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The expression GENERAL FORM, in mathematics, implies a form to which all expressions or equations of a certain type can be reduced. The only possible terms in a linear equation in one variable are the first-degree term and the constant term. Therefore, the general form of a linear equation in one variable is

ax + b = 0

By selecting various values for a and b, this form can represent any linear equation in one variable after such an equation has been simpli represents the numerical equation

    7x + 5 = 0

If a = 2m - n and b = p - q, then ax + b =0 represents the literal equation

    (2m-n)x + p - q = 0

This equation is solved as follows:


To solve a problem, we first translate the numerical sense of the problem into an equation. To see how this is accomplished, consider the following examples and their solutions.

EXAMPLE 1: Together Smith and Jones have $120. Jones has 5 times as much as Smith. How much has Smith?


Step 1. Get the problem clearly in mind. There are two parts to each problem-what is given (the facts) and what we want to know (the question). In this problem. we know that Jones has 5 times as much as Smith and together they have $120. We want to know how much Smith has.

Step 2. Express the unknown as a letter. Usually we express the unknown or number we know the least about as a letter (conventionally we use x). Here we know the least about Smiths money. Let x represent the number of dollars Smith has.

Step 3. Express the other facts in terms of the unknown. If x is the number of dollars Smith has and Jones has 5 times as much, then 5x is the number of dollars Jones has.

Step 4. Express the facts as an equation. The problem will express or imply a relation between the expressions in steps 2 and 3. Smiths dollars plus Jones dollars equal $120. Translating this statement into algebraic symbols, we have

    x + 5x = 120

Solving the equation for x,

        6x = 120
  x = 20

Thus Smith has $20.

Step 5. Check: See if the solution satisfies the original statement of the problem. Smith and Jones have $120.

EXAMPLE 2: Brown can do a piece of work in 5 hr. If Olsen can do it in 4 hr how long will it take them to do the work together?


Step 1. Given: Brown could do 5 hr. Olsen could do it in 4 hours. the work in
Unknown: How long it takes them to do the
work together.

Step 2. Let x represent the time it takes them to do the work together.

Step 3. Then 1/x is the amount they do together in 1 hr. Also, in 1 hour Brown does 1/5 of the work and Olsen does 1/4 of the work

Step 4. The amount done in 1 hr is equal to the part of the work done by Brown in 1 hr plus that done by Olsen in 1 hr.

Solving the equation, 

They complete the work together in 2 2/9: hours.

Practice problems. Use a linear equation in one variable to solve each of the following problems:

1. Find three numbers such that the second is twice the first and the third is three times as large as the first. Their sum is 180.

2. A seaman drew $75.00 pay in dollar bills and five-dollar bills. The number of dollar bills was three more than the number of five- draw? (Hint: If x is the number of five-dollar bills, then 5x is the number of dollars they represent.)

3. Airman A can complete a maintenance task in 4 hr. Airman B requires only 3 hr to do the same work. If they work together, how long should it take them to complete the job?


1. First number is 30.
Second number is 60.
Third number is 90.

2. Number of five-dollar bills is 12.
Number of one-dollar bills is 15.

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