CONDITIONAL AND EQUIVALENT EQUATIONS

OBJECTIVES

Upon completing this section you should be able to: An equation is a statement in symbols that two number expressions are equal.Equations can be classified in two main types:1. An identity is true for all values of the literal and arithmetical numbers ...

THE DIVISION RULE

OBJECTIVES

Upon completing this section you should be able to: As mentioned earlier, we wish to present an orderly procedure for solving equations. This procedure will involve the four basic operations, the first of which is presented in this section.If each term of an equation is divided by the same nonzero number, the resulting equation is equivalent to the original equation.To prepare to use the division rule for solving equations we must make note of the following process:(We usually write 1x as x with the coefficient 1 understood.)Example 1 Solve for x: 3x = 10SolutionOur goal is to obtain x = some number. The division rule allows us to divide each term of 3x = 10 by the same number, and our goal of finding a value of x would indicate that we divide by 3. This would give us a coefficient of 1 for x.Check: 3x = 10 and x = these equivalent equations?We substitute for x in the first equation obtainingThe equations are equivalent, so the solution is correct.Example 2 Solve for x: 5x = 20SolutionExample 3 Solve for x: 8x = 4SolutionExample 4 Solve for x: 0.5x = 6SolutionExample 6 The formula for finding the circumference (C) of a circle is C = 2πr, where π represents the radius of the circle and it is approximately 3.14. Find the radius of a circle if the circumference is measured to be 40.72 cm. Give the answer correct to two decimal places.SolutionTo solve a problem involving a formula we first use the substitution principle.

THE SUBTRACTION RULE

OBJECTIVES

Upon completing this section you should be able to use the subtraction rule to solve equations.The second step toward an orderly procedure for solving equations will be discussed in this section. You will use your knowledge of like terms from chapter l...

THE ADDITION RULE

OBJECTIVES

Upon completing this section you should be able to use the addition rule to solve equations.We now proceed to the next operation in our goal of developing an orderly procedure for solving equations. Once again, we will rely on previous knowledge.If the same quantity i...

THE MULTIPLICATION RULE

OBJECTIVES

Upon completing this section you should be able to: We now come to the last of the four basic operations in developing our procedure for solving equations. We will also introduce ratio and proportion and use the multiplication rule to solve proportio...

COMBINING RULES FOR SOLVING EQUATIONS

OBJECTIVES

Upon completing this section you should be able to: Many of the exercises in previous sections have required the use of more than one rule in the solution process. In fact, it is possible that a single problem could involve all the rulesThere is no mandatory process for solving equations involving more than one rule, but experience has shown that the following order gives a smoother, more mistake-free procedure.First Eliminate fractions, if any, by multiplying each term of the equation by the least common multiple of all denominators of fractions in the equation. Second Simplify by combining like terms on each side of the equation. Third Add or subtract the necessary quantities to obtain the unknown quantity on one side and the numbers of arithmetic on the other side. Fourth Divide by the coefficient of the unknown quantity. Fifth Check your answer.SolutionMultiplying each term by 15 yieldsYou may want to leave your answer as an improper fraction instead of a mixed number. Either form is correct, but the improper fraction form will be more useful in checking your solution.Example 3 The selling price (S) of a certain article was $30.00. If the margin (M) was one-fifth of the cost (C), find the cost of the article. Use the formula C + M = S.SolutionSince the margin was one-fifth of the cost, we may write

SUMMARY

Key Words

Procedures