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Writing & Solving Equations

Writing & Solving Equations. SOMETHING YOU MIGHT SAY…. In order to solve application problems, it is necessary to translate English phrases into algebraic symbols. The following are some common phrases and their mathematic translation. Applications.

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Writing & Solving Equations

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  1. Writing & Solving Equations

  2. SOMETHING YOU MIGHT SAY…

  3. In order to solve application problems, it is necessary to translate English phrases into algebraic symbols. The following are some common phrases and their mathematic translation.

  4. Applications Translating from Words to Mathematical Expressions Mathematical Expression (where x and y are numbers) Verbal Expression Addition The sum of a number and 2 x + 2 x + 3 3 more than a number 7 + x 7 plus a number x + 16 16 added to a number x + 9 A number increased by 9 The sum of two numbers x + y

  5. Applications Translating from Words to Mathematical Expressions Mathematical Expression (where x and y are numbers) Verbal Expression Subtraction 4 less than a number x – 4 10 – x 10 minus a number x – 5 A number decreased by 5 12 – x A number subtracted from 12 x – y The difference between two numbers

  6. 3 x of a number (used with fractions and percent) 4 3 4 Applications Translating from Words to Mathematical Expressions Mathematical Expression (where x and y are numbers) Verbal Expression Multiplication 14 times a number 14x 8x A number multiplied by 8 3x Triple (three times) a number xy The product of two numbers

  7. 6 (x ≠ 0) x x 15 Applications Translating from Words to Mathematical Expressions Mathematical Expression (where x and y are numbers) Verbal Expression Division The quotient of 6 and a number A number divided by 15 half a number

  8. x x 15 y Applications Caution Because subtraction and division are not commutative operations, be careful to correctly translate expressions involving them. For example, “5 less than a number” is translated as x – 5, not 5 – x. “A number subtracted from 12” is expressed as 12 – x, not x – 12. For division, the number by which we are dividing is the denominator, and the number into which we are dividing is the numerator. For example, “a number divided by 15” and “15 divided into x” both translate as . Similarly, “the quotient of x and y” is translated as .

  9. Applications Indicator Words for Equality Equality The symbol for equality, =, is often indicated by the word is. In fact, any words that indicate the idea of “sameness” translate to =.

  10. x x x + 6 8 Applications Translating Words into Equations Verbal Sentence Equation Twice a number, decreased by 4, is 32. 2x– 4 = 32 If the product of a number and 16 is decreased by 25, the result is 87. 16x– 25 = 87 The quotient of a number and the number plus 6 is 48. = 48 The quotient of a number and 8, plus the number,is 54. + x= 54

  11. Applications Distinguishing between Expressions and Equations Decide whether each is an expression or an equation. (a) 4(6 – x) + 2x – 1 There is no equals sign, so this is an expression. (b) 4(6 – x) + 2x – 1 = –15 Because of the equals sign, this is an equation.

  12. Applications Six Steps to Solving Application Problems Six Steps to Solving Application Problems Step 1Read the problem, several times if necessary, until you understand what is given and what is to be found. Step 2 If possible draw a picture or diagram to help visualize the problem. Step 3Assign a variable to represent the unknown value, using diagrams or tables as needed. Write down what the variable represents. Express any other unknown values in terms of the variable. Step 4 Write an equation using the variable expression(s). Step 5 Solve the equation. Step 6Check the answer in the words of the original problem.

  13. Now Lets Write & Solve Some Equations Example 1: Fifteen more than twice a number is – 23.

  14. Now Lets Write & Solve Some Equations Example 2: The quotient of a number and 9, increased by 10 is 11.

  15. Now Lets Write & Solve Some Equations Example 3: The difference between 5 times a number is 4 and 16.

  16. CARTOON

  17. Applications Solving a Geometry Problem The length of a rectangle is 2 ft more than three times the width. The perimeter of the rectangle is 124 ft. Find the length and the width of the rectangle. Step 1 Read the problem. We must find the length and width of the rectangle. The length is 2 ft more than three times the width and the perimeter is 124 ft. Step 2 Assign a variable. Let W = the width; then 2 + 3W = length. Make a sketch. W 2 + 3W Step 3 Write an equation. The perimeter of a rectangle is given by the formula P = 2L + 2W. Let L = 2 + 3W and P = 124. 124=2(2 + 3W) + 2W

  18. Applications Solving a Geometry Problem The length of a rectangle is 2 ft more than three times the width. The perimeter of the rectangle is 124 ft. Find the length and the width of the rectangle. Step 4 Solve the equation obtained in Step 3. 124=2(2 + 3W) + 2W 124=4 + 6W + 2W Remove parentheses Combine like terms. 124=4 + 8W Subtract 4. 124 – 4=4 + 8W– 4 120=8W 1208W = Divide by 8. 88 15= W

  19. Applications Solving a Geometry Problem The length of a rectangle is 2 ft more than three times the width. The perimeter of the rectangle is 124 ft. Find the length and the width of the rectangle. Step 5 State the answer. The width of the rectangle is 15 ft and the length is 2 + 3(15) = 47 ft. Step 6 Check the answer by substituting these dimensions into the words of the original problem.

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