GENERAL FORM OF A LINEAREQUATION
The expression GENERAL FORM, in mathematics, implies a form to which all expressionsor 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, thisform 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:
USING EQUATIONS TOSOLVE PROBLEMS
To solve a problem, we first translate thenumerical 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 Smith’s money. Let x represent the number of dollars Smith has.
Step 3. Express the other facts in terms ofthe 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. Smith’s dollars plus Jones’ dollars equal $120. Translating this statement into algebraic symbols, we have
x + 5x = 120
Solving the equation for x,
6x = 120
Thus Smith has $20.
Step 5. Check: See if the solution satisfiesthe original statement of the problem. Smith and Jones have $120.
EXAMPLE 2: Brown can do a piece of work in5 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 do5 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 takesthem 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 ofthe work and Olsen does 1/4 of the work
Step 4. The amount done in 1 hr is equal tothe 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 inone variable to solve each of the following problems:
1. Find three numbers such that the second istwice 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 billsand 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 taskin 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.
2. Number of five-dollar bills is 12.