Transcript Equations - Translating, Writing, & Solving

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
Translating from Words to Mathematical Expressions
Verbal Expression
Mathematical Expression
(where x and y are numbers)
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
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
Translating from Words to Mathematical Expressions
Verbal Expression
Mathematical Expression
(where x and y are numbers)
Multiplication
a) 14 times a number
14x
b) A number multiplied by 8
8x
3 of a number (used with
4
fractions and percent)
3 x
4
d) Triple (three times) a number
3x
e) The product of two numbers
xy
Applications
Translating from Words to Mathematical Expressions
Verbal Expression
Mathematical Expression
(where x and y are numbers)
Division
a) The quotient of 6 and a number
6 (x ≠ 0)
x
b) The quotient of a number and 6
x
6
c) A number divided by 15
d) half a number
x
15
x
2
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
x
the number into which we are dividing is the numerator.
15
x
y
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
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
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.
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.
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.
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
8 = 8
15 = W
Divide by 8.
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.