(as regards the axes) to the pair of values just found. The point (1, 4) is thus said to "belong to" the given equation. See P1 in the figure. Next, give x some other value; for example, x = 2. When this is done in the equation we find that y = 2. Explain this. Therefore, the point (2, 2) is another point which belongs to the equation, and is now to be plotted. See P2 in the figure. We proceed to plot several points in this way, first giving x a value in each case and then getting the corresponding y from the equation. In doing this it is to be observed that x may be given negative as well as positive values and y also may become negative, but we can plot the point in every case, as shown in § 121. The table below shows a variety of values of x and the corresponding value of y for each value of x. Note the signs. For example, when x=4, y=-2; when x1, y=8; etc. |12| 10 Graph of 9 2x+y=6 8 7 6% וס 4 3 P 2 1 10 2 3 4 5 6 The graph of the given equation is now obtained by drawing the straight line which passes through all the points thus plotted (see Fig. 66). This line may be extended indefinitely in either direction (see the arrows in the figure) and when so extended it constitutes the complete graph of the equation 2x+y=6. P 123. Linear Equations. Any equation which contains two letters (usually x and y) is called linear if these letters occur to no higher power than the first. Thus, the equation 2x+y=6 considered in § 122 is a linear equation in x and y; so also are the equations 6 x-3 y = 2, 1 x = 3 y-1, etc.; but x2-y21 is not linear. The graph of every linear equation is a straight line. It can be shown that if we take any point on this line its coördinates will satisfy the given equation. EXERCISES-GRAPHS OF LINEAR EQUATIONS Draw the graph of each of the following linear equations. In each case a table should be formed as in the Example of § 122. Plot at least three points in each case. 1. x+y=4. 2. x-y=3. 3. x+2y=6. 4. 2x+3y=1. 5. x+3y=3. 124. Plotting an Equation by Means of Two Points. We need find only two points to plot the line through them. Any two points will do, but it is usually most convenient to use the two points where the line crosses the axes. X 2 FIG. 67. EXAMPLE. Plot the equation 5x-4y=20. SOLUTION. Placing x=0 in the equation, gives y=-5. Placing y=0 in the equation, gives x= = 4. Plotting the two points thus obtained, namely (0, -5) and (4, 0), (which, by the Note in § 121 are the points where the graph crosses the axes) and drawing (with a ruler) the line through them, we get the graph required, as shown in Fig. 67. EXERCISES PLOTTING STRAIGHT LINES BY TWO POINTS Draw the graph of each of the following linear equations by plotting two points on it. 1. 3x-2y=-6. 3. 3x-2y-6=0. 5. 2x-5 y=10. 2. 3x+2y=-6. 4. 3x=2 y. 6. 4x=7y-14. 125. Simultaneous Equations. When we draw the graphs of the equations and x+y=6, 3x-2y=-2, -9 -8 3x-2y= 42 we find that the lines intersect (cut each other) in one point, namely (2, 4). Since this point is thus on both lines, it follows that the corresponding values, x=2, and y=4, must satisfy both equations. Test this and see that it is true. A set of values, such as (2, 4), that satisfies both of the two equations is said to be a solution of the equations. Any two equations considered together in this way are called simultaneous equations. 7 6 5 4. 2 X 0 1 2 3 4 FIG. 68. 126. Inconsistent Equations. If we draw the graphs of the simultaneous equations Y [x+y=3, we find that the lines do not intersect; in other words, the lines are parallel. Thus, the equations have no pair of values in common. Such a pair of simultaneous equations is called inconsistent. EXERCISES - SIMULTANEOUS EQUATIONS Draw the graphs of each of the following pairs of simultaneous equations. Determine which have a solution and which are inconsistent. Whenever there is a solution, find (from the graph) what it is and test your result in the equa * 127. Graphical Study of Motion. The way in which motion may be studied graphically is best seen from one or two examples. EXAMPLE 1. An automobile starts out and travels at the rate of 20 miles an hour, and two hours later another starts from the same place and goes in the same direction at 30 miles an hour. How long before the second overtakes the first? Miles 140 120 100 80 60 40 201 Y B 0 X 2c3 4 5 6 Y Hours SOLUTION. Take each unit on XX to . represent 1 hour and each unit on YY to represent 20 miles. The motion of the first automobile is then represented by a straight line starting at the origin and rising 1 unit for every unit it goes to the right; that is, it is represented by the line AB. As for the second automobile, it does not start until 2 hours have elapsed and then goes at the rate of 30 miles an hour. This means that its motion is represented by a straight line starting FIG. 70. at the 2-hour mark on XX and then rising 1 units for every unit it goes to the right; that is, it is represented by the line CB. The point B where these two lines meet corresponds graphically to the meeting of the automobiles, because at this point both have traveled the same distance as measured off on YY. The time that goes with this point, as measured off on XX, is seen to be 6 hours. Therefore, the second automobile will overtake the first one 6 hours after the first one starts, or, 4 hours after the second one starts. Ans. EXAMPLE 2. Two automobiles start from two towns that are 110 miles apart and travel towards each other, the one at 20 miles an hour and the other at 30 miles an hour. How long before they will be 10 miles apart? SOLUTION. Take the same units as in Example 1. Then the two motions will be represented by the two lines in Fig. 71. Note that the line representing the machine traveling at 20 miles per hour, starts at 0 hours and rises 1 unit for every unit it goes to the right; while the other line, which represents the machine traveling at 30 miles per hour, starts at the 0 hour also, but at the 110 mark, and it descends 1 units for every unit it goes to the right. Y 120 100 80 60 40 P 20 X 0 X 0 1 2 3 4 Y Hours Miles FIG. 71. Now, the question is, "When will the two machines be 10 miles apart?" Graphically, this means "What is the point on XX where the difference between the distance measured up to the lines will be 10 miles; that is, a unit?" is represented by PQ. The point on XX is seen to be 2. Hence, the answer to the example is 2 hours. * EXERCISES-MOTION This difference 1. A boy starts from home at noon and rides on a bicycle 10 miles an hour toward a certain town. At 3 o'clock in the afternoon a man starts out from the same place in an automobile and travels the same road at 30 miles an hour. Determine graphically how soon the boy will be overtaken, and also how far he will then be from home. 2. Answer the following question graphically: In a bicycle race |