Summary. Multiple.-When one quantity is exactly divisible by another, the former is said to be a multiple of the latter. Common Measure.—A number which exactly divides two or more given numbers is called a measure, or common measure of the numbers. Highest Common Factor.-The highest common factor (H.C.F.) of two or more algebraical expressions is the expression of highest dimensions which can be divided into each of the given expressions without a remainder; it is often called the greatest common measure (G.C.M.). The Lowest Common Multiple (L.C. M.) of two or more algebraical expressions is the expression of lowest dimensions into which each of the given quantities can be divided without a remainder. Evolution. The root of a given quantity is (as in Arithmetic) defined as that quantity which, when multiplied by itself the required number of times, produces the given quantity. CHAPTER VIII. SIMPLE EQUATIONS. SIMULTANEOUS EQUATIONS. Simple Equations.-When two algebraical expressions are connected together by the sign of equality the whole expression thus formed is called an equation, and the use of an equation consists in this, that from the relations expressed between certain known and unknown quantities we are able under proper conditions to find the unknown in terms of the known. Usually the earlier letters of the alphabet, a, b, c, d..., are used to represent known, and the concluding letters, x, y, z, to represent unknown quantities. The process of finding the value of the unknown quantity is called solving the equation, the value so found is the solution or the root of the equation. An equation which involves the unknown quantity to the first power or degree is called a simple equation; if it contains the square of the unknown quantity it is called a quadratic equation; if the cube of the unknown quantity, a cubic equation. If an equality involving only an algebraic operation exists between two quantities the expression is called an identity, thus (x+y)2= x2+2xy+y2 is an identity. In the example 4x+2=30 the statement intended is that if 2 be added to four times an unknown number x, the result will equal 30. By a process of trial, substituting the numbers 1, 2, 3 in turn for x, it will be found that the equation is true only when x=7. Then as 4×7=28, this value of x makes the expressions on the left- and right-hand sides of the sign of equality numerically equal, or, the equation is said to be satisfied. The solution of a simple equation depends upon the following truths. If two quantities are equal to one another they remain equal when: (a) The same quantity is added or subtracted from each side of the original equation. (b) When each side is multiplied or divided by the same quantity. (c) When each side is raised to the same power, or the same root of each is extracted. (d) The signs of all the terms in the equated expressions can be changed from + to if both sides are similarly altered. Ex. 1. Solve 4x+2=30. By (a) subtracting 2 from each side we get 4x=28; Instead of subtracting we can transpose the 2 in the preceding example from one side of the equation to the other by changing its sign; thus 4x=30 — 2 = 28. Ex. 2. Solve 4x+5=3x+8. Subtract 3x from both sides of the equation and we get 4x-3x+5=8, next subtract 5 from each side; .. x=8-5=3. In the above it is obvious that +3x and +5 on one side may be removed from one side to the other (or as it is called transposed) and appear on the opposite side with changed sign. Hence in the solution of equations, Transpose all the unknown quantities to one side, and all the known quantities to the other, simplify if necessary, and divide by the coefficient of the unknown quantity. The rules referred to above, under (a), (b), (c), (d), can perhaps be best illustrated by the consideration of a few simple examples. Ex. 3. Solve transposing, or Ex. 4. Solve transposing, RULE (a). 4x+2=3x+4; 4x-3x=4-2; .. x=2. 5(x+1)=3(x-5)+2, 5x+5=3x-15+2; 5x-3x=2-15-5; .. 2x-18; or x= -9. L.C.M. of denominators is 15. Hence multiplying both sides by 15, First clear fractions by multiplying all through by abx; transposing, .. a2-a3bx=ab3x - b2, - a3b≈ - ab3x= - a2 - b2, changing sign or multiplying by -1, x (a2+b2) ab=a2+b2, .. x= a2 + b2 1 ab (a+b2) ab = Abbreviated and Particular Methods.-In solving equations it often happens that an easy solution can be obtained by an advantageous arrangement of the terms on the two sides of an equation. Ex. 8. Solve √x+4+√x-1-5=0. It is well to notice that if we commence to solve the equation by squaring, the resulting equation will become very troublesome and difficult. If in an equation each side consists of a simple fraction, the work in many cases is much simplified by adding or subtracting the numerator and denominator of each fraction to form in each case a new numerator, and subtracting or adding the numerator and denominator to form a new denominator. then by adding unity to each side of the equation, we get or, = b d a+b b In a similar manner, subtracting unity from each side, ..(i) ...(ii) Hence, from (i) and (ii), since things which are equal to the same thing are equal to one another, |