198. Simultaneous Quadratic Equations. If from either of two equations which involve x and y the value of one of the unknowns can be expressed in terms of the other, then by substitution in the second equation we obtain a quadratic which may be solved by any one of the methods explained in this chapter.

From the first equation, we see that if y=3, then x=-4, and if y=-2, then x=3.

Homogeneous Equations of the Same Degree.

199. The most convenient method of solution is to substitute y=mx in each of the given equations. By division we eliminate x and obtain a quadratic to determine the values of m.

Example. Solve the simultaneous equations

5x2+3y2 = 32, x2-xy+2y2 = 16.

Put y = mx and substitute in each equation. Thus

(1) Take m = 3 and substitute in either (1) or (2).

.(1), .(2).

CHAPTER XXVI.

PROBLEMS LEADING TO QUADRATIC EQUATIONS.

200. WE shall now discuss some problems which give rise to quadratic equations.

Example 1. A train travels 300 miles at a uniform rate; if the speed had been 5 miles an hour more, the journey would have taken two hours less: find the rate of the train.

Suppose the train travels at the rate of x miles per hour, then the 300 time occupied is hours.

Hence the train travels 25 miles per hour, the negative value being inadmissible.

[For an explanation of the meaning of the negative value see Elementary Algebra.]

Example 2. A man buys a number of articles for $2.40, and sells for $2.52 all but two at 2 cents apiece more than they cost; how many did he buy?

Let x be the number of articles bought; then the cost price of each is cents, and the sale price is

240

х

252 x-2

cents.

Example 3. A cistern can be filled by two pipes in 333 minutes; if the larger pipe takes 15 minutes less than the smaller to fill the cistern, find in what time it will be filled by each pipe singly.

Suppose that the two pipes running singly would fill the cistern 1

1

in x and x - 15 minutes; then they will fill and

х

of the cistern х 15

respectively in one minute, and therefore when running together they

will fill But they fill

3

or

of the cistern in one minute.

33' 100

Thus the smaller pipe takes 75 minutes, the larger 60 minutes. The other solution 63 is inadmissible.

201. Sometimes it will be found convenient to use more than one unknown.

Example. Nine times the side of one square exceeds the perimeter of a second square by one foot, and six times the area of the second square exceeds twenty-nine times the area of the first by one square foot find the length of a side of each square.

Let x feet and y feet represent the sides of the two squares; then the perimeter of the second square is 4y feet; thus

9x-4y = 1.

The areas of the two squares are x2 and y2 square feet; thus

6y2 - 29x2 = 1.

whence x = 5, the negative value being inadmissible.

1. Find a number which is less than its square by 72.

2. Divide 16 into two parts such that the sum of their squares is 130.

3. Find two numbers differing by 5 such that the sum of their squares is equal to 233.

4. Find a number which when increased by 13 is 68 times the reciprocal of the number.

5. Find two numbers differing by 7 such that their product is 330.

6. The breadth of a rectangle is five yards shorter than the length, and the area is 374 square yards: find the sides.

7. One side of a rectangle is 7 yards longer than the other, and its diagonal is 13 yards: find the area.

8. Find two consecutive numbers the difference of whose reciprocals is 1 ਨੈਨ•

9. Find two consecutive even numbers the difference of whose reciprocals is 0.

10. The difference of the reciprocals of two consecutive odd numbers is find them.

:

11. A farmer bought a certain number of sheep for $315; through disease he lost 10, but by selling the remainder at 75 cents each more than he gave for them, he gained $75: how many did he buy?

12. By walking three-quarters of a mile more than his ordinary pace per hour, a man finds that he takes 1 hours less than usual to walk 29 miles: what is the ordinary rate?