in seconds, is given by one of the following equations. Construct the graph, and describe the motion, from t = 0 to t = 6.

20. Describe the motion of a ball rolling on the side of a hill if its velocity is given by one of the following equations, by interpreting the graph. What is the acceleration in each case?

scribe the motion of a body (a) if y represents the distance of a body from a certain station at the time x; (b) if y represents the velocity of a body at the time x.

23. Equation of a Straight Line. We have seen that the graph of y = mx + b is the straight line whose slope is m and whose intercept on the y-axis is b. Conversely, if a line is given whose slope is, say, 2 and whose intercept on the y-axis is, say, 3, then the equation of which it is the graph is y = 2x + 3; for the graph of this equation is known to be the given line.

The equation of which a given line or curve is the graph is called the equation of the line or curve.

To find the equation of a line whose slope and intercept on the y-axis are given, substitute the given values of m and b in the equation

This is called the slope-intercept form of the equation of a straight line.

To find the equation of a line determined by its slope and any point on it we apply the

Theorem. The equation of the line whose slope is m and which passes through the point P1(x1, y1) is

y - y1 = m(x − x1).
·

(2)

Let P(x, y) be any point on the given line.

Since the value

of Ay/Ax for any two points on a straight line is constant

(page 49), and equal to m (definition page 50), we have

This is called the point-slope form of the equation of a straight line.

To find the equation of a straight line determined by two points, find the slope and then apply (2).

EXAMPLE 1. Find the equation of the line determined by the points A (1, 4) and B (2, 3).

Check. The coördinates of A and of B satisfy this equation. EXAMPLE 2. Find the equation of the line through the point A (5, 2) which is parallel to

and hence the slope of the given line is m = {}, the coefficient of x (Corollary 2, to Theorem 2, Section 20). Hence the slope of the required line is (Theorem 2, Section 18).

EXAMPLE 3. The freezing point on a Centigrade thermometer is 0°, and on a Fahrenheit thermometer it is 32°. The boiling point on the first is 100° and on the second is 212°. Find the relation between any two corresponding readings on the two thermometers.

Since the interval between the freezing and boiling points is divided into equal parts on each thermometer, a change of one degree on one of them will always produce a definite, constant change on the other. Hence the graph will be a straight line.

In plotting, let abscissas denote readings on the Centigrade scale, and ordinates the corresponding readings on the Fahrenheit scale. Then the

The slope of the line, 1.8, is the rate of change of a reading on the Fahrenheit scale with respect to the Centigrade scale. As the rate of change is measured by the change on the Fahrenheit scale due to a unit change on the Centigrade scale, this means that 1°.8 Fahrenheit 1° Centigrade.

=

EXERCISES

1. Construct each of the lines indicated below. Find its equation and reduce it to the form Ax + By + C = 0. Check the result by substituting the coördinates of the given point or points.

(a) With the slope and the intercept on the y-axis 4.
(b) Passing through the point (3, 2) with the slope
(c) Determined by the points (2, 5) and (− 1, 2).
(d) Through the point (3, 4) parallel to the x-axis; the y-axis.

(e) Passing through the points (0, 4) and (3, 0).

=

2x 4.

(f) Through the point (1, 3) parallel to the line y (g) Through the point (3, 0) parallel to the line determined by the points (1, 2) and (5, 4).

(h) Through the point (−2, 3) parallel to the line x + 2y – 6 = 0. 2. What is the general form of the equation of a line parallel to the x-axis? the y-axis?

3. What is the value of A if the line Ax

3y+5

=

O is parallel to the

line 3x+4y-12 0? If its intercept on the x-axis is 10?

=

4. A medicine is 40% alcohol. Construct a graph showing the amount of alcohol in any amount of the medicine, and find its equation.

5. In an experiment in stretching a brass wire, it is assumed that the elongation E is connected with the tension T by a linear relation. If

=

==

T = 18 pounds when E 0.1 inch, and T = 58 pounds when E = 0.3 inches, construct the graph and find the relation.

6. The population of a town in 1900 was 1200 and in 1910 it was 1400. Assuming that the growth in population was uniform, construct the graph showing the population at any time, and find the relation between the population and the time.

7. A sum of money at simple interest amounts to $94.00 in 5 years, and $108.00 in 10 years. Find the sum of money and the rate of interest.

8. It is observed that the boiling point of water at sea level is 212° F., and that at an altitude of 2299 feet it is 208° F. What is the boiling point at an altitude of 5000 feet?

9. Find the coördinates of the point of intersection of each of the pairs of lines given below.

(a) x + y = 5, and x y = 1.

Let A be the point of intersection. Then since A is on both graphs, its coördinates must satisfy both equations, and hence these coördinates may be found by solving the equations simultaneously.

Adding the equations, we get 2x whence x = 3.

4

3

=

6,

Substituting in either equation we find that y = 2. Therefore the coördinates of A are (3,2).

=

(d) 0.3x0.8y + 5.3 0, and y

=

1.5.x.

10. Find the coördinates of the point of intersection of the line determined by the points (2, 1) and (4, 3) and the line through the point (1, 5) whose slope is

24. Application to the Solution of Problems. In the following examples and exercises it is seen how the graphical solution of problems solvable by functions which change uniformly may be used to suggest a method of algebraic solution.

EXAMPLE 1. Solve Example 1, Section 19, algebraically.

=

The graphical solution, page 54, shows that the value of DH must be found. But DH = OH - OD. It is known that OD 3, and since OH is the abscissa of G, it may be found by solving the equations of OC and EF for x.

Since OC passes through the origin and has the slope 0.2, its equation is

=

0.1x, its ordinate is y

=

The abscissa of E is x = 3, and since E lies on the line OB, whose equation is y 0.1 x 30.3. Hence the coördinates of E are (3, 0.3). Since EF was drawn parallel to OA its slope is m = and therefore its equation has the form (Theorem, Section 23)

0.5,

(2)

To find OH, the abscissa of G, solve (1) and (2) for x. Eliminating y, we have

whence

0.52 – 1.2,

1.2,

EXAMPLE 2. Solve Example 2, Section 19, algebraically.

The time of the freight train and the distance from A to B are represented by the coördinates of the point G (Figure 35, page 55), the point of intersection of the lines OC and FG. The equations of these lines are found to be respectively

The time of the express train is represented by

DI = FJ = OJ – OF = 62 – 4 = 2 hours.

EXERCISES

Solve each of the exercises below both graphically and algebraically using the graphic solution as the basis of the algebraic solution.

1. Two trains start toward each other from Buffalo and New York, respectively, 440 miles apart. The one from New York travels at the rate of 50 miles an hour, and the one from Buffalo at the rate of 40 miles an hour. When and where will they meet?

2. A man walks at the rate of 3 miles an hour. Three hours after he starts, another man starts from the same place and travels in the same direction at the rate of 10 miles per hour. When and where will the latter overtake the former?

3. An express train starts from a city and moves with a velocity of 40 miles per hour. A freight train is 90 miles ahead of the express at the start and is moving at the rate of 25 miles an hour. When and where will the express overtake the freight?