Equations and Problem Solving with Fractions. Variable. Expression. Equation. A variable is a letter used to represent a number.

Transcription

1 MAT 040: Basic Math Equations and Problem Solving with Fractions Variable A variable is a letter used to represent a number. Expression An algebraic expression is a combination of variables and/or numbers and symbols. An expression doesn t have an equal sign. Equation An equation is a statement of equality between two expressions. An equation has an equal sign. Key words help translate a problem into an equation or an expression. Key Word Translation is = (equals) of, product Sum, increase, total (multiply) + (add) difference, less, decrease (subtract) quotient (divide) twice half Multiply by 1/2 The following are examples of expressions. Page 1 of 10

2 Translating Words into an Expression You can use any variable to represent a number. Common letters used as variables are x, y, z, and n. Words five times a number the sum of 1/3 and a number seven less than a number twice a number Expression 5n or 1/3 + x x 2y or a number divided by 4 x or four fifths of a number y or Translating Words into an Equation Represent the unknown number with any variable. Use symbols to represent key words. Words Equation Three times a number is 21. The product of a number and 8 is 40. or or The difference between a number and ½ is 6. x Twice a number equals ¼. or Five more than a number is 20. One fourth less than a number is one third. The quotient of a number and 5 is ½. The sum of a number and 1/5 is 10. Page 2 of 10

3 Solving Equations You can use properties of equality to solve equations. Addition Property of Equality The addition property of equality allows you to add the same number to each side of an equation. Subtraction Property of Equality The subtraction property of equality allows you to subtract the same number from each side of an equation. Multiplication Property of Equality The multiplication property of equality allows you to multiply by the same non-zero number on each side of an equation. Division Property of Equality The division property of equality allows you to divide by the same non-zero number on each side of an equation. Basic Operations The four basic operations are addition, subtraction, multiplication and division. Inverse operations are used when solving equations. An inverse operation will undo the result of a previous operation. Example 1 The following illustrates an inverse operation. Start with 4. Add six to four to get = 10 Page 3 of 10

4 To undo this operation subtract 6 from = 4 Now you are back to the number you started with which is 4. We were able to undo adding 6 by subtracting 6. Inverse Operations Addition and subtraction are inverse operations. Multiplication and division are also inverse operations. Operation Addition Subtraction Multiplication Division Inverse Operation Subtraction Addition Division Subtraction To solve an equation you must get the variable alone on one side of the equation. You can do this by performing the inverse operation on each side of the equation. Example 2 Solve the following equation for x. The inverse of adding 9 is subtracting 9. Subtract 9 from each side of the equation. Example 3 Solve the following equation for n. The inverse of subtracting 5 is adding 5. Add 5 to each side of the equation. Example 4 Solve the following equation for x. Page 4 of 10 5x means 5 times x. The inverse of multiplying by 5 is dividing by 5. Divide each side by 5.

5 MAT 040: Basic Math The 5 s cancel on the left side of the equation. Example 5 Solve the following equation for x. means The inverse of dividing by 9 is multiplying by 9. Multiply each side by 9. The 9 s cancel on the left side of the equation. The following examples show how to translate word problems into equations and solve for a variable. Example 6 Solve. A number increased by is. Replace a number with Replace "increased" with +. Replace "is" with =." The inverse of adding is to subtract. Subtract from each side of the equation. Page 5 of 10

6 Example 7 Solve. less than a number is. To find less, subtract from the unknown number. Let the unknown number be n. Order is important in subtraction. would be wrong. The inverse of subtracting is adding. Add to each side of the equation. Notice: Example 8 The quotient of a number and 4 is. A number divided by 4 is represented as. The inverse of dividing by 4 is multiplying by 4. Multiply each side of the equation by 4. The 4 s cancel on the left side of the equation. Page 6 of 10

7 Example 9 of what number is 16? The word of represents multiplication. The inverse of multiplying by is dividing by. Divide each side of the equation by When dividing by invert and multiply by. Example 10 What fraction of is? Replace what fraction with Replace "of" with times. Replace "is" with =." Inverse of multiplying by is dividing by 3. Divide each side of the equation by. When dividing by, invert and multiply by Divide 2 and 4 by 2. Page 7 of 10

8 Example 11 Find a number such that ¾ of it is. The word of represents multiplication. The unknown number is represented by The inverse of multiplying by is dividing by. Divide each side of the equation by. When dividing by invert and multiply by. Divide 8 and 4 by 4. Example 12 Half of a number is. To find half of a number, multiply by. The unknown number is represented by The inverse of multiplying by is dividing by. Divide each side of the equation by. When dividing by invert and multiply by. Divide 8 and 4 by 2. Page 8 of 10

9 Example 13 of what number is 5? The word of represents multiplication. The unknown number is represented by Change each mixed number to an improper fraction. The inverse of multiplying by is dividing by. Divide each side of the equation by. When dividing by invert and multiply by. Divide 16 and 8 by 8. Divide 3 and 3 by 3. Example 14 What fraction of is? Replace what fraction with n. Replace "of " with times. The inverse of multiplying by is dividing by. Divide each side of the equation by. When dividing by, invert and multiply by. Page 9 of 10

10 Example 15 Find of. Replace "of " with times. Let the unknown number be represented by Brenda Moore and Indian Hills Community College Page 10 of 10