5. EQUATIONS.

1. What is an equation? Illustrate by a familiar example, and thus show that the equation is not destroyed if both sides are similarly dealt with.

2. Give instances of the great applicability of equations to the solution of questions, &c.

3. Explain what is exactly meant by removing a quantity from one side of the equation to the other with its sign changed.

4. Name various kinds of equations, and explain the terms applied to them, as "simultaneous," and "indeterminate," &c.

5. Explain the words "dimension" and "homogeneous," as applied to algebraical quantities.

6. Give some account of the usual course of proceeding in the solution of an equation.

7. Solve the following equations, writing opposite each step of the operation an explanation of it:

Also verify each equation by substituting the value of x.

8. In applying equations to the solution of questions, what is first to be considered, and what is thus the first thing to be done; and after this, what is meant by setting down or stating the equation?

9. Illustrate this, by writing down accurately all that is needful in the following questions, and then setting down the equations, but not working them out :

A. A post is one-fourth in the mud, one-third in the water, and 10 feet out of the water; required its length.

B. A has half as much money again as B, and A and B together have twice as much as C; now they have altogether 357. How much has each ?

C. A and B begin to play with equal numbers of marbles. A wins 4 from B, and now has as many again as B. How many had each at first?

10. Work out the three former equations, and verify by seeing that the obtained result fulfils the conditions.

11. When two or more quantities are unknown, there are three methods of solution. What are they? Explain them.

12. Illustrate these three methods, by solving each of the following equations three times, and seeing that the same results are obtained :— A. 2x + 3y = 12..

13. What is the course of proceeding with a question having two or more unknowns; and how many equations are needed for solution? 14. Illustrate by stating and solving the following:—

A. What fraction is that, to the numerator of which if 3 be added it is equal to, but if 2 be taken from the denominator it is equal to ?

B. There is a number of three digits; the sum of the digits is 16; the sum of the first and second less the third is 2, and the sum of the third and first less the second is 6: required the number.

A. There is a number consisting of two digits; the sum of the digits is 12, and the number is eleven times the first digit less four; what is the number?

B. What number is that to which if its third and fourth parts be added, the sum will exceed its half by 13?

C. A labourer was engaged for 24 days at 2s. 6d. per day, but was to forfeit 1s. 6d. for every day he was absent; he received at the end of his time 17. 12s.; how many days did he work? D. The length of a floor exceeds its breadth by 4 feet, and if each is increased 1 foot, the area will be 27 square feet more; required the length and breadth.

3. Solve the following equations with two unknown quantities:

4. Find the solution of the following questions:

A. There is a number of two digits which divided by the sum
of the digits is five less than the number inverted divided
by the difference of the digits; and the inverted number
is 18 less than the number; required the number.
B. Some soldiers being drawn up in a solid rectangle, it was
observed that if there were 116 men more the length of
the rectangle would be 4 men more, and the breadth 5;
but if there were 113 more, then the length would be 5
more and the breadth 4; required the number of men and
the number in each rank.

C. Find three numbers such that the first with half the second,
the second with one-third of the third, and the third with
one-fourth of the first, may each be 1000.

D. Find three numbers, the sums of each two of which are a, b and c.

E. Find three numbers whose products taken two and two together are a2, b2, c2.

F. Divide 72 into three such parts, that the first, the second, and the third, shall be equal to one another.

G. C and D playing at cards, says C to D, "If you will give me a guinea, I will bet you half-a-crown to eighteen-pence on every game, and will play 36 games." D won his guinea back again, and 17. 17s. besides. How many games did each win?

7. QUADRATIC EQUATIONS, &c.

1. What is a quadratic equation? Show that √2 has two results, and therefore that every quadratic has two roots.

2. Distinguish a "pure" quadratic from an "adfected" quadratic, and give an example of each. Also show to what form all quadratic equations can be reduced.

3. Explain the method of completing the square in adfected quadratics, and illustrate it by completing the square in the following equations, and solving them, marking accurately the two results:

A. 2x2+3x+1 = x2 + 5.

4. There is also another method of completing the square; explain it, and apply it in the following cases :

5. Sometimes it is advisable to have recourse to artifices in preparing an equation for solution or in solving it: illustrate such by the solution of the following:

6. Find a solution of the following quadratic equations with two unknown quantities :