Reduction of Equations by Division

When the unknown quantity is multiplied into any known quantity, the equation is reduced by dividing both sides by this known quantity.

Ex. 1. Reduce the equation      ax + b - 3h = d
  By transposition      ax = d + 3h - b
  Dividing by a       x = ( + 3h - b)/a.

Ex 2. Reduce the equation      2x = a/c - d/h + 4b
Clearing of fractions      2chx = ah - cd + 4bch
Dividing by 2ch       x = (ah - cd + 4bch)/2ch.

If the unknown quantity has coefficients in several terms, the equation must be divided by all these coefficients, connected by their signs.

Ex. 3. Reduce the equation      ax + x = h - 4
  Dividing by a + 1      x = (h - 4)/(a + 1)

Ex. 4. Reduce the equation      x - (x - b)/h = (a + d)/4
  Clearing of fractions    4hx - 4x = ah + dh - 4b
  Dividing by 4h - 4      x = (ah + dh -4b)/(4h - 4)

If any quantity, either known or unknown, is found as a factor in every term, the equation may be divided by it. On the other hand, if any quantity is a dkmsor in every termy the equation may be multiplied by it. In this way, the factor or divisor will be removed, so as to render the expression more simple.

Ex. 5. Reduce the equation      ax + 3ab = 6ad + a
  Dividing by a       x + 3b = 6d + 1
  And        x = 6d + 1- 3b.

Ex. 6. Reduce the equation      x.(a + b) - a - b = d.(a + b)
Dividing by a + b    x - 1 = d
And         x = d + 1.

Sometimes the conditions of a problem are at first stated, not in an equation, but by means of a proportion. To show how this may be reduced to an equation, it will be necessary to anticipate the subject of a future section, so far as to admit the principle that " when four quantities are in geometrical proportion, the product of the two extremes is equal to the.product of the two means :w a principle which is at the foundation of the Rule of Three in arithmetic.

     Thus, if a:b = c:d,      then ad = bc.
     And if 3:4 = 6:8,      then 3.8 = 4.6. Hence,

A proportion is converted into an equation by making the product op the extremes, one side op the equation; and the product of the means, the other side.

Ex. 1. Reduce to an equation      ax:b = ch:d.
The product of the extremes is     adx
The product of the means is      bch
The equation is, therefore      adx=bch.

2. Reduce to an equation      a + b:c = h - m:y.
The equation is       ay + by = ch - cm.

On the other hand, an equation may be converted into a proportion, by resolving one side of the equation into two fractors, for the middle terms of the proportion: and the other side into two fractors, for the extremes.

As a quantity may often be resolved into different pairs of factors, a variety of proportions may frequently be derived from the same equation.

Ex. 1. Reduce to a proportion      abc = deh.
The side abc maybe resolved into     a.bc, or ab.c, or ac.b.
And deh may be resolved into      d.eh, or de.h or dh.e.

Therefore a:d :: eh:bc      And ac:dh = e:b
And ab:de = h:c       And ac:d = eh:b, &c.

For in each of these instances, the product of the extremes is abc, and the product of the means deh.

Ex 2. Reduce to a proportion      ax + bx = cd - ch
The first member may be resolved into    x.(a + b)
And the second into       c.(d - h)
TherefoVe x:c = (d - h):(a + b)
And d - h:x = a + b:c, &c.

If for any term or terms in an equation, any other expression of the same value be substituted, it is manifest that the equality of the sides will not be affected.

Thus, instead of 16, we may write 2.8, or 64/4, or 25 - 9.

For these are only different forms of expression for the same quantity.

It will generally be well to have the several steps, in the reduction of equations, succeed each other in the following order.

First, Clear the equation of fractions.
Secondly, Transpose and unite the terms.
Thirdly, Divide by the co-efficients of the unknown quantity.

Examples.

1. Reduce the equation      3x/4 + 6 = 5x/8 + 7
Clearing of fractions      24x + 192 = 20x + 224
Transp. and uniting terms     4x = 32
Dividing by 4       x = 8.

2. Reduce the equation      x/a + h = x/b - x/c + d
Clearing of fractions      bcx + abx - acx = abcd - abch
Dividing        x = (abcd - abch)/(bc + ab - ac)

3. Reduce      40 - 6x - 16 = 120 - 14x.      Ans. x = 12.

4. Reduce      x/3 + x/5 = 20 - x/4.

5. Reduce      (1 - a)/x - 4 = 5.

6. Reduce      6x/(x + 4) = 1.

7. Reduce      x + x/2 + x/3 = 11.

8. Reduce      (x - 5)/4 + 6x = (284 - x)/5.

9. Reduce      3x + (2x + 6)/5 = 5 + (11x - 37)/2

10. Reduce      (6x - 4)/3 - 2 = (18- 4x)/3 + x.

11. Reduce      3x - (x - 4)/4 - 4 = (5x + 14)/3 - 1/12.

12. Reduce      (7x + 5)/3 - (16 + 4x)/5 + 6 = (3x + 9)/2.

13. Reduce      x - (3x - 3)/5 + 4 = (20 - x)/2 - (6x - 8)/7 + (4x - 4)/5.

14. Reduce     (6x + 7)/9 + (7x - 13)/(6x + 3) = (2x + 4)/3.

15. Reduce      [(5x + 4)/2]:[(18 - x)/4] = 7:4.


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