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Algebra-Equations. 170. An equation is the expression of the equality of two numbers or combinations of numbers.

EQUATIONS.
(1) 2 + 4 + 6 = 3 + 5 + 4
(2) a + b + c = 40 – 12

(3) 2 x + x + 4 10 + 9 1. Every equation is made up of two members. The part of the equation which is on the left of the sign of equality is called the first member ; the part on the right of the sign of equality, the second member.

2. The first member of equation No. 1 (above) is It is made up of terms.

3. If the same number be added to each member of an equation, the equality will not be destroyed.

If x =

8, then x + 4 8 + 4.

If a + b = 16, then a + b + c = 16 + 0. 4. If the same number be subtracted from each member of an equation, the equality will not be destroyed.

8, then x 3 8 – 3. If a + b 16, then a + b - c = 16 – C. 5. If each member of an equation be multiplied by the same number, the equality will not be destroyed.

8, then 4 x 4 times 8, or 32. If a + b 16, then 4 a + 4 6 4 times 16, or 64. 6. If each member of an equation be divided by the same number, the equality will not be destroyed.

8 If x = 8, then

4 4

b 16 If a + b = 16, then

If x =

If x =

х

=

or 2.

a

+

or 4.

If x + y

Algebra-Equations. 7. Any term in an equation may be transposed from one member of the equation to the other; but its sign must be changed when the transposition is made.

If x + 5 15, then x = 15 – 5, or 10.*
If y - 6 = 27, then y = 27 + 6, or 33.
If a + b + c = 18, then a + b = 18 – C.
25, then x + y

25 +2. 171. To find the number for which x stands, in an equation in which there is no other unknown number.

EXAMPLE No. 1.
Equation, x + 2 x + 3 x – 5 13
Transposing, x + 2 x + 3

x =

13 + 5 Uniting,

18 Dividing,

X =

3

6 x

EXAMPLE No. 2.

5 x

Equation,
Transposing,
Uniting,
Dividing,

2 x + 3 x + 6 - 2 x + 18
2 x + 3 x – 5 x + 2 x = 18 - 6

12
6

2 x =

PROBLEMS. Find the value of x. 1. x +4 12

6. 3 x + 2 x - 4 = x + 16 2. x + 3 x 8

7. 5 - 7 3 x + 5 2 = 23

8. 7 x + 2 x

x = 3 x + 35 4. 3 x

9. 5

3 x 6 x 41 5. 7 x + x = 144.

10. 6 x 8 2 x 3 x + 5 (a) Find the sum of the ten results.

3. 5 x

x = 44

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4 x

+Observe that 5 is subtracted from each member of the equation. t Observe that 6 is added to each member of the equation.

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1. Two of the sides of a trapezoid are parallel and two are not parallel. In the trapezoid represented above the side ac is parallel to the side

2. No two of the bounding lines of a trapezium are parallel.

3. In the trapezoid represented above no one of the angles is a right angle. Name the angles that are greater than right angles; the angles that are less than right angles.

4. Draw a trapezoid two of whose angles are right angles.

5. Can you draw a trapezoid having one and only one right angle?

6. Draw a trapezium one of whose angles is a right angle.

7. Can you draw a trapezium having more than one right angle?

8. Every quadrilateral (trapezium, trapezoid, or parallelogram) may be divided into two triangles. Remember that the sum of the angles of two triangles is equal to four right angles. Observe that the sum of the angles of the two triangles is equal to the sum of the angles of the quadrilateral. So the sum of the angles of a quadrilateral is equal to four right angles.

173. MISCELLANEOUS REVIEW. 1. If two of the angles of a trapezoid are right angles and the third is an angle of 60°, how many degrees in the fourth angle? Draw such a trapezoid. *

2. If the sum of three of the angles of a trapezium is 298°, how many degrees in the fourth angle? Draw such a trapezium. *

3. If one of the angles of a triangle is an angle of 80°, and the other two angles are equal, how many degrees in each of the other angles ? Draw the figure. *

4. If one of the angles of a quadrilateral is a right angle, and the other three angles are equal, what kind of a quadrilateral is the figure?

5. One of the angles of a quadrilateral is a degrees; another is b degrees; the third is c degrees. How many degrees in the fourth angle

6. The smallest angle of a triangle is x degrees; another angle is 2 x degrees, and the third is 3 x degrees: Then x + 2 x + 3 x =

180.
Find the value of x; of 2 x; of 3 x.

7. 643,265,245,350. Without performing the division tell whether this number is exactly divisible by 9; by 5; by 10; by 25; by 50; by 12; by 18; by 6; by 15; by 30; by 90; by 163.†

8. A number is made up of the following prime factors: 2, 2, 3, 3, 5, 7, 11. Is the number exactly divisible by 18? by 26 ? by 35 ? by 77? by 21 ? by 30 ? by 45 ? by 8?

* It is not expected that this drawing will be accurate in its angular measurements-simply an approximation to accuracy, to aid the pupil in recognizing the comparative size of angles.

+ A careful study of pages 71-75 inclusive will enable the pupil to make the statements called for, with little hesitation.

See problem 3, page 70.

FRACTIONS. .

174. A fraction may be expressed by two numbers, one of them being written above and the other below a short horizontal line ; thus, 5, 11, 14%.

175. The number above the line is the numerator of the fraction; the number below the line, the denominator of the fraction.

176. KINDS OF FRACTIONS. 1. A fraction whose numerator is less than its denominator is a proper fraction.

% 14, are proper fractions.

2. A fraction whose numerator is equal to or greater than its denominator is an improper fraction.

Š, &, 47, are improper fractions.

NOTE.-The fraction .7 is a proper fraction. 2.7 may be regarded as an improper fraction or as a mixed number. If it is to be considered an improper fraction it should be read, 27 tenths; if a mixed number, 2 and 7 tenths.

3. Such expressions as the following are compound fractions:

of q, of }, Š of 11.

4. A fraction whose numerator or denominator is itself a fraction or a mixed number, is a complex fraction.

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