an equation, we will make a picture that represents the solutions to the equation. A graph gives us that picture. The rectangular coordinate plane, where we place our graph, is created by a horizontal number line (x-axis) and a vertical number line (y-axis). Where the two number lines meet in the center, at x=0 and y=0. is called the origin. Notice that the rectangular coordinate plane has four sections which are called quadrants. y 6 II 4 2 I 6 4 2 2 4 6 III 2 4 IV x 6 A point is an ordered pair given as (x, y). The first number is the value on the x-axis. This is the distance the point moves right (if positive) or left (if negative) from the origin. The second number is the value on the y-axis. This is the distance the point moves up (if positive) or down (if negative) from the origin.
Example 1: Give the coordinates of the points on the graph. y A 6 4 2 D E C B 6 4 2 2 4 6 2 4 6 x Point A: From the point down to the x-axis = -4 From the point right to the y-axis = 2 The coordinates are (-4, 2) Point B: From the point down to the x-axis = 4 From the point left to the y-axis = 5 The coordinates are (-4, 5) Point C: The point is on the x-axis = 3 The point left to the y-axis = 0 The coordinates are (3, 0) Point D: From the point up to the x-axis = 1 From the point left to the y-axis = -3 The coordinates are (1, -3) Point E: From the point up to the x-axis = 0 The point is on the y-axis = -5 The coordinates are (0, -5)
In a similar manner, we can go backwards and plot points on the plane. Example 2: Plot the points: A=(5, 4), B=(-2, -5), C=(-4, 5), D=(0, 2), E=(3, 0) C y 6 4 A 2 D E 6 4 2 2 4 6 2 x B 4 6 A is at (5, 4), so x=5 (right 5) and y=4 (up 4) B is at (-2, -5), so x=-2 (left 2) and y=-5 (down 5) C is at (-4, 5), so x=-4 (left 4) and y=5 (up 5) D is at (0, 2), so x=0 (no movement) and y=2 (up 2) E is at (3, 0), so x=3 (right 3) and y=0 (no movement) The main purpose of graphs is not to plot random points, but rather to give a picture of the solutions to an equation. Each point on the graph of a line is solution to that linear equation. To check whether a point is a solution to the linear equation, substitute the x-value of the point into the equation in place of x, and substitute the y-value of the point into the equation in place of y. Then simplify. A true final statement indicates that the point is a solution. A false final statement indicates that the point is not a solution.
Example 3: Are the points (-1, 7) and (-2, 3) solutions to the equation y = -2x + 5 For (-1, 7): 7 = -2(-1) + 5 7 = 2 + 5 7 = 7 True, (-1, 7) is a solution For (-2, 3): 3 = -2(-2) + 5 3 = 4 + 5 3 9 False, (-2, 3) is not a solution Example 4: Are the point (-1, 1) and 3x 5y = 8 solutions to the equation For (1, 1): 3(1) 5(1) = 8-3 5 = 8-8 8 False, (-1, 1) is not a solution For : 3(3) 5( ) = 8 9 1 = 8 8 = 8 True, (3, ) is a solution To find the points of a linear equation, we need to construct ordered pairs. If we are given an x- value, we substitute that x-value into the linear equation and solve for y. This gives the y-value and completes the ordered pairs. In a similar manner, if we are given the y-value, we substitute to find the x- value and complete the ordered pair.
Example 5: Complete the following ordered pairs for 2x 3y = 6 (-3, ), (0, ), (, 1), (3, ) 2(-3) 3y = 6-6 3y = 6 +6 +6 Add 6 to both sides - 3y = 12-3 -3 Divide both sides by -3 y = -4 The ordered pair is (-3, -4) 2(0) 3y = 6 0-3y = 6-3y = 6-3 -3 Divide both sides by -3 y = -2 The ordered pair is (0, -2) 2x - 3(1) = 6 2x - 3 = 6 +3 +3 Add 3 to both sides 2x = 9 2 2 Divide both sides by 2 x = The ordered pair is (, 1) 2(3) 3y = 6 6 3y = 6-6 - 6 Subtract 6 from both sides - 3y = 0-3 -3 Divide both sides by -3 y = 0 The ordered pair is (3, 0)
Example 6: Complete the ordered pairs for y = 2x 1 (0, ), (-4, ), (, 1), (, 0) y = 2(0) 1 y = 0 1 y = - 1 The ordered pair is (0, -1) y = 2(-4) 1 y = - 8 1 y = - 9 The ordered pair is (-4, -9) 1 = 2x 1 + 1 + 1 Add 1 to both sides 2 = 2x 2 2 Divide both sides by 2 1 = x The ordered pair is (1, 1) 0 = 2x 1 + 1 + 1 Add 1 to both sides 1 = 2x 2 2 Divide both sides by 2 = x The ordered pair is (, 0)