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GENERAL FORM OF A LINEAR EQUATION 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 firstdegree 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 (2mn)x + p  q = 0 This equation is solved as follows: USING EQUATIONS TO SOLVE PROBLEMS 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? SOLUTION: Step 1. Get the problem clearly in mind. There are two parts to each problemwhat 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 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. 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 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? SOLUTION: Step 1. Given: Brown could do 5
hr. Olsen could do it in 4 hours. the
work in 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 fivedollar bills. The number of dollar bills was three more than the number of five draw? (Hint: If x is the number of fivedollar 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? Answers: 1. First number is 30. 2. Number of fivedollar bills is 12. 