Writing & Solving Equations

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

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

Applications Subtraction Translating from Words to Mathematical Expressions Verbal Expression (ORDER DOES MATTER!) Mathematical Expression (where x and y are numbers) Subtraction a) 4 less than a number x – 4 b) 10 minus a number 10 – x c) A number decreased by 6 x – 6 d) A number subtracted from 12 12 – x e) The difference between two numbers x – y

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

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

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: X/15 . Similarly, “the quotient of x and y” is translated as : X / Y x 15 x y

Indicator Words for Equality 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 =.

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

Distinguishing between Expressions 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.

Six Steps to Solving Application Problems Applications Six Steps to Solving Application Problems Six Steps to Solving Application Problems Step 1 Read 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 3 Assign 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 6 Check the answer in the words of the original problem.

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

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

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

Solving a Geometry Problem 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. 124 = 2(2 + 3W) + 2W Let L = 2 + 3W and P = 124.

Solving a Geometry Problem 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 124 = 4 + 8W Combine like terms. 124 – 4 = 4 + 8W – 4 Subtract 4. 120 = 8W 120 8W = Divide by 8. 8 8 15 = W

Solving a Geometry Problem 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.