4. In what time will $100 amount to $200 at 6%? at 8%? at 7% ?

5. At what rate per cent will $100 amount to $135 in 7 years?

6. At what rate per cent will $100 double itself in 20 years? in 25 years? in 12 years?

7. At what rate per cent will $100 amount to $124 in 4 years?

8. Find the time in which a sum of money amounts to 11⁄2 times itself at 5%? at 6%? at 8% ?

9. The cost price of an article is $ C and the profit is r%. Find the selling price.

10. Find the selling price if the cost price of an article is $ C and it was sold at a loss of r %.

CHAPTER VI

SIMULTANEOUS EQUATIONS

65. In solving problems requiring two answers, it is generally best to use two letters to represent the unknown quantities, and then from the conditions of the problem to form two equations and find from these equations the values of the unknown quantities.

As an illustration take the following problem :

Example 1. The sum of two numbers is 30 and their difference is 6. Find the numbers.

Adding the first members, and also the second members, of the equations (1) and (2), the result in one case is 2 x and in the other case 36.

66. An equation involving two unknown quantities, such as x+y= 30, has no definite solution. For if x = 1, then y = 29. If x=2, then y = 28. If x=5, then y=25.

The equation, x + y = 30, states in algebraic language that the sum of two numbers is 30. Nothing more is known about the two numbers until another condition is given. When that other condition is given, then, and not before, can the numbers be determined.

Equations containing two or more letters, the same letters representing the same numbers in each equation, are called simultaneous equations (Latin, simul, at the same time, and teneo, I hold).

Simultaneous equations, then, express different conditions of the same problem and are so called because the same letter has the same value in each of the equations.

Example 2. The difference of two numbers is 15 and the greater number exceeds twice the less by 1. Find

Subtract the members of equation (2) from those of

equation (1) and get

y = 14.

(3)

Step 1. Multiply the members of equation (1) by 3 and the members of equation (2) by 5,

6x+15y = 81.

35x + 15 y = 110.

(3)

(4)

Step 2. Subtract the members of equation (4) from those of equation (3), and get by axiom 3,

Step 3. Substitute in either of the original equations 1 Substituting 1 for x in equation (1), the result is

for x.

Check by substituting the values of x and y in equa

tion (2),

7+ 3 x 5 = 22.

Example 4.

5x-4y=-4.

3x+2y= 24.

(1)

(2)

Step 1. Make the coefficients of y in both equations equal in absolute value. This is done by multiplying the members of equation (2) by 2,

6x+4y = 48.

5x-4y=-4.

Step 2. Add the members of (3) and (1),

11 x = 44.

x = 4.

Step 3. Substitute in equation (1) 4 for x,

(3)

(1)

(4)

(5)

NOTE. Since equations are not numbers or quantities, it is incorrect to say add, subtract, multiply, or divide equations. The members of equations are quantities, hence they can be added, subtracted, multiplied, or divided.

67. To solve two simultaneous equations :

(1) Make the coefficients of the same letter in both equations equal in absolute value.

(2) Add if the coefficients of this letter in the two equa tions have unlike signs, subtract if they have like signs. (3) Solve the resulting equation.

(4) Substitute the value of the letter thus found in either of the original equations.