Multiplying the num' and den' of the left-hand side of the equation by √(a+x)+(a−x), we get

squaring both sides of the equation, we obtain

a2x2-b2x2-2abx+a2,

or x2+b2x2=2abx,

This formula may advantageously be applied to the solution of equans similar in form to the last, where the unknown quantity occurs ly on one side of the equation.

ocess.

PROBLEMS.

105. Stated Rules have been given for the solution of Equations, but the solution of Problems producing equations these rules will not guide in forming the equations themselves. Indeed, each particular question quires a distinct process of reasoning to bring it into an algebraical m, and practice alone can produce expertness in conducting this The best advice which can be given to the Student is this: . to read over the problem carefully, and understand clearly what is en, and what is required to be determined; then to represent the antity to be determined by x, and to express the conditions of the blem in algebraical language, using a wherever the unknown quantity ers. An equation will thus be formed, from which a may be deterned, and when x has been found from it the solution of the problem rrived at.

Ex. XLVI.

Problems worked out.

Ex. 1. Find a number, which being multiplied by 6, and having added to the product, the sum shall be 66.

Let x represent the number required. Multiplying it by 6 and adding o the product, we have 6x+12.

By the question, 6x+12=66,

or 6x=66-12=54;

.. x=9.

Ex. 2. After paying away 1th, and th of my money, I had £170 left; what money had I at first?

Let a represent the money at first, in pounds.

Then

+ represents the money paid away;

.. x − ( x + x)

By the question

whence we obtain

left.

x-(x+4x)=170;

x= £280.

Ex. 3. A person left £700 to be divided among three persons, in such a way that the first was to receive double of what the second received, and the second double of what the third received. Find each person's share.

Let x= share of 3rd person, in pounds.

Then 2x=

And 4x=

2nd .........

1st

By the question

x+2x+4x=700;

or 7x=700;

.. x=100.

hence the shares are £400, £200, £100, respectively.

Ex. 4. Two persons A and B lay out equal sums of money in trade; A gains £126, and B loses £87, and A's money is now double of B's: what did each lay out?

Let x represent the money laid out, in pounds.

Then by the question x+126=2(x−87),

whence we obtain

00= = £300.

Ex. 5. Divide the number 30 into two parts, such that one of them shall exceed the other by 13.

Let x represent one part; then +13 represents the other.

By the question

x+(x+13) = 30;

whence we obtain x=8, and x+13=211.

Ex. 6. What two numbers are those, whose difference is 10, and if 15 be added to their sum, the whole will be 43?

Let x the smaller number.

Then x+10= the greater number.

By the question

x+x+10+15=43;

or 2x=18;

.. x=9.

.. the numbers are 9 and 19.